Optimal. Leaf size=136 \[ -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{b} \sqrt {a x^2+b}}{\sqrt {a}}-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}}{x \sqrt [4]{a x^2+b}}\right )}{2 a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{a x^2+b}}{2 \sqrt {b} \sqrt {a x^2+b}+a x^2}\right )}{2 a^{3/2} \sqrt [4]{b}} \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {441} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b^{3/4} \left (1-\frac {\sqrt {a x^2+b}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {b^{3/4} \left (\frac {\sqrt {a x^2+b}}{\sqrt {b}}+1\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 441
Rubi steps
\begin {align*} \int \frac {x^2}{\left (b+a x^2\right )^{3/4} \left (2 b+a x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {b^{3/4} \left (1+\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{b+a x^2}}\right )}{a^{3/2} \sqrt [4]{b}}+\frac {\tanh ^{-1}\left (\frac {b^{3/4} \left (1-\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{b+a x^2}}\right )}{a^{3/2} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 67, normalized size = 0.49 \begin {gather*} \frac {x^3 \left (\frac {a x^2+b}{b}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {a x^2}{b},-\frac {a x^2}{2 b}\right )}{6 b \left (a x^2+b\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.39, size = 136, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {b+a x^2}}{\sqrt {a}}}{x \sqrt [4]{b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b+a x^2}}{a x^2+2 \sqrt {b} \sqrt {b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 207, normalized size = 1.52 \begin {gather*} -2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{4} b x \sqrt {\frac {a^{4} x^{2} \sqrt {-\frac {1}{a^{6} b}} + 2 \, \sqrt {a x^{2} + b}}{x^{2}}} \left (-\frac {1}{a^{6} b}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a x^{2} + b\right )}^{\frac {1}{4}} a^{4} b \left (-\frac {1}{a^{6} b}\right )^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\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 {x^{2}}{{\left (a x^{2} + 2 \, b\right )} {\left (a x^{2} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{2}+b \right )^{\frac {3}{4}} \left (a \,x^{2}+2 b \right )}\, dx\]
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 {x^{2}}{{\left (a x^{2} + 2 \, b\right )} {\left (a x^{2} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (a\,x^2+b\right )}^{3/4}\,\left (a\,x^2+2\,b\right )} \,d x \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 {x^{2}}{\left (a x^{2} + b\right )^{\frac {3}{4}} \left (a x^{2} + 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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