Optimal. Leaf size=652 \[ \frac{40 \sqrt{2} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{9 \sqrt [4]{3} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}-\frac{20 \sqrt{2+\sqrt{3}} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{3}{a x^3 \sqrt [6]{a+b x^2}} \]
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Rubi [A] time = 0.653809, antiderivative size = 652, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {290, 325, 238, 198, 235, 304, 219, 1879} \[ -\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 \sqrt{2} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{20 \sqrt{2+\sqrt{3}} b \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{3}{a x^3 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 238
Rule 198
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^{7/6}} \, dx &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}+\frac{10 \int \frac{1}{x^4 \sqrt [6]{a+b x^2}} \, dx}{a}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}-\frac{(40 b) \int \frac{1}{x^2 \sqrt [6]{a+b x^2}} \, dx}{9 a^2}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac{\left (80 b^2\right ) \int \frac{1}{\sqrt [6]{a+b x^2}} \, dx}{27 a^3}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac{\left (40 b^2\right ) \int \frac{1}{\left (a+b x^2\right )^{7/6}} \, dx}{27 a^2}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac{\left (40 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{27 a^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac{\left (20 b \sqrt{-\frac{b x^2}{a+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{9 a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac{\left (20 b \sqrt{-\frac{b x^2}{a+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{9 a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac{\left (20 \sqrt{2 \left (2+\sqrt{3}\right )} b \sqrt{-\frac{b x^2}{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 )}{9 a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac{3}{a x^3 \sqrt [6]{a+b x^2}}-\frac{40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac{10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac{40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac{40 b \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt{-1+\frac{a}{a+b x^2}}}{9 a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \left (1-\sqrt{3}-\sqrt [3]{\frac{a}{a+b x^2}}\right )}-\frac{20 \sqrt{2+\sqrt{3}} b \sqrt{-\frac{b x^2}{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}} E\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 )}{3\ 3^{3/4} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{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}}}+\frac{40 \sqrt{2} b \sqrt{-\frac{b x^2}{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 )}{9 \sqrt [4]{3} a^2 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{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 = 0.0088248, size = 54, normalized size = 0.08 \[ -\frac{\sqrt [6]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{3}{2},\frac{7}{6};-\frac{1}{2};-\frac{b x^2}{a}\right )}{3 a x^3 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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{1}{{\left (b x^{2} + a\right )}^{\frac{7}{6}} x^{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{5}{6}}}{b^{2} x^{8} + 2 \, a b x^{6} + a^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.00891, size = 32, normalized size = 0.05 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{7}{6} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac{7}{6}} x^{3}} \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{7}{6}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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