Optimal. Leaf size=74 \[ \frac{\left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^3}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0557358, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{\left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^3}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2}}{x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x) (c+d x)^{3/2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{c (b c-a d) (c+d x)^{3/2}}{d^2}+\frac{(-2 b c+a d) (c+d x)^{5/2}}{d^2}+\frac{b (c+d x)^{7/2}}{d^2}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{c (b c-a d) \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^3}+\frac{(2 b c-a d) \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^3}\\ \end{align*}
Mathematica [A] time = 0.0250015, size = 71, normalized size = 0.96 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (9 a d x^2 \left (2 c x^2-5 d\right )+b \left (-8 c^2 x^4+20 c d x^2-35 d^2\right )\right )}{315 d^3 x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 70, normalized size = 1. \begin{align*}{\frac{ \left ( 18\,acd{x}^{4}-8\,b{c}^{2}{x}^{4}-45\,a{d}^{2}{x}^{2}+20\,bcd{x}^{2}-35\,b{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{315\,{d}^{3}{x}^{6}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.937936, size = 113, normalized size = 1.53 \begin{align*} -\frac{1}{35} \,{\left (\frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{2}} - \frac{7 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{2}}\right )} a - \frac{1}{315} \,{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{3}} - \frac{90 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{3}} + \frac{63 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{3}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59785, size = 239, normalized size = 3.23 \begin{align*} -\frac{{\left (2 \,{\left (4 \, b c^{4} - 9 \, a c^{3} d\right )} x^{8} -{\left (4 \, b c^{3} d - 9 \, a c^{2} d^{2}\right )} x^{6} + 35 \, b d^{4} + 3 \,{\left (b c^{2} d^{2} + 24 \, a c d^{3}\right )} x^{4} + 5 \,{\left (10 \, b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, d^{3} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 11.8264, size = 194, normalized size = 2.62 \begin{align*} - \frac{a c \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} - \frac{a \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{2}} - \frac{b c \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} - \frac{b \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 4.18228, size = 581, normalized size = 7.85 \begin{align*} \frac{4 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 840 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{7}{2}} d \mathrm{sgn}\left (x\right ) + 1260 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{9}{2}} d \mathrm{sgn}\left (x\right ) + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{7}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{9}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{7}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 504 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{9}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 441 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{7}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{9}{2}} d^{4} \mathrm{sgn}\left (x\right ) - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{7}{2}} d^{5} \mathrm{sgn}\left (x\right ) - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{9}{2}} d^{5} \mathrm{sgn}\left (x\right ) + 81 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{7}{2}} d^{6} \mathrm{sgn}\left (x\right ) + 4 \, b c^{\frac{9}{2}} d^{6} \mathrm{sgn}\left (x\right ) - 9 \, a c^{\frac{7}{2}} d^{7} \mathrm{sgn}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]