Optimal. Leaf size=126 \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]
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Rubi [A] time = 0.0573729, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {279, 321, 310, 281, 335, 275, 196} \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 310
Rule 281
Rule 335
Rule 275
Rule 196
Rubi steps
\begin{align*} \int x^6 \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{1}{10} (3 a) \int \frac{x^6}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^2\right ) \int \frac{x^2}{\sqrt [4]{a+b x^4}} \, dx}{20 b}\\ &=-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{\left (3 a^3\right ) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{40 b}\\ &=-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{\left (3 a^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{40 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{40 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{80 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}-\frac{3 a^{5/2} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0481878, size = 64, normalized size = 0.51 \[ \frac{x^3 \left (a+b x^4\right )^{3/4} \left (-\frac{a \, _2F_1\left (-\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\left (\frac{b x^4}{a}+1\right )^{3/4}}+a+b x^4\right )}{10 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{x}^{6} \left ( b{x}^{4}+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^{4} + a\right )}^{\frac{3}{4}} x^{6}\,{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^{4} + a\right )}^{\frac{3}{4}} x^{6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.22726, size = 39, normalized size = 0.31 \begin{align*} \frac{a^{\frac{3}{4}} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\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^{4} + a\right )}^{\frac{3}{4}} x^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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