\(\int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx\) [1082]

Optimal result

Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=-\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\arcsin (x) \]

[Out]

2/3*(1+x)^(3/2)/(1-x)^(3/2)+arcsin(x)-2*(1+x)^(1/2)/(1-x)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=\arcsin (x)+\frac {2 (x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 \sqrt {x+1}}{\sqrt {1-x}} \]

[In]

Int[(1 + x)^(3/2)/(1 - x)^(5/2),x]

[Out]

(-2*Sqrt[1 + x])/Sqrt[1 - x] + (2*(1 + x)^(3/2))/(3*(1 - x)^(3/2)) + ArcSin[x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}-\int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {2 (1+x)^{3/2}}{3 (1-x)^{3/2}}+\sin ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=\frac {4 (-1+2 x) \sqrt {1-x^2}}{3 (-1+x)^2}-2 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(5/2),x]

[Out]

(4*(-1 + 2*x)*Sqrt[1 - x^2])/(3*(-1 + x)^2) - 2*ArcTan[Sqrt[1 - x^2]/(-1 + x)]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(31)=62\).

Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85

method	result	size
risch	\(-\frac {4 \left (2 x^{2}+x -1\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(76\)

[In]

int((1+x)^(3/2)/(1-x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-4/3*(2*x^2+x-1)/(-1+x)/(-(-1+x)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)+((1+x)*(1-x))^(1/2)/
(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).

Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, x^{2} - 2 \, {\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 4 \, x + 2\right )}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(2*x^2 - 2*(2*x - 1)*sqrt(x + 1)*sqrt(-x + 1) + 3*(x^2 - 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x
) - 4*x + 2)/(x^2 - 2*x + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.97 (sec) , antiderivative size = 498, normalized size of antiderivative = 12.15 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=\begin {cases} - \frac {6 i \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {3 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 i \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {6 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {8 i \left (x + 1\right )^{8}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 i \left (x + 1\right )^{7}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {6 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {8 \left (x + 1\right )^{8}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 \left (x + 1\right )^{7}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((1+x)**(3/2)/(1-x)**(5/2),x)

[Out]

Piecewise((-6*I*sqrt(x - 1)*(x + 1)**(15/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(x - 1)*(x + 1)**(15/2) - 6*sq
rt(x - 1)*(x + 1)**(13/2)) + 3*pi*sqrt(x - 1)*(x + 1)**(15/2)/(3*sqrt(x - 1)*(x + 1)**(15/2) - 6*sqrt(x - 1)*(
x + 1)**(13/2)) + 12*I*sqrt(x - 1)*(x + 1)**(13/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(x - 1)*(x + 1)**(15/2)
- 6*sqrt(x - 1)*(x + 1)**(13/2)) - 6*pi*sqrt(x - 1)*(x + 1)**(13/2)/(3*sqrt(x - 1)*(x + 1)**(15/2) - 6*sqrt(x
- 1)*(x + 1)**(13/2)) + 8*I*(x + 1)**8/(3*sqrt(x - 1)*(x + 1)**(15/2) - 6*sqrt(x - 1)*(x + 1)**(13/2)) - 12*I
*(x + 1)**7/(3*sqrt(x - 1)*(x + 1)**(15/2) - 6*sqrt(x - 1)*(x + 1)**(13/2)), Abs(x + 1) > 2), (6*sqrt(1 - x)*(
x + 1)**(15/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/2)) - 1
2*sqrt(1 - x)*(x + 1)**(13/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x +
1)**(13/2)) - 8*(x + 1)**8/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/2)) + 12*(x + 1)**7/(3*
sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/2)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + \arcsin \left (x\right ) \]

[In]

integrate((1+x)^(3/2)/(1-x)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 1)^(3/2)/(x^3 - 3*x^2 + 3*x - 1) + 2/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 7/3*sqrt(-x^2 + 1)/(x - 1
) + arcsin(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=\frac {4 \, {\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[In]

integrate((1+x)^(3/2)/(1-x)^(5/2),x, algorithm="giac")

[Out]

4/3*(2*x - 1)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{{\left (1-x\right )}^{5/2}} \,d x \]

[In]

int((x + 1)^(3/2)/(1 - x)^(5/2),x)

[Out]

int((x + 1)^(3/2)/(1 - x)^(5/2), x)