Optimal. Leaf size=65 \[ \frac{a^3}{4 b^4 \left (a+b x^2\right )^2}-\frac{3 a^2}{2 b^4 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2}{2 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0458384, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^3}{4 b^4 \left (a+b x^2\right )^2}-\frac{3 a^2}{2 b^4 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 b^3}+\frac{a^3}{4 b^4 \left (a+b x^2\right )^2}-\frac{3 a^2}{2 b^4 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0544749, size = 48, normalized size = 0.74 \[ -\frac{\frac{a^2 \left (5 a+6 b x^2\right )}{\left (a+b x^2\right )^2}+6 a \log \left (a+b x^2\right )-2 b x^2}{4 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 58, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{3}}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.33507, size = 89, normalized size = 1.37 \begin{align*} -\frac{6 \, a^{2} b x^{2} + 5 \, a^{3}}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac{x^{2}}{2 \, b^{3}} - \frac{3 \, a \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.19138, size = 186, normalized size = 2.86 \begin{align*} \frac{2 \, b^{3} x^{6} + 4 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.538994, size = 66, normalized size = 1.02 \begin{align*} - \frac{3 a \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{5 a^{3} + 6 a^{2} b x^{2}}{4 a^{2} b^{4} + 8 a b^{5} x^{2} + 4 b^{6} x^{4}} + \frac{x^{2}}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.71611, size = 84, normalized size = 1.29 \begin{align*} \frac{x^{2}}{2 \, b^{3}} - \frac{3 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{9 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} + 4 \, a^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]