Optimal. Leaf size=100 \[ \frac{24 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{12 a x}{5 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0314686, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {285, 197, 196} \[ \frac{24 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a+b x^2}}-\frac{12 a x}{5 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 285
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}}-\frac{(6 a) \int \frac{x^2}{\left (a+b x^2\right )^{5/4}} \, dx}{5 b}\\ &=-\frac{12 a x}{5 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}}+\frac{\left (12 a^2\right ) \int \frac{1}{\left (a+b x^2\right )^{5/4}} \, dx}{5 b^2}\\ &=-\frac{12 a x}{5 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}}+\frac{\left (12 a \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{5 b^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{12 a x}{5 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt [4]{a+b x^2}}+\frac{24 a^{3/2} \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0203067, size = 65, normalized size = 0.65 \[ \frac{2 \left (-6 a x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+6 a x+b x^3\right )}{5 b^2 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.819233, size = 27, normalized size = 0.27 \begin{align*} \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{5}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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