Optimal. Leaf size=124 \[ -\frac{16 a^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a-b x^2}}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}+\frac{8 a x \left (a-b x^2\right )^{3/4}}{3 b^3}+\frac{2 x^5}{b \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.0474167, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {288, 321, 229, 228} \[ -\frac{16 a^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a-b x^2}}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}+\frac{8 a x \left (a-b x^2\right )^{3/4}}{3 b^3}+\frac{2 x^5}{b \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a-b x^2\right )^{5/4}} \, dx &=\frac{2 x^5}{b \sqrt [4]{a-b x^2}}-\frac{10 \int \frac{x^4}{\sqrt [4]{a-b x^2}} \, dx}{b}\\ &=\frac{2 x^5}{b \sqrt [4]{a-b x^2}}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}-\frac{(20 a) \int \frac{x^2}{\sqrt [4]{a-b x^2}} \, dx}{3 b^2}\\ &=\frac{2 x^5}{b \sqrt [4]{a-b x^2}}+\frac{8 a x \left (a-b x^2\right )^{3/4}}{3 b^3}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}-\frac{\left (8 a^2\right ) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{3 b^3}\\ &=\frac{2 x^5}{b \sqrt [4]{a-b x^2}}+\frac{8 a x \left (a-b x^2\right )^{3/4}}{3 b^3}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}-\frac{\left (8 a^2 \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{3 b^3 \sqrt [4]{a-b x^2}}\\ &=\frac{2 x^5}{b \sqrt [4]{a-b x^2}}+\frac{8 a x \left (a-b x^2\right )^{3/4}}{3 b^3}+\frac{20 x^3 \left (a-b x^2\right )^{3/4}}{9 b^2}-\frac{16 a^{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 )}{3 b^{7/2} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0287999, size = 78, normalized size = 0.63 \[ -\frac{2 x \left (12 a^2 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-12 a^2+2 a b x^2+b^2 x^4\right )}{9 b^3 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{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{x^{6}}{{\left (-b x^{2} + a\right )}^{\frac{5}{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^{2} + a\right )}^{\frac{3}{4}} x^{6}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.93804, size = 29, normalized size = 0.23 \begin{align*} \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7 a^{\frac{5}{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^{6}}{{\left (-b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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