Optimal. Leaf size=555 \[ \frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{3 x}{\left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.394179, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {198, 235, 304, 219, 1879} \[ -\frac{3 x}{\left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 198
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{7/6}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{\left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=-\frac{\left (3 \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 )}{2 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac{\left (3 \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 )}{2 b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac{\left (3 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \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 )}{b x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac{3 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt{-1+\frac{a}{a+b x^2}}}{b 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{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \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 )}{2 b 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{\sqrt{2} 3^{3/4} \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 )}{b 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 = 0.007328, size = 49, normalized size = 0.09 \[ \frac{x \sqrt [6]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right )}{a \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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{7}{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{5}{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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.03937, size = 24, normalized size = 0.04 \begin{align*} \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{6} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{7}{6}}} \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{7}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]