Optimal. Leaf size=324 \[ -\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{128 b^4 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b} \]
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Rubi [A] time = 0.292085, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 241, 236, 219} \[ \frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{128 b^4 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 241
Rule 236
Rule 219
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a+b x^2\right )^{5/6}} \, dx &=\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac{(15 a) \int \frac{x^4}{\left (a+b x^2\right )^{5/6}} \, dx}{16 b}\\ &=-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}+\frac{\left (27 a^2\right ) \int \frac{x^2}{\left (a+b x^2\right )^{5/6}} \, dx}{32 b^2}\\ &=\frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac{\left (81 a^3\right ) \int \frac{1}{\left (a+b x^2\right )^{5/6}} \, dx}{128 b^3}\\ &=\frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac{\left (81 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-b x^2\right )^{2/3}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b^3 \sqrt [3]{\frac{a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=\frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}+\frac{\left (243 a^3 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt [6]{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{256 b^4 x \sqrt [3]{\frac{a}{a+b x^2}}}\\ &=\frac{81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{1+\sqrt [3]{\frac{a}{a+b x^2}}+\left (\frac{a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}}{1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}}\right )|-7+4 \sqrt{3}\right )}{128 b^4 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}\right )^2}} \sqrt{-1+\frac{a}{a+b x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0313999, size = 89, normalized size = 0.27 \[ \frac{3 x \left (-27 a^3 \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )+15 a^2 b x^2+27 a^3-4 a b^2 x^4+8 b^3 x^6\right )}{128 b^3 \left (a+b x^2\right )^{5/6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{{x}^{6} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{6}}}}\, 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}{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{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{5}{6}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.976871, size = 27, normalized size = 0.08 \begin{align*} \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac{5}{6}}} \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}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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