Optimal. Leaf size=80 \[ -\frac{8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}+\frac{4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140599, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ -\frac{8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}+\frac{4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=-\frac{\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac{(4 c) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx}{9 b}\\ &=-\frac{\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac{4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}+\frac{\left (8 c^2\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx}{63 b^2}\\ &=-\frac{\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac{4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac{8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}\\ \end{align*}
Mathematica [A] time = 0.0150695, size = 46, normalized size = 0.57 \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (35 b^2-20 b c x^2+8 c^2 x^4\right )}{315 b^3 x^{14}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 50, normalized size = 0.6 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 8\,{c}^{2}{x}^{4}-20\,bc{x}^{2}+35\,{b}^{2} \right ) }{315\,{x}^{12}{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3423, size = 142, normalized size = 1.78 \begin{align*} -\frac{{\left (8 \, c^{4} x^{8} - 4 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} + 50 \, b^{3} c x^{2} + 35 \, b^{4}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{3} x^{10}} \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{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{13}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34965, size = 278, normalized size = 3.48 \begin{align*} \frac{16 \,{\left (210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 441 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} b^{2} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 126 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} b^{3} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{4} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{5} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + b^{6} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]