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

Optimal. Leaf size=61 \[ \frac {2 \sqrt {x+1}}{15 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{15 (1-x)^{3/2}}+\frac {\sqrt {x+1}}{5 (1-x)^{5/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 37} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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 45

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*} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.49 \begin {gather*} \frac {\sqrt {x+1} \left (2 x^2-6 x+7\right )}{15 (1-x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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))

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IntegrateAlgebraic [A]  time = 0.06, size = 48, normalized size = 0.79 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {3 (x+1)^2}{(1-x)^2}+\frac {10 (x+1)}{1-x}+15\right )}{60 \sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(15 + (10*(1 + x))/(1 - x) + (3*(1 + x)^2)/(1 - x)^2))/(60*Sqrt[1 - x])

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fricas [A]  time = 1.18, size = 56, normalized size = 0.92 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.69, size = 29, normalized size = 0.48 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.00, size = 25, normalized size = 0.41 \begin {gather*} \frac {\sqrt {x +1}\, \left (2 x^{2}-6 x +7\right )}{15 \left (-x +1\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.03, size = 64, normalized size = 1.05 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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mupad [B]  time = 0.32, size = 55, normalized size = 0.90 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [C]  time = 7.89, size = 332, normalized size = 5.44 \begin {gather*} \begin {cases} - \frac {2 i \left (x + 1\right )^{2}}{- 15 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 60 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 60 i \sqrt {-1 + \frac {2}{x + 1}}} + \frac {10 i \left (x + 1\right )}{- 15 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 60 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 60 i \sqrt {-1 + \frac {2}{x + 1}}} - \frac {15 i}{- 15 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 60 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 60 i \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {2 \left (x + 1\right )^{2}}{15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 i \sqrt {1 - \frac {2}{x + 1}}} - \frac {10 \left (x + 1\right )}{15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 i \sqrt {1 - \frac {2}{x + 1}}} + \frac {15}{15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 60 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 60 i \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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