Optimal. Leaf size=115 \[ -\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 = {452}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac {\text {ArcTan}\left (\frac {a^{3/4} \left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 452
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.09, size = 121, normalized size = 1.05 \begin {gather*} \frac {\tan ^{-1}\left (\frac {b x^2-2 \sqrt {a} \sqrt {a+b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}\right )-\tanh ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}{b x^2+2 \sqrt {a} \sqrt {a+b x^2}}\right )}{2 \sqrt [4]{a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (b \,x^{2}+2 a \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (87) = 174\).
time = 1.34, size = 207, normalized size = 1.80 \begin {gather*} -2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{4}\right )^{\frac {3}{4}} a b^{4} x \sqrt {\frac {b^{4} x^{2} \sqrt {-\frac {1}{a b^{6}}} + 2 \, \sqrt {b x^{2} + a}}{x^{2}}} \left (-\frac {1}{a b^{6}}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (b x^{2} + a\right )}^{\frac {1}{4}} a b^{4} \left (-\frac {1}{a b^{6}}\right )^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) \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 + b x^{2}\right )^{\frac {3}{4}} \cdot \left (2 a + b x^{2}\right )}\, dx \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*} \text {could not integrate} \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 (b\,x^2+a\right )}^{3/4}\,\left (b\,x^2+2\,a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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