Optimal. Leaf size=80 \[ \frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{x^2 (b c-a d)^2}{2 b^3}-\frac{a (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^6}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0863416, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{x^2 (b c-a d)^2}{2 b^3}-\frac{a (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^6}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (c+d x)^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(b c-a d)^2}{b^3}+\frac{d (2 b c-a d) x}{b^2}+\frac{d^2 x^2}{b}-\frac{a (-b c+a d)^2}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 x^2}{2 b^3}+\frac{d (2 b c-a d) x^4}{4 b^2}+\frac{d^2 x^6}{6 b}-\frac{a (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0385701, size = 82, normalized size = 1.02 \[ \frac{b x^2 \left (6 a^2 d^2-3 a b d \left (4 c+d x^2\right )+2 b^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right )-6 a (b c-a d)^2 \log \left (a+b x^2\right )}{12 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 124, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{6}}{6\,b}}-{\frac{{x}^{4}a{d}^{2}}{4\,{b}^{2}}}+{\frac{c{x}^{4}d}{2\,b}}+{\frac{{x}^{2}{a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{2}cd}{{b}^{2}}}+{\frac{{x}^{2}{c}^{2}}{2\,b}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) cd}{{b}^{3}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.982187, size = 136, normalized size = 1.7 \begin{align*} \frac{2 \, b^{2} d^{2} x^{6} + 3 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{4} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{12 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \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 = 1.64973, size = 212, normalized size = 2.65 \begin{align*} \frac{2 \, b^{3} d^{2} x^{6} + 3 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{4} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2} - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.536022, size = 83, normalized size = 1.04 \begin{align*} - \frac{a \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{d^{2} x^{6}}{6 b} - \frac{x^{4} \left (a d^{2} - 2 b c d\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14728, size = 144, normalized size = 1.8 \begin{align*} \frac{2 \, b^{2} d^{2} x^{6} + 6 \, b^{2} c d x^{4} - 3 \, a b d^{2} x^{4} + 6 \, b^{2} c^{2} x^{2} - 12 \, a b c d x^{2} + 6 \, a^{2} d^{2} x^{2}}{12 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]