Optimal. Leaf size=297 \[ \frac{12 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}+\frac{24 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{24 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.227211, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {277, 329, 305, 220, 1196} \[ \frac{24 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{12 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 277
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{(c x)^{7/2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac{(6 b) \int \frac{\sqrt{a+b x^2}}{(c x)^{3/2}} \, dx}{5 c^2}\\ &=-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac{\left (12 b^2\right ) \int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx}{5 c^4}\\ &=-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac{\left (24 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^5}\\ &=-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}+\frac{\left (24 \sqrt{a} b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^4}-\frac{\left (24 \sqrt{a} b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a} c}}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^4}\\ &=-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}+\frac{24 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{5 c (c x)^{5/2}}-\frac{24 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}+\frac{12 \sqrt [4]{a} b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{7/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.012294, size = 57, normalized size = 0.19 \[ -\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 (c x)^{7/2} \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 216, normalized size = 0.7 \begin{align*}{\frac{2}{5\,{x}^{2}{c}^{3}} \left ( 12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-7\,{b}^{2}{x}^{4}-8\,ab{x}^{2}-{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{7}{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 (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}}{c^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 62.7262, size = 53, normalized size = 0.18 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{7}{2}} x^{\frac{5}{2}} \Gamma \left (- \frac{1}{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 (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]