Optimal. Leaf size=38 \[ -\frac {2}{5} (1-x)^{5/2}+\frac {8}{3} (1-x)^{3/2}-8 \sqrt {1-x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6129, 43} \[ -\frac {2}{5} (1-x)^{5/2}+\frac {8}{3} (1-x)^{3/2}-8 \sqrt {1-x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} (1+x)^{3/2} \, dx &=\int \frac {(1+x)^2}{\sqrt {1-x}} \, dx\\ &=\int \left (\frac {4}{\sqrt {1-x}}-4 \sqrt {1-x}+(1-x)^{3/2}\right ) \, dx\\ &=-8 \sqrt {1-x}+\frac {8}{3} (1-x)^{3/2}-\frac {2}{5} (1-x)^{5/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.61 \[ -\frac {2}{15} \sqrt {1-x} \left (3 x^2+14 x+43\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 26, normalized size = 0.68 \[ -\frac {2 \, {\left (3 \, x^{2} + 14 \, x + 43\right )} \sqrt {-x^{2} + 1}}{15 \, \sqrt {x + 1}} \]
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 = 30, normalized size = 0.79 \[ \frac {2 \left (-1+x \right ) \left (3 x^{2}+14 x +43\right ) \sqrt {1+x}}{15 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 24, normalized size = 0.63 \[ \frac {2 \, {\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt {-x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.88, size = 45, normalized size = 1.18 \[ -\frac {6\,x^2\,\sqrt {1-x^2}+28\,x\,\sqrt {1-x^2}+86\,\sqrt {1-x^2}}{15\,\sqrt {x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x + 1\right )^{\frac {5}{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________