Optimal. Leaf size=61 \[ -\frac{b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac{(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3}+\frac{B x^2}{2 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0697961, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 446, 77} \[ -\frac{b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac{(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3}+\frac{B x^2}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^7 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^3 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (A+B x)}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{B}{c^2}+\frac{b (b B-A c)}{c^2 (b+c x)^2}+\frac{-2 b B+A c}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{B x^2}{2 c^2}-\frac{b (b B-A c)}{2 c^3 \left (b+c x^2\right )}-\frac{(2 b B-A c) \log \left (b+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0360475, size = 50, normalized size = 0.82 \[ \frac{\frac{b (A c-b B)}{b+c x^2}+(A c-2 b B) \log \left (b+c x^2\right )+B c x^2}{2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 74, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) Bb}{{c}^{3}}}+{\frac{Ab}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}-{\frac{B{b}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.26542, size = 81, normalized size = 1.33 \begin{align*} \frac{B x^{2}}{2 \, c^{2}} - \frac{B b^{2} - A b c}{2 \,{\left (c^{4} x^{2} + b c^{3}\right )}} - \frac{{\left (2 \, B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.689542, size = 165, normalized size = 2.7 \begin{align*} \frac{B c^{2} x^{4} + B b c x^{2} - B b^{2} + A b c -{\left (2 \, B b^{2} - A b c +{\left (2 \, B b c - A c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{2 \,{\left (c^{4} x^{2} + b c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.715123, size = 56, normalized size = 0.92 \begin{align*} \frac{B x^{2}}{2 c^{2}} - \frac{- A b c + B b^{2}}{2 b c^{3} + 2 c^{4} x^{2}} - \frac{\left (- A c + 2 B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29789, size = 95, normalized size = 1.56 \begin{align*} \frac{B x^{2}}{2 \, c^{2}} - \frac{{\left (2 \, B b - A c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} + \frac{2 \, B b c x^{2} - A c^{2} x^{2} + B b^{2}}{2 \,{\left (c x^{2} + b\right )} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]