Optimal. Leaf size=67 \[ -\frac{c}{b^3 \left (b+c x^2\right )}-\frac{c}{4 b^2 \left (b+c x^2\right )^2}+\frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{1}{2 b^3 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0605118, antiderivative size = 67, normalized size of antiderivative = 1., 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 = {1584, 266, 44} \[ -\frac{c}{b^3 \left (b+c x^2\right )}-\frac{c}{4 b^2 \left (b+c x^2\right )^2}+\frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{1}{2 b^3 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^3}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{1}{x^3 \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^2}-\frac{3 c}{b^4 x}+\frac{c^2}{b^2 (b+c x)^3}+\frac{2 c^2}{b^3 (b+c x)^2}+\frac{3 c^2}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left (b+c x^2\right )^2}-\frac{c}{b^3 \left (b+c x^2\right )}-\frac{3 c \log (x)}{b^4}+\frac{3 c \log \left (b+c x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0567536, size = 59, normalized size = 0.88 \[ -\frac{\frac{b \left (2 b^2+9 b c x^2+6 c^2 x^4\right )}{x^2 \left (b+c x^2\right )^2}-6 c \log \left (b+c x^2\right )+12 c \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{b}^{3}{x}^{2}}}-{\frac{c}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{c}{{b}^{3} \left ( c{x}^{2}+b \right ) }}-3\,{\frac{c\ln \left ( x \right ) }{{b}^{4}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.988503, size = 104, normalized size = 1.55 \begin{align*} -\frac{6 \, c^{2} x^{4} + 9 \, b c x^{2} + 2 \, b^{2}}{4 \,{\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} + \frac{3 \, c \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac{3 \, c \log \left (x^{2}\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62646, size = 247, normalized size = 3.69 \begin{align*} -\frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3} - 6 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (c x^{2} + b\right ) + 12 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.980319, size = 78, normalized size = 1.16 \begin{align*} - \frac{2 b^{2} + 9 b c x^{2} + 6 c^{2} x^{4}}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} - \frac{3 c \log{\left (x \right )}}{b^{4}} + \frac{3 c \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2833, size = 89, normalized size = 1.33 \begin{align*} \frac{3 \, c \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac{3 \, c \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]