\(\int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx\) [1113]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x}}{5 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{15 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{15 \sqrt {1-x}} \]

[Out]

1/5*(1+x)^(1/2)/(1-x)^(5/2)+2/15*(1+x)^(1/2)/(1-x)^(3/2)+2/15*(1+x)^(1/2)/(1-x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=\frac {2 \sqrt {x+1}}{15 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{15 (1-x)^{3/2}}+\frac {\sqrt {x+1}}{5 (1-x)^{5/2}} \]

[In]

Int[1/((1 - x)^(7/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(5*(1 - x)^(5/2)) + (2*Sqrt[1 + x])/(15*(1 - x)^(3/2)) + (2*Sqrt[1 + x])/(15*Sqrt[1 - x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x}}{5 (1-x)^{5/2}}+\frac {2}{5} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx \\ & = \frac {\sqrt {1+x}}{5 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{15 (1-x)^{3/2}}+\frac {2}{15} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx \\ & = \frac {\sqrt {1+x}}{5 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{15 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{15 \sqrt {1-x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x} \left (7-6 x+2 x^2\right )}{15 (1-x)^{5/2}} \]

[In]

Integrate[1/((1 - x)^(7/2)*Sqrt[1 + x]),x]

[Out]

(Sqrt[1 + x]*(7 - 6*x + 2*x^2))/(15*(1 - x)^(5/2))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41

method	result	size
gosper	\(\frac {\sqrt {1+x}\, \left (2 x^{2}-6 x +7\right )}{15 \left (1-x \right )^{\frac {5}{2}}}\)	\(25\)
default	\(\frac {\sqrt {1+x}}{5 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{15 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{15 \sqrt {1-x}}\)	\(44\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{3}-4 x^{2}+x +7\right )}{15 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(54\)

[In]

int(1/(1-x)^(7/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/15/(1-x)^(5/2)*(1+x)^(1/2)*(2*x^2-6*x+7)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=\frac {7 \, x^{3} - 21 \, x^{2} - {\left (2 \, x^{2} - 6 \, x + 7\right )} \sqrt {x + 1} \sqrt {-x + 1} + 21 \, x - 7}{15 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/15*(7*x^3 - 21*x^2 - (2*x^2 - 6*x + 7)*sqrt(x + 1)*sqrt(-x + 1) + 21*x - 7)/(x^3 - 3*x^2 + 3*x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.84 (sec) , antiderivative size = 303, normalized size of antiderivative = 4.97 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=\begin {cases} \frac {2 \left (x + 1\right )^{2}}{15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {10 \left (x + 1\right )}{15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {-1 + \frac {2}{x + 1}}} + \frac {15}{15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \left (x + 1\right )^{2}}{15 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {1 - \frac {2}{x + 1}}} + \frac {10 i \left (x + 1\right )}{15 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {1 - \frac {2}{x + 1}}} - \frac {15 i}{15 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(1-x)**(7/2)/(1+x)**(1/2),x)

[Out]

Piecewise((2*(x + 1)**2/(15*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 60*sqrt(-1 + 2/(x + 1))*(x + 1) + 60*sqrt(-1 + 2
/(x + 1))) - 10*(x + 1)/(15*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 60*sqrt(-1 + 2/(x + 1))*(x + 1) + 60*sqrt(-1 + 2
/(x + 1))) + 15/(15*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 60*sqrt(-1 + 2/(x + 1))*(x + 1) + 60*sqrt(-1 + 2/(x + 1)
)), 1/Abs(x + 1) > 1/2), (-2*I*(x + 1)**2/(15*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 60*sqrt(1 - 2/(x + 1))*(x + 1)
+ 60*sqrt(1 - 2/(x + 1))) + 10*I*(x + 1)/(15*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 60*sqrt(1 - 2/(x + 1))*(x + 1) +
60*sqrt(1 - 2/(x + 1))) - 15*I/(15*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 60*sqrt(1 - 2/(x + 1))*(x + 1) + 60*sqrt(
1 - 2/(x + 1))), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{15 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{15 \, {\left (x - 1\right )}} \]

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-1/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 2/15*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 2/15*sqrt(-x^2 + 1)/(x - 1
)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=-\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 4\right )} + 15\right )} \sqrt {x + 1} \sqrt {-x + 1}}{15 \, {\left (x - 1\right )}^{3}} \]

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

-1/15*(2*(x + 1)*(x - 4) + 15)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx=-\frac {x\,\sqrt {1-x}+7\,\sqrt {1-x}-4\,x^2\,\sqrt {1-x}+2\,x^3\,\sqrt {1-x}}{15\,{\left (x-1\right )}^3\,\sqrt {x+1}} \]

[In]

int(1/((1 - x)^(7/2)*(x + 1)^(1/2)),x)

[Out]

-(x*(1 - x)^(1/2) + 7*(1 - x)^(1/2) - 4*x^2*(1 - x)^(1/2) + 2*x^3*(1 - x)^(1/2))/(15*(x - 1)^3*(x + 1)^(1/2))