Optimal. Leaf size=297 \[ \frac{4 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.223943, 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 = {279, 329, 305, 220, 1196} \[ \frac{4 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \sqrt{c x} \left (a+b x^2\right )^{3/2} \, dx &=\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac{1}{3} (2 a) \int \sqrt{c x} \sqrt{a+b x^2} \, dx\\ &=\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac{1}{15} \left (4 a^2\right ) \int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx\\ &=\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac{\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{15 c}\\ &=\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}+\frac{\left (8 a^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{15 \sqrt{b}}-\frac{\left (8 a^{5/2}\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 )}{15 \sqrt{b}}\\ &=\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c}-\frac{8 a^{9/4} \sqrt{c} \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 )}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{4 a^{9/4} \sqrt{c} \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 )}{15 b^{3/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0118379, size = 57, normalized size = 0.19 \[ \frac{2 a x \sqrt{c x} \sqrt{a+b x^2} \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )}{3 \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 218, normalized size = 0.7 \begin{align*}{\frac{2}{45\,bx}\sqrt{cx} \left ( 5\,{b}^{3}{x}^{6}+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 ){a}^{3}-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 ){a}^{3}+16\,a{b}^{2}{x}^{4}+11\,{a}^{2}b{x}^{2} \right ){\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{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}\,{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 ({\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 4.2133, size = 46, normalized size = 0.15 \begin{align*} \frac{a^{\frac{3}{2}} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{7}{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{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]