Optimal. Leaf size=151 \[ -\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.0610959, antiderivative size = 151, 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 = {290, 325, 229, 228} \[ -\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}+\frac{11 \int \frac{1}{x^6 \sqrt [4]{a-b x^2}} \, dx}{a}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac{(77 b) \int \frac{1}{x^4 \sqrt [4]{a-b x^2}} \, dx}{10 a^2}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}+\frac{\left (77 b^2\right ) \int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a^3}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{\left (77 b^3\right ) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^4}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{\left (77 b^3 \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{40 a^4 \sqrt [4]{a-b x^2}}\\ &=\frac{2}{a x^5 \sqrt [4]{a-b x^2}}-\frac{11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac{77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac{77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac{77 b^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0102303, size = 55, normalized size = 0.36 \[ -\frac{\sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{5}{2},\frac{5}{4};-\frac{3}{2};\frac{b x^2}{a}\right )}{5 a x^5 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \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{1}{{\left (-b x^{2} + a\right )}^{\frac{5}{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 (\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{b^{2} x^{10} - 2 \, a b x^{8} + a^{2} x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.78238, size = 34, normalized size = 0.23 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac{5}{4}} x^{5}} \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{1}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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