Optimal. Leaf size=71 \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.105496, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac{\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(b c-a d)^2 (a+b x)^2}{b^2}+\frac{2 d (b c-a d) (a+b x)^3}{b^2}+\frac{d^2 (a+b x)^4}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \left (a+b x^2\right )^3}{6 b^3}+\frac{d (b c-a d) \left (a+b x^2\right )^4}{4 b^3}+\frac{d^2 \left (a+b x^2\right )^5}{10 b^3}\\ \end{align*}
Mathematica [A] time = 0.0218552, size = 81, normalized size = 1.14 \[ \frac{1}{60} x^2 \left (10 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+30 a^2 c^2+15 b d x^6 (a d+b c)+30 a c x^2 (a d+b c)+6 b^2 d^2 x^8\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 90, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{10}}{10}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{4}}{4}}+{\frac{{a}^{2}{c}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0141, size = 115, normalized size = 1.62 \begin{align*} \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac{1}{2} \, a^{2} c^{2} x^{2} + \frac{1}{2} \,{\left (a b c^{2} + a^{2} c d\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.0144, size = 220, normalized size = 3.1 \begin{align*} \frac{1}{10} x^{10} d^{2} b^{2} + \frac{1}{4} x^{8} d c b^{2} + \frac{1}{4} x^{8} d^{2} b a + \frac{1}{6} x^{6} c^{2} b^{2} + \frac{2}{3} x^{6} d c b a + \frac{1}{6} x^{6} d^{2} a^{2} + \frac{1}{2} x^{4} c^{2} b a + \frac{1}{2} x^{4} d c a^{2} + \frac{1}{2} x^{2} c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.07623, size = 94, normalized size = 1.32 \begin{align*} \frac{a^{2} c^{2} x^{2}}{2} + \frac{b^{2} d^{2} x^{10}}{10} + x^{8} \left (\frac{a b d^{2}}{4} + \frac{b^{2} c d}{4}\right ) + x^{6} \left (\frac{a^{2} d^{2}}{6} + \frac{2 a b c d}{3} + \frac{b^{2} c^{2}}{6}\right ) + x^{4} \left (\frac{a^{2} c d}{2} + \frac{a b c^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14301, size = 127, normalized size = 1.79 \begin{align*} \frac{1}{10} \, b^{2} d^{2} x^{10} + \frac{1}{4} \, b^{2} c d x^{8} + \frac{1}{4} \, a b d^{2} x^{8} + \frac{1}{6} \, b^{2} c^{2} x^{6} + \frac{2}{3} \, a b c d x^{6} + \frac{1}{6} \, a^{2} d^{2} x^{6} + \frac{1}{2} \, a b c^{2} x^{4} + \frac{1}{2} \, a^{2} c d x^{4} + \frac{1}{2} \, a^{2} c^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]