Optimal. Leaf size=68 \[ \frac{\sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{d^3}+\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0521264, antiderivative size = 68, 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} \[ \frac{\sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{d^3}+\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{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)}{d^2 (c+d x)^{3/2}}+\frac{-2 b c+a d}{d^2 \sqrt{c+d x}}+\frac{b \sqrt{c+d x}}{d^2}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}+\frac{(2 b c-a d) \sqrt{c+\frac{d}{x^2}}}{d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0226575, size = 60, normalized size = 0.88 \[ \frac{b \left (8 c^2 x^4+4 c d x^2-d^2\right )-3 a d x^2 \left (2 c x^2+d\right )}{3 d^3 x^4 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 69, normalized size = 1. \begin{align*} -{\frac{ \left ( 6\,acd{x}^{4}-8\,b{c}^{2}{x}^{4}+3\,a{d}^{2}{x}^{2}-4\,bcd{x}^{2}+b{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{3\,{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 = 0.956154, size = 109, normalized size = 1.6 \begin{align*} -\frac{1}{3} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{3}} - \frac{6 \, \sqrt{c + \frac{d}{x^{2}}} c}{d^{3}} - \frac{3 \, c^{2}}{\sqrt{c + \frac{d}{x^{2}}} d^{3}}\right )} - a{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{c}{\sqrt{c + \frac{d}{x^{2}}} d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58677, size = 150, normalized size = 2.21 \begin{align*} \frac{{\left (2 \,{\left (4 \, b c^{2} - 3 \, a c d\right )} x^{4} - b d^{2} +{\left (4 \, b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \,{\left (c d^{3} x^{4} + d^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.43081, size = 61, normalized size = 0.9 \begin{align*} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d^{3}} - \frac{c \left (a d - b c\right )}{d^{3} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (a d - 2 b c\right )}{d^{3}} \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{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]