3.12.34 \(\int \frac {1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\) [1134]

Optimal. Leaf size=63 \[ \frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{15 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

1/5/(1-x)^(5/2)/(1+x)^(3/2)+4/15*x/(1-x)^(3/2)/(1+x)^(3/2)+8/15*x/(1-x)^(1/2)/(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 40, 39} \begin {gather*} \frac {8 x}{15 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(7/2)*(1 + x)^(5/2)),x]

[Out]

1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(15*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*x)/(15*Sqrt[1 - x]*Sqrt[1 + x]
)

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

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} (1+x)^{5/2}} \, dx &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4}{5} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8}{15} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{15 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 40, normalized size = 0.63 \begin {gather*} \frac {3+12 x-12 x^2-8 x^3+8 x^4}{15 (1-x)^{5/2} (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*(1 + x)^(5/2)),x]

[Out]

(3 + 12*x - 12*x^2 - 8*x^3 + 8*x^4)/(15*(1 - x)^(5/2)*(1 + x)^(3/2))

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Maple [A]
time = 0.16, size = 72, normalized size = 1.14

method	result	size
gosper	\(\frac {8 x^{4}-8 x^{3}-12 x^{2}+12 x +3}{15 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {5}{2}}}\)	\(35\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{4}-8 x^{3}-12 x^{2}+12 x +3\right )}{15 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(61\)
default	\(\frac {1}{5 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{15 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{15 \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{15 \sqrt {1+x}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(7/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/5/(1-x)^(5/2)/(1+x)^(3/2)+4/15/(1-x)^(3/2)/(1+x)^(3/2)+4/5/(1-x)^(1/2)/(1+x)^(3/2)-8/15*(1-x)^(1/2)/(1+x)^(3
/2)-8/15*(1-x)^(1/2)/(1+x)^(1/2)

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Maxima [A]
time = 0.35, size = 52, normalized size = 0.83 \begin {gather*} \frac {8 \, x}{15 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{15 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{5 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

8/15*x/sqrt(-x^2 + 1) + 4/15*x/(-x^2 + 1)^(3/2) - 1/5/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

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Fricas [A]
time = 0.99, size = 84, normalized size = 1.33 \begin {gather*} \frac {3 \, x^{5} - 3 \, x^{4} - 6 \, x^{3} + 6 \, x^{2} - {\left (8 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, x - 3}{15 \, {\left (x^{5} - x^{4} - 2 \, x^{3} + 2 \, x^{2} + x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/15*(3*x^5 - 3*x^4 - 6*x^3 + 6*x^2 - (8*x^4 - 8*x^3 - 12*x^2 + 12*x + 3)*sqrt(x + 1)*sqrt(-x + 1) + 3*x - 3)/
(x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 18.17, size = 425, normalized size = 6.75 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {40 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac {60 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {20 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {5 \sqrt {-1 + \frac {2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {40 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac {60 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {20 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {5 i \sqrt {1 - \frac {2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \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)**(5/2),x)

[Out]

Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120)
+ 40*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) - 60*sqrt
(-1 + 2/(x + 1))*(x + 1)**2/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 20*sqrt(-1 + 2/(
x + 1))*(x + 1)/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 5*sqrt(-1 + 2/(x + 1))/(-120
*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120), 1/Abs(x + 1) > 1/2), (-8*I*sqrt(1 - 2/(x + 1))*(x
+ 1)**4/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 40*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/
(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) - 60*I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-120*x
+ 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 20*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-120*x + 15*(x + 1
)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 5*I*sqrt(1 - 2/(x + 1))/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3
+ 180*(x + 1)**2 - 120), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (45) = 90\).
time = 1.32, size = 119, normalized size = 1.89 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{384 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {15 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{128 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {45 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{384 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left (73 \, x - 247\right )} {\left (x + 1\right )} + 360\right )} \sqrt {x + 1} \sqrt {-x + 1}}{240 \, {\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)^(5/2),x, algorithm="giac")

[Out]

1/384*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 15/128*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/384*(x + 1)^(
3/2)*(45*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/240*((73*x - 247)*(x + 1) + 36
0)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3

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Mupad [B]
time = 0.38, size = 75, normalized size = 1.19 \begin {gather*} -\frac {12\,x\,\sqrt {1-x}+3\,\sqrt {1-x}-12\,x^2\,\sqrt {1-x}-8\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{\left (15\,x+15\right )\,{\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)^(5/2)),x)

[Out]

-(12*x*(1 - x)^(1/2) + 3*(1 - x)^(1/2) - 12*x^2*(1 - x)^(1/2) - 8*x^3*(1 - x)^(1/2) + 8*x^4*(1 - x)^(1/2))/((1
5*x + 15)*(x - 1)^3*(x + 1)^(1/2))

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