Optimal. Leaf size=88 \[ -\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {92, 81, 52, 41,
222} \begin {gather*} \frac {7 \text {ArcSin}(x)}{8}-\frac {1}{4} \sqrt {1-x} x (x+1)^{5/2}-\frac {1}{6} \sqrt {1-x} (x+1)^{5/2}-\frac {7}{24} \sqrt {1-x} (x+1)^{3/2}-\frac {7}{8} \sqrt {1-x} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 52
Rule 81
Rule 92
Rule 222
Rubi steps
\begin {align*} \int \frac {x^2 (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}-\frac {1}{4} \int \frac {(-1-2 x) (1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{12} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {7}{8} \sqrt {1-x} \sqrt {1+x}-\frac {7}{24} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{6} \sqrt {1-x} (1+x)^{5/2}-\frac {1}{4} \sqrt {1-x} x (1+x)^{5/2}+\frac {7}{8} \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 63, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {1-x} \left (32+53 x+37 x^2+22 x^3+6 x^4\right )}{24 \sqrt {1+x}}-\frac {7}{4} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 80, normalized size = 0.91
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (-6 x^{3} \sqrt {-x^{2}+1}-16 x^{2} \sqrt {-x^{2}+1}-21 x \sqrt {-x^{2}+1}+21 \arcsin \left (x \right )-32 \sqrt {-x^{2}+1}\right )}{24 \sqrt {-x^{2}+1}}\) | \(80\) |
risch | \(\frac {\left (6 x^{3}+16 x^{2}+21 x +32\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {7 \arcsin \left (x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8 \sqrt {1-x}\, \sqrt {1+x}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.62, size = 56, normalized size = 0.64 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \frac {2}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {7}{8} \, \sqrt {-x^{2} + 1} x - \frac {4}{3} \, \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.30, size = 52, normalized size = 0.59 \begin {gather*} -\frac {1}{24} \, {\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 65.45, size = 291, normalized size = 3.31 \begin {gather*} 2 \left (\begin {cases} - \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \sqrt {1 - x} \sqrt {x + 1} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 4 \left (\begin {cases} - \frac {3 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} - 2 \sqrt {1 - x} \sqrt {x + 1} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) + 2 \left (\begin {cases} - \frac {7 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {2 \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{3} + \frac {\sqrt {1 - x} \sqrt {x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt {1 - x} \sqrt {x + 1} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.76, size = 46, normalized size = 0.52 \begin {gather*} -\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )} + 7\right )} {\left (x + 1\right )} + 21\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________