Optimal. Leaf size=38 \[ a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {41, 194} \[ -\frac {2}{3} a^2 b^2 c^2 x^3+a^4 c^2 x+\frac {1}{5} b^4 c^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 194
Rubi steps
\begin {align*} \int (a+b x)^2 (a c-b c x)^2 \, dx &=\int \left (a^2 c-b^2 c x^2\right )^2 \, dx\\ &=\int \left (a^4 c^2-2 a^2 b^2 c^2 x^2+b^4 c^2 x^4\right ) \, dx\\ &=a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 38, normalized size = 1.00 \[ a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 34, normalized size = 0.89 \[ \frac {1}{5} x^{5} c^{2} b^{4} - \frac {2}{3} x^{3} c^{2} b^{2} a^{2} + x c^{2} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 34, normalized size = 0.89 \[ \frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 35, normalized size = 0.92 \[ \frac {1}{5} b^{4} c^{2} x^{5}-\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+a^{4} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 34, normalized size = 0.89 \[ \frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 31, normalized size = 0.82 \[ \frac {c^2\,x\,\left (15\,a^4-10\,a^2\,b^2\,x^2+3\,b^4\,x^4\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 36, normalized size = 0.95 \[ a^{4} c^{2} x - \frac {2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac {b^{4} c^{2} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________