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 = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = 0.01, size = 50, normalized size = 1.00 \[ \frac {1}{5} d x^5 (a d+2 b c)+\frac {1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac {1}{7} b d^2 x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 50, normalized size = 1.00 \[ \frac {1}{7} x^{7} d^{2} b + \frac {2}{5} x^{5} d c b + \frac {1}{5} x^{5} d^{2} a + \frac {1}{3} x^{3} c^{2} b + \frac {2}{3} x^{3} d c a + x c^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 50, normalized size = 1.00 \[ \frac {1}{7} \, b d^{2} x^{7} + \frac {2}{5} \, b c d x^{5} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{3} \, b c^{2} x^{3} + \frac {2}{3} \, a c d x^{3} + a c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 49, normalized size = 0.98 \[ \frac {b \,d^{2} x^{7}}{7}+\frac {\left (a \,d^{2}+2 b c d \right ) x^{5}}{5}+a \,c^{2} x +\frac {\left (2 a c d +b \,c^{2}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, b d^{2} x^{7} + \frac {1}{5} \, {\left (2 \, b c d + a d^{2}\right )} x^{5} + a c^{2} x + \frac {1}{3} \, {\left (b c^{2} + 2 \, a c d\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 48, normalized size = 0.96 \[ x^3\,\left (\frac {b\,c^2}{3}+\frac {2\,a\,d\,c}{3}\right )+x^5\,\left (\frac {a\,d^2}{5}+\frac {2\,b\,c\,d}{5}\right )+\frac {b\,d^2\,x^7}{7}+a\,c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.07, size = 53, normalized size = 1.06 \[ a c^{2} x + \frac {b d^{2} x^{7}}{7} + x^{5} \left (\frac {a d^{2}}{5} + \frac {2 b c d}{5}\right ) + x^{3} \left (\frac {2 a c d}{3} + \frac {b c^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________