Optimal. Leaf size=41 \[ \frac {b B-A c}{2 c^2 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {1598, 455, 45}
\begin {gather*} \frac {b B-A c}{2 c^2 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 455
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {-b B+A c}{c (b+c x)^2}+\frac {B}{c (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b B-A c}{2 c^2 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} \frac {b B-A c}{2 c^2 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 38, normalized size = 0.93
method | result | size |
default | \(\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{2}}-\frac {A c -B b}{2 c^{2} \left (c \,x^{2}+b \right )}\) | \(38\) |
norman | \(\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{2}}-\frac {A c -B b}{2 c^{2} \left (c \,x^{2}+b \right )}\) | \(38\) |
risch | \(\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{2}}-\frac {A}{2 c \left (c \,x^{2}+b \right )}+\frac {B b}{2 c^{2} \left (c \,x^{2}+b \right )}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 40, normalized size = 0.98 \begin {gather*} \frac {B b - A c}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} + \frac {B \log \left (c x^{2} + b\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.51, size = 44, normalized size = 1.07 \begin {gather*} \frac {B b - A c + {\left (B c x^{2} + B b\right )} \log \left (c x^{2} + b\right )}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.18, size = 36, normalized size = 0.88 \begin {gather*} \frac {B \log {\left (b + c x^{2} \right )}}{2 c^{2}} + \frac {- A c + B b}{2 b c^{2} + 2 c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.33, size = 37, normalized size = 0.90 \begin {gather*} \frac {B \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} - \frac {B x^{2} + A}{2 \, {\left (c x^{2} + b\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 37, normalized size = 0.90 \begin {gather*} \frac {B\,\ln \left (c\,x^2+b\right )}{2\,c^2}-\frac {A\,c-B\,b}{2\,c^2\,\left (c\,x^2+b\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________