Optimal. Leaf size=92 \[ -\frac{6 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{6 a x}{5 \sqrt [4]{a+b x^2}}+\frac{2}{5} x \left (a+b x^2\right )^{3/4} \]
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Rubi [A] time = 0.0209407, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {195, 229, 227, 196} \[ -\frac{6 a^{3/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{6 a x}{5 \sqrt [4]{a+b x^2}}+\frac{2}{5} x \left (a+b x^2\right )^{3/4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^{3/4} \, dx &=\frac{2}{5} x \left (a+b x^2\right )^{3/4}+\frac{1}{5} (3 a) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx\\ &=\frac{2}{5} x \left (a+b x^2\right )^{3/4}+\frac{\left (3 a \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{5 \sqrt [4]{a+b x^2}}\\ &=\frac{6 a x}{5 \sqrt [4]{a+b x^2}}+\frac{2}{5} x \left (a+b x^2\right )^{3/4}-\frac{\left (3 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 \sqrt [4]{a+b x^2}}\\ &=\frac{6 a x}{5 \sqrt [4]{a+b x^2}}+\frac{2}{5} x \left (a+b x^2\right )^{3/4}-\frac{6 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 \sqrt{b} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0047591, size = 46, normalized size = 0.5 \[ \frac{x \left (a+b x^2\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\left (\frac{b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{{\frac{3}{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{\left (b x^{2} + a\right )}^{\frac{3}{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 ({\left (b x^{2} + a\right )}^{\frac{3}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.896532, size = 26, normalized size = 0.28 \begin{align*} a^{\frac{3}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \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{\left (b x^{2} + a\right )}^{\frac{3}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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