Optimal. Leaf size=126 \[ -\frac{2 a^{7/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0546435, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {335, 195, 220} \[ -\frac{2 a^{7/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 335
Rule 195
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \left (a+b x^4\right )^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x}-\frac{1}{7} (6 a) \operatorname{Subst}\left (\int \sqrt{a+b x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x}-\frac{1}{7} \left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x}-\frac{2 a^{7/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}
Mathematica [C] time = 0.0161156, size = 52, normalized size = 0.41 \[ -\frac{b \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{a x^4}{b}\right )}{7 x^5 \sqrt{\frac{a x^4}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.014, size = 157, normalized size = 1.3 \begin{align*} -{\frac{1}{7\,x \left ( a{x}^{4}+b \right ) ^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( -4\,{a}^{2}\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{7}+3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}+4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}ab+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{2} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \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{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}{x^{2}}\,{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 (a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.87011, size = 39, normalized size = 0.31 \begin{align*} - \frac{a^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \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{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]