Optimal. Leaf size=50 \[ a^2 c x+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{5} a x^5 (a d+2 b c)+\frac {1}{13} b^2 d x^{13} \]
[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} \[ a^2 c x+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{5} a x^5 (a d+2 b c)+\frac {1}{13} b^2 d x^{13} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^4\right )^2 \left (c+d x^4\right ) \, dx &=\int \left (a^2 c+a (2 b c+a d) x^4+b (b c+2 a d) x^8+b^2 d x^{12}\right ) \, dx\\ &=a^2 c x+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{9} b (b c+2 a d) x^9+\frac {1}{13} b^2 d x^{13}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ a^2 c x+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{5} a x^5 (a d+2 b c)+\frac {1}{13} b^2 d x^{13} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 50, normalized size = 1.00 \[ \frac {1}{13} x^{13} d b^{2} + \frac {1}{9} x^{9} c b^{2} + \frac {2}{9} x^{9} d b a + \frac {2}{5} x^{5} c b a + \frac {1}{5} x^{5} d a^{2} + x c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 50, normalized size = 1.00 \[ \frac {1}{13} \, b^{2} d x^{13} + \frac {1}{9} \, b^{2} c x^{9} + \frac {2}{9} \, a b d x^{9} + \frac {2}{5} \, a b c x^{5} + \frac {1}{5} \, a^{2} d x^{5} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 49, normalized size = 0.98 \[ \frac {b^{2} d \,x^{13}}{13}+\frac {\left (2 a b d +b^{2} c \right ) x^{9}}{9}+\frac {\left (a^{2} d +2 a b c \right ) x^{5}}{5}+a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 48, normalized size = 0.96 \[ \frac {1}{13} \, b^{2} d x^{13} + \frac {1}{9} \, {\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{5} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 48, normalized size = 0.96 \[ x^5\,\left (\frac {d\,a^2}{5}+\frac {2\,b\,c\,a}{5}\right )+x^9\,\left (\frac {c\,b^2}{9}+\frac {2\,a\,d\,b}{9}\right )+\frac {b^2\,d\,x^{13}}{13}+a^2\,c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 53, normalized size = 1.06 \[ a^{2} c x + \frac {b^{2} d x^{13}}{13} + x^{9} \left (\frac {2 a b d}{9} + \frac {b^{2} c}{9}\right ) + x^{5} \left (\frac {a^{2} d}{5} + \frac {2 a b c}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________