Optimal. Leaf size=311 \[ -\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{a^2 x^2-2 a b x-b^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{a^2 x^2-a b x-b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{2 \sqrt {b}}-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^3-b^2 x}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}}{\sqrt {a^2 x^3-b^2 x}}\right )}{3 \sqrt {a} \sqrt {b}} \]
________________________________________________________________________________________
Rubi [C] time = 8.75, antiderivative size = 1295, normalized size of antiderivative = 4.16, number of steps used = 209, number of rules used = 20, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2056, 1586, 6715, 6725, 1729, 1209, 1201, 224, 221, 1200, 1199, 424, 1219, 1218, 1248, 735, 844, 217, 206, 725}
result too large to display
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 221
Rule 224
Rule 424
Rule 725
Rule 735
Rule 844
Rule 1199
Rule 1200
Rule 1201
Rule 1209
Rule 1218
Rule 1219
Rule 1248
Rule 1586
Rule 1729
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {-b^6+a^6 x^6}{\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (b^6+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {\sqrt {-b^2+a^2 x^2} \left (b^4+a^2 b^2 x^2+a^4 x^4\right )}{\sqrt {x} \left (b^6+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4} \left (b^4+a^2 b^2 x^4+a^4 x^8\right )}{b^6+a^6 x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-i \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+i \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt [12]{-a^6} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-i \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+i \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [3]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [3]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-(-1)^{5/6} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+(-1)^{5/6} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [6]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [6]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-(-1)^{2/3} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+(-1)^{2/3} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}\\ &=2 \frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-\sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+\sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-(-1)^{2/3} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+(-1)^{2/3} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}\\ &=2 \left (-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+(-1)^{2/3} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-(-1)^{2/3} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+\sqrt [3]{-1} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-\sqrt [3]{-1} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a-\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a+\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a-\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a+\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (\frac {\sqrt {a} \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (-\frac {\sqrt [3]{-1} \sqrt {a} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {a} \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {(-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt [3]{-1} \sqrt {a} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.06, size = 294, normalized size = 0.95 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {1-\frac {b^2}{a^2 x^2}} \left (3 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (i;\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{3 \sqrt {-\frac {b}{a}} \sqrt {a^2 x^3-b^2 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.83, size = 311, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.10, size = 2045, normalized size = 6.58
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 475, normalized size = 1.53
method | result | size |
default | \(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {2 b^{2} \left (-\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}+\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\right )}{3}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b}\) | \(475\) |
elliptic | \(\frac {b \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {b \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -b}{2 b}, \frac {\sqrt {2}}{2}\right )\right )}{6 a \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {b^{2} \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{6 a^{2} \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a^{3} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{4} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 b \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {2 b \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {2 b^{2} \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\) | \(847\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________