3.1113 \(\int \frac{1}{(1-x)^{7/2} \sqrt{1+x}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0078398, antiderivative size = 61, normalized size of antiderivative = 1., 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} \[ \frac{2 \sqrt{x+1}}{15 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{15 (1-x)^{3/2}}+\frac{\sqrt{x+1}}{5 (1-x)^{5/2}} \]

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 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])

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]

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*}

Mathematica [A]  time = 0.0093584, size = 30, normalized size = 0.49 \[ \frac{\sqrt{x+1} \left (2 x^2-6 x+7\right )}{15 (1-x)^{5/2}} \]

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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Maple [A]  time = 0.003, size = 25, normalized size = 0.4 \begin{align*}{\frac{2\,{x}^{2}-6\,x+7}{15}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.4832, size = 86, normalized size = 1.41 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.86757, size = 139, normalized size = 2.28 \begin{align*} \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{align*}

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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Sympy [B]  time = 35.3623, size = 301, normalized size = 4.93 \begin{align*} \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{2}{\left |{x + 1}\right |} > 1 \\- \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} \end{align*}

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*(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)
)), 2/Abs(x + 1) > 1), (-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) + 6
0*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))

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Giac [A]  time = 1.08045, size = 39, normalized size = 0.64 \begin{align*} -\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{align*}

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