Optimal. Leaf size=133 \[ \frac{40 a^{7/2} \left (1-\frac{b x^4}{a}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b} \]
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Rubi [A] time = 0.085963, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {275, 321, 233, 232} \[ -\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}+\frac{40 a^{7/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b} \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 233
Rule 232
Rubi steps
\begin{align*} \int \frac{x^{13}}{\left (a-b x^4\right )^{3/4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{11 b}\\ &=-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac{\left (30 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2}\\ &=-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac{\left (20 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^3}\\ &=-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac{\left (20 a^3 \left (1-\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{77 b^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac{x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac{40 a^{7/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0334611, size = 92, normalized size = 0.69 \[ \frac{x^2 \left (20 a^3 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )+10 a^2 b x^4-20 a^3+3 a b^2 x^8+7 b^3 x^{12}\right )}{77 b^3 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{{x}^{13} \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 \frac{x^{13}}{{\left (-b x^{4} + 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 (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{13}}{b x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.54669, size = 29, normalized size = 0.22 \begin{align*} \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{14 a^{\frac{3}{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^{13}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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