Optimal. Leaf size=326 \[ -\frac{224 \sqrt{2-\sqrt{3}} b^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{135 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{\sqrt [6]{a+b x^2}}{5 a x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.278822, antiderivative size = 326, 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 = {325, 241, 236, 219} \[ -\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}-\frac{224 \sqrt{2-\sqrt{3}} b^2 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{135 \sqrt [4]{3} a^3 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{\sqrt [6]{a+b x^2}}{5 a x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 241
Rule 236
Rule 219
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^2\right )^{5/6}} \, dx &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}-\frac{(14 b) \int \frac{1}{x^4 \left (a+b x^2\right )^{5/6}} \, dx}{15 a}\\ &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}+\frac{\left (112 b^2\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{5/6}} \, dx}{135 a^2}\\ &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}-\frac{\left (224 b^3\right ) \int \frac{1}{\left (a+b x^2\right )^{5/6}} \, dx}{405 a^3}\\ &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}-\frac{\left (224 b^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 )}{405 a^3 \sqrt [3]{\frac{a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}+\frac{\left (112 b^2 \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 )}{135 a^3 x \sqrt [3]{\frac{a}{a+b x^2}}}\\ &=-\frac{\sqrt [6]{a+b x^2}}{5 a x^5}+\frac{14 b \sqrt [6]{a+b x^2}}{45 a^2 x^3}-\frac{112 b^2 \sqrt [6]{a+b x^2}}{135 a^3 x}-\frac{224 \sqrt{2-\sqrt{3}} b^2 \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 )}{135 \sqrt [4]{3} a^3 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 = 0.0096064, size = 51, normalized size = 0.16 \[ -\frac{\left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (-\frac{5}{2},\frac{5}{6};-\frac{3}{2};-\frac{b x^2}{a}\right )}{5 x^5 \left (a+b x^2\right )^{5/6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{6}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{6}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{6}}}{b x^{8} + a x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.9733, size = 32, normalized size = 0.1 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{6} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{5}{6}} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{6}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]