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

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

[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)

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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 = {47, 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 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 {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 = 0.05, size = 30, normalized size = 0.49 \begin {gather*} \frac {\sqrt {1+x} \left (7-6 x+2 x^2\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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 8.26, size = 211, normalized size = 3.46 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {7-6 x+2 x^2}{15 \left (1-2 x+x^2\right ) \sqrt {\frac {1-x}{1+x}}},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-15 I}{-60 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}+15 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+60 \sqrt {1-\frac {2}{1+x}}}-\frac {2 I \left (1+x\right )^2}{-60 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}+15 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+60 \sqrt {1-\frac {2}{1+x}}}+\frac {I 10 \left (1+x\right )}{-60 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}+15 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+60 \sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{(7 - 6 x + 2 x ^ 2) / (15 (1 - 2 x + x ^ 2) Sqrt[(1 - x) / (1 + x)]), 1 / Abs[1 + x] > 1 / 2}}, -1
5 I / (-60 (1 + x) Sqrt[1 - 2 / (1 + x)] + 15 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] + 60 Sqrt[1 - 2 / (1 + x)]) -
2 I (1 + x) ^ 2 / (-60 (1 + x) Sqrt[1 - 2 / (1 + x)] + 15 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] + 60 Sqrt[1 - 2 /
(1 + x)]) + I 10 (1 + x) / (-60 (1 + x) Sqrt[1 - 2 / (1 + x)] + 15 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] + 60 Sqrt
[1 - 2 / (1 + x)])]

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Maple [A]
time = 0.14, size = 44, normalized size = 0.72

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 \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{3}-4 x^{2}+x +7\right )}{15 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) \(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]
time = 0.35, 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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Fricas [A]
time = 0.30, 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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Sympy [C] Result contains complex when optimal does not.
time = 5.54, size = 303, normalized size = 4.97 \begin {gather*} \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} \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*(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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (43) = 86\).
time = 0.01, size = 222, normalized size = 3.64 \begin {gather*} 2 \left (\frac {\frac {1}{5}\cdot 4294967296 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}+\frac {1}{3}\cdot 21474836480 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {21474836480 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{1099511627776}+\frac {-150 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-25 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-3}{3840 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}}\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)

[Out]

-1/640*(sqrt(2) - sqrt(x + 1))^5/(-x + 1)^(5/2) - 5/384*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 5/64*(sqrt(
2) - sqrt(x + 1))/sqrt(-x + 1) + 1/1920*(150*(sqrt(2) - sqrt(x + 1))^4/(x - 1)^2 - 25*(sqrt(2) - sqrt(x + 1))^
2/(x - 1) + 3)*(-x + 1)^(5/2)/(sqrt(2) - sqrt(x + 1))^5

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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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