Optimal. Leaf size=167 \[ -\frac {2 \sqrt {a^2 x^3+b^2 x}}{3 \left (a^2 x^2+b^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]
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Rubi [C] time = 20.80, antiderivative size = 2697, normalized size of antiderivative = 16.15, number of steps used = 27, number of rules used = 10, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2056, 6715, 6725, 220, 2073, 414, 523, 409, 1217, 1707}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 220
Rule 409
Rule 414
Rule 523
Rule 1217
Rule 1707
Rule 2056
Rule 2073
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 (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-b^6+a^6 x^{12}}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )} \, 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 {1}{\sqrt {b^2+a^2 x^4}}-\frac {2 b^6}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )}\right ) \, 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 \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^6 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^6 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 a^4}{\sqrt {3} \sqrt {-a^4} b^2 \left (b^2+a^2 x^4\right )^{3/2} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )}-\frac {2 a^4}{\sqrt {3} \sqrt {-a^4} b^2 \left (b^2+a^2 x^4\right )^{3/2} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 \sqrt {-a^4} b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b^2+a^2 x^4\right )^{3/2} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 \sqrt {-a^4} b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b^2+a^2 x^4\right )^{3/2} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 \left (5 a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2-2 a^6 x^4}{\sqrt {b^2+a^2 x^4} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} a^2 \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-a^2 \left (5 a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2+2 a^6 x^4}{\sqrt {b^2+a^2 x^4} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} a^2 \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \sqrt {-a^4} \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 )}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \sqrt {-a^4} \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 )}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 a^2 \sqrt {-a^4} b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 a^2 \sqrt {-a^4} b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2-2 a^4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \left (1-\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}\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 )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (1-\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^2 \sqrt {-a^4} \left (1+\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}\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 )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (1-\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^3 \sqrt {-a^4} \left (2 a-\sqrt {2} \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (8 a^3 \sqrt {-a^4} \left (2 a+\sqrt {2} \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \left (1-\frac {\sqrt {2} a}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}\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 )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (1-\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 \sqrt {-a^4} \left (1+\frac {\sqrt {2} a}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}\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 )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (1-\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^3 \sqrt {-a^4} \left (2 a-\sqrt {2} \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (1-\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^3 \sqrt {-a^4} \left (2 a+\sqrt {2} \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (1+\frac {\sqrt {2} a^2 x^2}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}} b}\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2+\sqrt {3} \sqrt {-a^4}\right ) \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}\\ &=\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt {-a^4} x}{\sqrt {3} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt [4]{2} a^6 \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {3 a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {3} \sqrt {-a^4} \left (a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/2} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt [4]{2} a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {-3 a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {3} \left (a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (-3 a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/2} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt [4]{2} a^6 \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {-3 a^2-\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {3} \sqrt {-a^4} \left (-3 a^2-\sqrt {3} \sqrt {-a^4}\right )^{3/2} \left (a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/4} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {4 \sqrt [4]{2} a^2 \sqrt {-a^4} \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {3 a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b} \sqrt {x}}{\sqrt [4]{2} \sqrt [4]{a^2+\sqrt {3} \sqrt {-a^4}} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {3} \left (a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right )^{3/2} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \sqrt {-a^4} \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {a} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 a^{7/2} \left (1-\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {-a^4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 a^{7/2} \left (1+\frac {\sqrt {2} a}{\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {-a^4} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {-a^4} \left (\frac {\sqrt {3}}{\sqrt {a}}+\frac {\sqrt {6} \sqrt {a}}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{3 \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 a^{7/2} \left (1-\frac {\sqrt {2} a}{\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {3} \sqrt {-a^4} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {-a^4} \left (2 a+\sqrt {2} \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (-\frac {\left (\sqrt {2} a-\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{4 \sqrt {3} a^{5/2} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\sqrt {-a^4} \left (2 a-\sqrt {2} \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {\left (\sqrt {2} a+\sqrt {a^2-\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2-\sqrt {3} \sqrt {-a^4}}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{4 \sqrt {3} a^{5/2} \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\sqrt {-a^4} \left (2 a+\sqrt {2} \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (-\frac {\left (\sqrt {2} a-\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{4 \sqrt {3} a^{5/2} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\sqrt {-a^4} \left (2 a-\sqrt {2} \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \Pi \left (\frac {\left (\sqrt {2} a+\sqrt {a^2+\sqrt {3} \sqrt {-a^4}}\right )^2}{4 \sqrt {2} a \sqrt {a^2+\sqrt {3} \sqrt {-a^4}}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )|\frac {1}{2}\right )}{4 \sqrt {3} a^{5/2} \left (3 a^2+\sqrt {3} \sqrt {-a^4}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [C] time = 2.27, size = 345, normalized size = 2.07 \begin {gather*} \frac {2 \left (-x^{3/2}-\frac {i x^2 \sqrt {\frac {b^2}{a^2 x^2}+1} \left (-\Pi \left (-\frac {i \sqrt {2} a}{\sqrt {\frac {\left (1-i \sqrt {3}\right ) a^2}{b^2}} b};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2} a}{\sqrt {\frac {\left (1-i \sqrt {3}\right ) a^2}{b^2}} b};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i \sqrt {2} a}{\sqrt {\frac {\left (1+i \sqrt {3}\right ) a^2}{b^2}} b};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2} a}{\sqrt {\frac {\left (1+i \sqrt {3}\right ) a^2}{b^2}} b};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i b}{a}}}\right )}{3 \sqrt {x} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 167, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {b^2 x+a^2 x^3}}{3 \left (b^2+a^2 x^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 456, normalized size = 2.73 \begin {gather*} -\frac {4 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {3 \, \left (\frac {1}{3}\right )^{\frac {3}{4}} \sqrt {a^{2} x^{3} + b^{2} x} a^{2} b^{2} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}}{a^{2} x^{2} + b^{2}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} + b^{2}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} + a^{2} b^{4} x\right )} \sqrt {\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} + a^{2} b^{4}\right )} \left (\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} + b^{2} x}}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2} x}}{6 \, {\left (a^{2} x^{2} + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [C] time = 0.29, size = 341, normalized size = 2.04
method | result | size |
elliptic | \(-\frac {2 x}{3 \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-a^{2} b^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 i b^{3}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 i \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 b^{3}}{3 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{9 b \,a^{2}}\) | \(341\) |
default | \(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}-\frac {2 b^{2} \left (\frac {x}{b^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {i \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 b a \sqrt {a^{2} x^{3}+b^{2} x}}\right )}{3}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-a^{2} b^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 i b^{3}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -2 i \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-2 b^{3}}{3 b^{3}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{9 b \,a^{2}}\) | \(447\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 5.79, size = 201, normalized size = 1.20 \begin {gather*} \frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2-6\,\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}+3^{3/4}\,a^2\,x^2+3\,3^{1/4}\,a\,b\,x}{a^2\,x^2-\sqrt {3}\,a\,b\,x+b^2}\right )}{9\,\sqrt {a}\,\sqrt {b}}-\frac {2\,\sqrt {a^2\,x^3+b^2\,x}}{3\,\left (a^2\,x^2+b^2\right )}+\frac {3^{3/4}\,\ln \left (\frac {3^{3/4}\,b^2+3^{3/4}\,a^2\,x^2-3\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+\sqrt {3}\,a\,b\,x+b^2}\right )\,1{}\mathrm {i}}{9\,\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2} x^{2} + b^{2}\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.
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