Optimal. Leaf size=143 \[ -\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{8 a^3 x}{65 b^2 \sqrt [4]{a+b x^2}}-\frac{8 a^{7/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b} \]
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Rubi [A] time = 0.0514642, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {279, 321, 229, 227, 196} \[ -\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{8 a^3 x}{65 b^2 \sqrt [4]{a+b x^2}}-\frac{8 a^{7/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int x^4 \left (a+b x^2\right )^{3/4} \, dx &=\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}+\frac{1}{13} (3 a) \int \frac{x^4}{\sqrt [4]{a+b x^2}} \, dx\\ &=\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}-\frac{\left (2 a^2\right ) \int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx}{13 b}\\ &=-\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}+\frac{\left (4 a^3\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{65 b^2}\\ &=-\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}+\frac{\left (4 a^3 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{65 b^2 \sqrt [4]{a+b x^2}}\\ &=\frac{8 a^3 x}{65 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}-\frac{\left (4 a^3 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{65 b^2 \sqrt [4]{a+b x^2}}\\ &=\frac{8 a^3 x}{65 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a^2 x \left (a+b x^2\right )^{3/4}}{65 b^2}+\frac{2 a x^3 \left (a+b x^2\right )^{3/4}}{39 b}+\frac{2}{13} x^5 \left (a+b x^2\right )^{3/4}-\frac{8 a^{7/2} \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 )}{65 b^{5/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0522239, size = 93, normalized size = 0.65 \[ \frac{2 x \left (a+b x^2\right )^{3/4} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \left (-2 a^2+a b x^2+3 b^2 x^4\right )+2 a^2 \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )\right )}{39 b^2 \left (\frac{b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \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}} x^{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^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.35365, size = 29, normalized size = 0.2 \begin{align*} \frac{a^{\frac{3}{4}} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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