Optimal. Leaf size=121 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}{a^2 x^2+b}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}{a^2 x^2+b}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}} \]
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Rubi [C] time = 6.23, antiderivative size = 1906, normalized size of antiderivative = 15.75, number of steps used = 23, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2056, 1586, 6715, 6725, 406, 220, 409, 1217, 1707}
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Warning: Unable to verify antiderivative.
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Rule 220
Rule 406
Rule 409
Rule 1217
Rule 1586
Rule 1707
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b^2+a^4 x^4}{\sqrt {b x+a^2 x^3} \left (b^2+a^4 x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {-b^2+a^4 x^4}{\sqrt {x} \sqrt {b+a^2 x^2} \left (b^2+a^4 x^4\right )} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^2}}{\sqrt {x} \left (b^2+a^4 x^4\right )} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+a^2 x^4\right ) \sqrt {b+a^2 x^4}}{b^2+a^4 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-a^4} \left (a^2 b-\sqrt {-a^4} b\right ) \sqrt {b+a^2 x^4}}{2 a^4 b \left (b-\sqrt {-a^4} x^4\right )}+\frac {\sqrt {-a^4} \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {b+a^2 x^4}}{2 a^4 b \left (b+\sqrt {-a^4} x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}}\\ &=-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a^2 x^4}}{b-\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a^2 x^4}}{b+\sqrt {-a^4} x^4} \, dx,x,\sqrt {x}\right )}{a^4 b \sqrt {b x+a^2 x^3}}\\ &=\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^4} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4} \left (b-\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^4} \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4} \left (b+\sqrt {-a^4} x^4\right )} \, dx,x,\sqrt {x}\right )}{a^6 \sqrt {b x+a^2 x^3}}+\frac {\left (\left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{a^2 b \sqrt {b x+a^2 x^3}}\\ &=\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{5/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {-a^4} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt {-a^4} b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}\\ &=\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{5/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {-a^4} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {-a^4} \left (a-\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt [4]{-a^4} \left (a-\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {-a^4} \left (a+\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}-\frac {\left (\left (a^2+\sqrt {-a^4}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^2 \sqrt [4]{-a^4} \left (a+\sqrt [4]{-a^4}\right ) b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (-a^4\right )^{3/4} \left (\sqrt [4]{-a^4}+\frac {a \sqrt {-b}}{\sqrt {b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1+\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a+\frac {\sqrt [4]{-a^4} \sqrt {-b}}{\sqrt {b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^5 b \sqrt {b x+a^2 x^3}}+\frac {\left (\left (-a^4\right )^{3/4} \left (\sqrt [4]{-a^4}+\frac {a \sqrt {b}}{\sqrt {-b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{\sqrt {b}}}{\left (1-\frac {\sqrt [4]{-a^4} x^2}{\sqrt {-b}}\right ) \sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^6 b \sqrt {b x+a^2 x^3}}+\frac {\left (\sqrt {-a^4} \left (a+\frac {\sqrt [4]{-a^4} \sqrt {b}}{\sqrt {-b}}\right ) \left (a^2 b+\sqrt {-a^4} b\right ) \sqrt {x} \sqrt {b+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{2 a^5 b \sqrt {b x+a^2 x^3}}\\ &=-\frac {\sqrt [8]{-a^4} \sqrt {x} \sqrt {b+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {a^2-\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {b+a^2 x^2}}\right )}{2 \sqrt {a^2-\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {b x+a^2 x^3}}+\frac {\sqrt [8]{-a^4} \sqrt [4]{-b} \sqrt {-b^2} \sqrt {x} \sqrt {b+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {-a^2+\sqrt {-a^4}} \sqrt [4]{-b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {b+a^2 x^2}}\right )}{2 \sqrt {-a^2+\sqrt {-a^4}} b^{3/2} \sqrt {b x+a^2 x^3}}-\frac {a \sqrt {x} \sqrt {b+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt [8]{-a^4} \sqrt {-a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {x}}{a \sqrt {b+a^2 x^2}}\right )}{2 \sqrt [8]{-a^4} \sqrt {-a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (a^2+\sqrt {-a^4}\right )^{3/2} \sqrt {x} \sqrt {b+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {a^2+\sqrt {-a^4}} \sqrt [4]{b} \sqrt {x}}{\sqrt [8]{-a^4} \sqrt {b+a^2 x^2}}\right )}{4 a^2 \left (-a^4\right )^{3/8} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (\sqrt [4]{-a^4} \sqrt {-b}+a \sqrt {b}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{11/2} b^{3/4} \sqrt {b x+a^2 x^3}}-\frac {\sqrt {a} \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \left (a-\sqrt [4]{-a^4}\right ) \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\sqrt {a} \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \left (a+\sqrt [4]{-a^4}\right ) \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 a^{5/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a^2+\sqrt {-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {a} \sqrt {-a^4} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\sqrt {-a^4} \left (a^2+\sqrt {-a^4}\right ) \left (a+\frac {\sqrt [4]{-a^4} \sqrt {b}}{\sqrt {-b}}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{11/2} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a+\sqrt [4]{-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \Pi \left (\frac {a^3 \left (a-\sqrt [4]{-a^4}\right )^2}{4 \left (-a^4\right )^{5/4}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {a} \left (a-\sqrt [4]{-a^4}\right ) \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (a-\sqrt [4]{-a^4}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \Pi \left (\frac {\left (a+\sqrt [4]{-a^4}\right )^2}{4 a \sqrt [4]{-a^4}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt {a} \left (a+\sqrt [4]{-a^4}\right ) \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\sqrt [4]{-a^4} \left (a^2 \sqrt {-b}-\sqrt {-a^4} \sqrt {-b}+a \sqrt [4]{-a^4} \sqrt {b}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \Pi \left (\frac {a^3 \left (a \sqrt {-b}-\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 \left (-a^4\right )^{5/4} \sqrt {-b^2}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{7/2} b^{3/4} \sqrt {b x+a^2 x^3}}+\frac {\left (-a^4\right )^{3/4} \left (a^3 b+a \sqrt {-a^4} b+\left (-a^4\right )^{3/4} \sqrt {-b^2}\right ) \sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} \Pi \left (\frac {\left (a \sqrt {-b}+\sqrt [4]{-a^4} \sqrt {b}\right )^2}{4 a \sqrt [4]{-a^4} \sqrt {-b^2}};2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{13/2} \sqrt {-b} b^{3/4} \sqrt {b x+a^2 x^3}}\\ \end {align*}
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Mathematica [C] time = 1.29, size = 234, normalized size = 1.93 \begin {gather*} -\frac {i x^{3/2} \sqrt {\frac {b}{a^2 x^2}+1} \left (2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {b}}{a}} \sqrt {x \left (a^2 x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 121, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}{a^2 x^2+b}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}}{a^2 x^2+b}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 344, normalized size = 2.84 \begin {gather*} -\left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a^{2} x^{3} + b x} a^{2} b \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}}{a^{2} x^{2} + b}\right ) - \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} + 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + b^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} + 4 \, a^{2} b x^{2} + b^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (a^{4} b x^{3} + a^{2} b^{2} x\right )} \sqrt {\frac {1}{a^{2} b}} - 4 \, \sqrt {a^{2} x^{3} + b x} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{2} b x \left (\frac {1}{a^{2} b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{4} b x^{2} + a^{2} b^{2}\right )} \left (\frac {1}{a^{2} b}\right )^{\frac {3}{4}}\right )}}{a^{4} x^{4} + 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^{4} x^{4} - b^{2}}{{\left (a^{4} x^{4} + b^{2}\right )} \sqrt {a^{2} x^{3} + b x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 303, normalized size = 2.50 \begin {gather*} \frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}+b^{2}\right )}{\sum }\frac {\sqrt {-b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}\, \sqrt {-\frac {x a}{\sqrt {-b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}-b \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}+b \sqrt {-b}\right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b}}{a}\right ) a}{\sqrt {-b}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {-b}\, a^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -\underline {\hspace {1.25 ex}}\alpha \sqrt {-b}\, a b -b^{2}}{2 b^{2}}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x \left (a^{2} x^{2}+b \right )}}\right )}{4 a^{4}} \end {gather*}
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^{4} x^{4} - b^{2}}{{\left (a^{4} x^{4} + b^{2}\right )} \sqrt {a^{2} x^{3} + b x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.63, size = 164, normalized size = 1.36 \begin {gather*} \frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-4\,\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}+2\,2^{1/4}\,a\,\sqrt {b}\,x}{b+a^2\,x^2-\sqrt {2}\,a\,\sqrt {b}\,x}\right )}{4\,\sqrt {a}\,b^{1/4}}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2^{3/4}\,a^2\,x^2-2\,2^{1/4}\,a\,\sqrt {b}\,x+\sqrt {a}\,b^{1/4}\,\sqrt {a^2\,x^3+b\,x}\,4{}\mathrm {i}}{b+a^2\,x^2+\sqrt {2}\,a\,\sqrt {b}\,x}\right )\,1{}\mathrm {i}}{4\,\sqrt {a}\,b^{1/4}} \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^{2} x^{2} - b\right ) \left (a^{2} x^{2} + b\right )}{\sqrt {x \left (a^{2} x^{2} + b\right )} \left (a^{4} x^{4} + b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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