Optimal. Leaf size=352 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (c x^2\right )^{5/2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}\right ),-7-4 \sqrt{3}\right )}{20 a x^5 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.185679, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {368, 277, 325, 218} \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (c x^2\right )^{5/2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{20 a x^5 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 368
Rule 277
Rule 325
Rule 218
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^6} \, dx &=\frac{\left (c x^2\right )^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^3}}{x^6} \, dx,x,\sqrt{c x^2}\right )}{x^5}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5}+\frac{\left (3 b \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x^3}} \, dx,x,\sqrt{c x^2}\right )}{10 x^5}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{\left (3 b^2 \left (c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^3}} \, dx,x,\sqrt{c x^2}\right )}{40 a x^5}\\ &=-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (c x^2\right )^{5/2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}\right )|-7-4 \sqrt{3}\right )}{20 a x^5 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}\\ \end{align*}
Mathematica [C] time = 0.0168909, size = 69, normalized size = 0.2 \[ -\frac{\sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{5 x^5 \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, 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{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{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{\sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x^{6}}\, dx \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{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]