Optimal. Leaf size=40 \[ \frac {2 (c-a c x)^{9/2}}{9 a c}-\frac {4 (c-a c x)^{7/2}}{7 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6130, 21, 43} \[ \frac {2 (c-a c x)^{9/2}}{9 a c}-\frac {4 (c-a c x)^{7/2}}{7 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 43
Rule 6130
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=\int \frac {(1+a x) (c-a c x)^{7/2}}{1-a x} \, dx\\ &=c \int (1+a x) (c-a c x)^{5/2} \, dx\\ &=c \int \left (2 (c-a c x)^{5/2}-\frac {(c-a c x)^{7/2}}{c}\right ) \, dx\\ &=-\frac {4 (c-a c x)^{7/2}}{7 a}+\frac {2 (c-a c x)^{9/2}}{9 a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 34, normalized size = 0.85 \[ \frac {2 c^3 (a x-1)^3 (7 a x+11) \sqrt {c-a c x}}{63 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 60, normalized size = 1.50 \[ \frac {2 \, {\left (7 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} + 26 \, a c^{3} x - 11 \, c^{3}\right )} \sqrt {-a c x + c}}{63 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 205, normalized size = 5.12 \[ -\frac {2 \, {\left (90 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 378 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 630 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 945 \, \sqrt {-a c x + c} c^{3} + 210 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c\right )} c^{2} - \frac {35 \, {\left (a c x - c\right )}^{4} \sqrt {-a c x + c} + 180 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} c + 378 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c^{2} - 420 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {-a c x + c} c^{4}}{c}\right )}}{315 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 21, normalized size = 0.52 \[ -\frac {2 \left (-a c x +c \right )^{\frac {7}{2}} \left (7 a x +11\right )}{63 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 32, normalized size = 0.80 \[ \frac {2 \, {\left (7 \, {\left (-a c x + c\right )}^{\frac {9}{2}} - 18 \, {\left (-a c x + c\right )}^{\frac {7}{2}} c\right )}}{63 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 32, normalized size = 0.80 \[ \frac {2\,{\left (c-a\,c\,x\right )}^{9/2}}{9\,a\,c}-\frac {4\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 31.79, size = 172, normalized size = 4.30 \[ c^{3} \left (\begin {cases} \sqrt {c} x & \text {for}\: a = 0 \\0 & \text {for}\: c = 0 \\- \frac {2 \left (- a c x + c\right )^{\frac {3}{2}}}{3 a c} & \text {otherwise} \end {cases}\right ) - \frac {2 c \left (- \frac {c \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a} + \frac {2 \left (\frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a} + \frac {2 \left (- \frac {c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (- a c x + c\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (- a c x + c\right )^{\frac {7}{2}}}{7} + \frac {\left (- a c x + c\right )^{\frac {9}{2}}}{9}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________