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

Optimal. Leaf size=41 \[ \frac {(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac {(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac {(x+1)^{3/2}}{5 (1-x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(7/2),x]

[Out]

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

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 {\sqrt {1+x}}{(1-x)^{7/2}} \, dx &=\frac {(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac {1}{5} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac {(1+x)^{3/2}}{15 (1-x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(x-4) (x+1)^{3/2}}{15 (1-x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*((-4 + x)*(1 + x)^(3/2))/(1 - x)^(5/2)

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IntegrateAlgebraic [A]  time = 0.06, size = 34, normalized size = 0.83 \begin {gather*} \frac {(x+1)^{3/2} \left (\frac {3 (x+1)}{1-x}+5\right )}{30 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(3/2)*(5 + (3*(1 + x))/(1 - x)))/(30*(1 - x)^(3/2))

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fricas [A]  time = 0.92, size = 53, normalized size = 1.29 \begin {gather*} \frac {4 \, x^{3} - 12 \, x^{2} + {\left (x^{2} - 3 \, x - 4\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12 \, x - 4}{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+x)^(1/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

1/15*(4*x^3 - 12*x^2 + (x^2 - 3*x - 4)*sqrt(x + 1)*sqrt(-x + 1) + 12*x - 4)/(x^3 - 3*x^2 + 3*x - 1)

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giac [A]  time = 1.05, size = 22, normalized size = 0.54 \begin {gather*} \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (x - 4\right )} \sqrt {-x + 1}}{15 \, {\left (x - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.44 \begin {gather*} -\frac {\left (x +1\right )^{\frac {3}{2}} \left (x -4\right )}{15 \left (-x +1\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(x+1)^(3/2)*(x-4)/(-x+1)^(5/2)

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maxima [B]  time = 1.40, size = 64, normalized size = 1.56 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.24, size = 50, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {x\,\sqrt {x+1}}{5}+\frac {4\,\sqrt {x+1}}{15}-\frac {x^2\,\sqrt {x+1}}{15}\right )}{x^3-3\,x^2+3\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x)^(7/2),x)

[Out]

-((1 - x)^(1/2)*((x*(x + 1)^(1/2))/5 + (4*(x + 1)^(1/2))/15 - (x^2*(x + 1)^(1/2))/15))/(3*x - 3*x^2 + x^3 - 1)

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sympy [B]  time = 6.55, size = 173, normalized size = 4.22 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {5}{2}}}{15 \sqrt {x - 1} \left (x + 1\right )^{2} - 60 \sqrt {x - 1} \left (x + 1\right ) + 60 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{15 \sqrt {x - 1} \left (x + 1\right )^{2} - 60 \sqrt {x - 1} \left (x + 1\right ) + 60 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {\left (x + 1\right )^{\frac {5}{2}}}{15 \sqrt {1 - x} \left (x + 1\right )^{2} - 60 \sqrt {1 - x} \left (x + 1\right ) + 60 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{15 \sqrt {1 - x} \left (x + 1\right )^{2} - 60 \sqrt {1 - x} \left (x + 1\right ) + 60 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*(x + 1)**(5/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)) - 5*I*(x + 1
)**(3/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-(x + 1)**
(5/2)/(15*sqrt(1 - x)*(x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)) + 5*(x + 1)**(3/2)/(15*sqrt(1 - x)
*(x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)), True))

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