3.11.71 \(\int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx\) [1071]

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

[Out]

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

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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 = {47, 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 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*} \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 = 0.04, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(-4+x) (1+x)^{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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 5.66, size = 125, normalized size = 3.05 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-4+x\right ) \left (1+x\right )^{\frac {3}{2}}}{15 \sqrt {-1+x} \left (1-2 x+x^2\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},-\frac {\left (1+x\right )^{\frac {5}{2}}}{-60 \left (1+x\right ) \sqrt {1-x}+15 \left (1+x\right )^2 \sqrt {1-x}+60 \sqrt {1-x}}+\frac {5 \left (1+x\right )^{\frac {3}{2}}}{-60 \left (1+x\right ) \sqrt {1-x}+15 \left (1+x\right )^2 \sqrt {1-x}+60 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{I / 15 (-4 + x) (1 + x) ^ (3 / 2) / (Sqrt[-1 + x] (1 - 2 x + x ^ 2)), Abs[1 + x] > 2}}, -(1 + x) ^
(5 / 2) / (-60 (1 + x) Sqrt[1 - x] + 15 (1 + x) ^ 2 Sqrt[1 - x] + 60 Sqrt[1 - x]) + 5 (1 + x) ^ (3 / 2) / (-6
0 (1 + x) Sqrt[1 - x] + 15 (1 + x) ^ 2 Sqrt[1 - x] + 60 Sqrt[1 - x])]

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Maple [A]
time = 0.15, size = 44, normalized size = 1.07

method	result	size
gosper	\(-\frac {\left (1+x \right )^{\frac {3}{2}} \left (x -4\right )}{15 \left (1-x \right )^{\frac {5}{2}}}\)	\(18\)
default	\(\frac {2 \sqrt {1+x}}{5 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{15 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{15 \sqrt {1-x}}\)	\(44\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{3}-2 x^{2}-7 x -4\right )}{15 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
time = 0.25, 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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Fricas [A]
time = 0.30, 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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Sympy [C] Result contains complex when optimal does not.
time = 4.77, size = 172, normalized size = 4.20 \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}\: \left |{x + 1}\right | > 2 \\- \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), (-(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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Giac [A]
time = 0.01, size = 59, normalized size = 1.44 \begin {gather*} \frac {2 \left (\frac 1{6}-\frac {1}{30} \sqrt {x+1} \sqrt {x+1}\right ) \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {-x+1}}{\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)

[Out]

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

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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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