Optimal. Leaf size=124 \[ \frac{8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac{16 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0434474, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {285, 197, 196} \[ \frac{8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac{16 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 285
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac{(10 a) \int \frac{x^4}{\left (a+b x^2\right )^{5/4}} \, dx}{9 b}\\ &=-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}}+\frac{\left (4 a^2\right ) \int \frac{x^2}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^2}\\ &=\frac{8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac{\left (8 a^3\right ) \int \frac{1}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^3}\\ &=\frac{8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac{\left (8 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{3 b^3 \sqrt [4]{a+b x^2}}\\ &=\frac{8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac{16 a^{5/2} \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 )}{3 b^{7/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0284226, size = 78, normalized size = 0.63 \[ \frac{2 \left (12 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-12 a^2 x-2 a b x^3+b^2 x^5\right )}{9 b^3 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, 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 = 0.902676, size = 27, normalized size = 0.22 \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^{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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