Optimal. Leaf size=173 \[ -\frac{4 c^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{77 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237128, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2020, 2025, 2032, 329, 220} \[ -\frac{4 c^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{19/2}} \, dx &=-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}+\frac{1}{11} (6 c) \int \frac{\sqrt{b x^2+c x^4}}{x^{11/2}} \, dx\\ &=-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}+\frac{1}{77} \left (12 c^2\right ) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}-\frac{\left (4 c^3\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{77 b}\\ &=-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}-\frac{\left (4 c^3 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{77 b \sqrt{b x^2+c x^4}}\\ &=-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}-\frac{\left (8 c^3 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{77 b \sqrt{b x^2+c x^4}}\\ &=-\frac{12 c \sqrt{b x^2+c x^4}}{77 x^{9/2}}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{77 b x^{5/2}}-\frac{2 \left (b x^2+c x^4\right )^{3/2}}{11 x^{17/2}}-\frac{4 c^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0178096, size = 58, normalized size = 0.34 \[ -\frac{2 b \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};-\frac{c x^2}{b}\right )}{11 x^{13/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.19, size = 156, normalized size = 0.9 \begin{align*} -{\frac{2}{77\, \left ( c{x}^{2}+b \right ) ^{2}b} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{5}{c}^{2}+4\,{c}^{3}{x}^{6}+17\,b{c}^{2}{x}^{4}+20\,{b}^{2}c{x}^{2}+7\,{b}^{3} \right ){x}^{-{\frac{17}{2}}}} \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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{19}{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{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{x^{\frac{15}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}{x^{\frac{19}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]