Optimal. Leaf size=50 \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0274234, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \, dx &=\int \left (a c^2+c (b c+2 a d) x^2+d (2 b c+a d) x^4+b d^2 x^6\right ) \, dx\\ &=a c^2 x+\frac{1}{3} c (b c+2 a d) x^3+\frac{1}{5} d (2 b c+a d) x^5+\frac{1}{7} b d^2 x^7\\ \end{align*}
Mathematica [A] time = 0.0108334, size = 50, normalized size = 1. \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 49, normalized size = 1. \begin{align*}{\frac{b{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{3}}{3}}+a{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01954, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{7} \, b d^{2} x^{7} + \frac{1}{5} \,{\left (2 \, b c d + a d^{2}\right )} x^{5} + a c^{2} x + \frac{1}{3} \,{\left (b c^{2} + 2 \, a c d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5615, size = 120, normalized size = 2.4 \begin{align*} \frac{1}{7} x^{7} d^{2} b + \frac{2}{5} x^{5} d c b + \frac{1}{5} x^{5} d^{2} a + \frac{1}{3} x^{3} c^{2} b + \frac{2}{3} x^{3} d c a + x c^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.064517, size = 53, normalized size = 1.06 \begin{align*} a c^{2} x + \frac{b d^{2} x^{7}}{7} + x^{5} \left (\frac{a d^{2}}{5} + \frac{2 b c d}{5}\right ) + x^{3} \left (\frac{2 a c d}{3} + \frac{b c^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07553, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{7} \, b d^{2} x^{7} + \frac{2}{5} \, b c d x^{5} + \frac{1}{5} \, a d^{2} x^{5} + \frac{1}{3} \, b c^{2} x^{3} + \frac{2}{3} \, a c d x^{3} + a c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]