Optimal. Leaf size=71 \[ \frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 649
Rule 657
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=c \int \sqrt {c-a c x} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}+\frac {1}{5} \left (4 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 44, normalized size = 0.62 \[ -\frac {2 c (a x+1)^{3/2} (3 a x-7) \sqrt {c-a c x}}{15 a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 52, normalized size = 0.73 \[ \frac {2 \, {\left (3 \, a^{2} c x^{2} - 4 \, a c x - 7 \, c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{15 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 47, normalized size = 0.66 \[ \frac {2 \left (a x +1\right )^{2} \left (3 a x -7\right ) \left (-a c x +c \right )^{\frac {3}{2}}}{15 a \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 82, normalized size = 1.15 \[ -\frac {2 \, {\left (a^{3} c^{\frac {3}{2}} x^{3} - 2 \, a^{2} c^{\frac {3}{2}} x^{2} + 3 \, a c^{\frac {3}{2}} x + 6 \, c^{\frac {3}{2}}\right )}}{5 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (a^{2} c^{\frac {3}{2}} x^{2} - 4 \, a c^{\frac {3}{2}} x - 5 \, c^{\frac {3}{2}}\right )}}{3 \, \sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.93, size = 49, normalized size = 0.69 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {22\,c\,x}{15}+\frac {14\,c}{15\,a}-\frac {2\,a^2\,c\,x^3}{5}+\frac {2\,a\,c\,x^2}{15}\right )}{\sqrt {1-a^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________