Optimal. Leaf size=54 \[ \frac {1}{4 b \left (b+c x^2\right )^2}+\frac {1}{2 b^2 \left (b+c x^2\right )}+\frac {\log (x)}{b^3}-\frac {\log \left (b+c x^2\right )}{2 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 46}
\begin {gather*} -\frac {\log \left (b+c x^2\right )}{2 b^3}+\frac {\log (x)}{b^3}+\frac {1}{2 b^2 \left (b+c x^2\right )}+\frac {1}{4 b \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^3 x}-\frac {c}{b (b+c x)^3}-\frac {c}{b^2 (b+c x)^2}-\frac {c}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4 b \left (b+c x^2\right )^2}+\frac {1}{2 b^2 \left (b+c x^2\right )}+\frac {\log (x)}{b^3}-\frac {\log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 43, normalized size = 0.80 \begin {gather*} \frac {\frac {b \left (3 b+2 c x^2\right )}{\left (b+c x^2\right )^2}+4 \log (x)-2 \log \left (b+c x^2\right )}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 59, normalized size = 1.09
method | result | size |
risch | \(\frac {\frac {c \,x^{2}}{2 b^{2}}+\frac {3}{4 b}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\ln \left (x \right )}{b^{3}}-\frac {\ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(46\) |
norman | \(\frac {-\frac {c \,x^{7}}{b^{2}}-\frac {3 c^{2} x^{9}}{4 b^{3}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}+\frac {\ln \left (x \right )}{b^{3}}-\frac {\ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(55\) |
default | \(-\frac {c \left (-\frac {b}{c \left (c \,x^{2}+b \right )}+\frac {\ln \left (c \,x^{2}+b \right )}{c}-\frac {b^{2}}{2 c \left (c \,x^{2}+b \right )^{2}}\right )}{2 b^{3}}+\frac {\ln \left (x \right )}{b^{3}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 60, normalized size = 1.11 \begin {gather*} \frac {2 \, c x^{2} + 3 \, b}{4 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{2} + b^{4}\right )}} - \frac {\log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac {\log \left (x^{2}\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 90, normalized size = 1.67 \begin {gather*} \frac {2 \, b c x^{2} + 3 \, b^{2} - 2 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \log \left (x\right )}{4 \, {\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{2} + b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.21, size = 56, normalized size = 1.04 \begin {gather*} \frac {3 b + 2 c x^{2}}{4 b^{4} + 8 b^{3} c x^{2} + 4 b^{2} c^{2} x^{4}} + \frac {\log {\left (x \right )}}{b^{3}} - \frac {\log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.55, size = 59, normalized size = 1.09 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, b^{3}} - \frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac {3 \, c^{2} x^{4} + 8 \, b c x^{2} + 6 \, b^{2}}{4 \, {\left (c x^{2} + b\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 56, normalized size = 1.04 \begin {gather*} \frac {\ln \left (x\right )}{b^3}+\frac {\frac {3}{4\,b}+\frac {c\,x^2}{2\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}-\frac {\ln \left (c\,x^2+b\right )}{2\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________