Optimal. Leaf size=37 \[ \frac {c^5 (1-a x)^6}{6 a}-\frac {2 c^5 (1-a x)^5}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac {c^5 (1-a x)^6}{6 a}-\frac {2 c^5 (1-a x)^5}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx &=c^5 \int (1-a x)^4 (1+a x) \, dx\\ &=c^5 \int \left (2 (1-a x)^4-(1-a x)^5\right ) \, dx\\ &=-\frac {2 c^5 (1-a x)^5}{5 a}+\frac {c^5 (1-a x)^6}{6 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 23, normalized size = 0.62 \[ \frac {c^5 (a x-1)^5 (5 a x+7)}{30 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 59, normalized size = 1.59 \[ \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 59, normalized size = 1.59 \[ \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 45, normalized size = 1.22 \[ c^{5} \left (\frac {1}{6} x^{6} a^{5}-\frac {3}{5} a^{4} x^{5}+\frac {1}{2} x^{4} a^{3}+\frac {2}{3} x^{3} a^{2}-\frac {3}{2} a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 59, normalized size = 1.59 \[ \frac {1}{6} \, a^{5} c^{5} x^{6} - \frac {3}{5} \, a^{4} c^{5} x^{5} + \frac {1}{2} \, a^{3} c^{5} x^{4} + \frac {2}{3} \, a^{2} c^{5} x^{3} - \frac {3}{2} \, a c^{5} x^{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 59, normalized size = 1.59 \[ \frac {a^5\,c^5\,x^6}{6}-\frac {3\,a^4\,c^5\,x^5}{5}+\frac {a^3\,c^5\,x^4}{2}+\frac {2\,a^2\,c^5\,x^3}{3}-\frac {3\,a\,c^5\,x^2}{2}+c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.09, size = 66, normalized size = 1.78 \[ \frac {a^{5} c^{5} x^{6}}{6} - \frac {3 a^{4} c^{5} x^{5}}{5} + \frac {a^{3} c^{5} x^{4}}{2} + \frac {2 a^{2} c^{5} x^{3}}{3} - \frac {3 a c^{5} x^{2}}{2} + c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________