3.12.23 \(\int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\) [1123]

Optimal. Leaf size=62 \[ \frac {1}{5 (1-x)^{5/2} \sqrt {1+x}}+\frac {1}{5 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{5 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 62, 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, 39} \begin {gather*} \frac {2 x}{5 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{5 (1-x)^{3/2} \sqrt {x+1}}+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

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

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Mathematica [A]
time = 0.05, size = 33, normalized size = 0.53 \begin {gather*} \frac {2+x-4 x^2+2 x^3}{5 (-1+x)^2 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 13.12, size = 206, normalized size = 3.32 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-2-x+4 x^2-2 x^3\right ) \sqrt {\frac {1-x}{1+x}}}{5 \left (-1+3 x-3 x^2+x^3\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-15 I \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{20+60 x-30 \left (1+x\right )^2+5 \left (1+x\right )^3}-\frac {2 I \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{20+60 x-30 \left (1+x\right )^2+5 \left (1+x\right )^3}+\frac {I 5 \sqrt {1-\frac {2}{1+x}}}{20+60 x-30 \left (1+x\right )^2+5 \left (1+x\right )^3}+\frac {I 10 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{20+60 x-30 \left (1+x\right )^2+5 \left (1+x\right )^3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]')

[Out]

Piecewise[{{(-2 - x + 4 x ^ 2 - 2 x ^ 3) Sqrt[(1 - x) / (1 + x)] / (5 (-1 + 3 x - 3 x ^ 2 + x ^ 3)), 1 / Abs[1
+ x] > 1 / 2}}, -15 I (1 + x) Sqrt[1 - 2 / (1 + x)] / (20 + 60 x - 30 (1 + x) ^ 2 + 5 (1 + x) ^ 3) - 2 I (1 +
x) ^ 3 Sqrt[1 - 2 / (1 + x)] / (20 + 60 x - 30 (1 + x) ^ 2 + 5 (1 + x) ^ 3) + I 5 Sqrt[1 - 2 / (1 + x)] / (20
+ 60 x - 30 (1 + x) ^ 2 + 5 (1 + x) ^ 3) + I 10 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] / (20 + 60 x - 30 (1 + x) ^
2 + 5 (1 + x) ^ 3)]

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Maple [A]
time = 0.16, size = 58, normalized size = 0.94

method	result	size
gosper	\(\frac {2 x^{3}-4 x^{2}+x +2}{5 \sqrt {1+x}\, \left (1-x \right )^{\frac {5}{2}}}\)	\(28\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{3}-4 x^{2}+x +2\right )}{5 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(54\)
default	\(\frac {1}{5 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {1}{5 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {2}{5 \sqrt {1-x}\, \sqrt {1+x}}-\frac {2 \sqrt {1-x}}{5 \sqrt {1+x}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(7/2)/(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 79, normalized size = 1.27 \begin {gather*} \frac {2 \, x}{5 \, \sqrt {-x^{2} + 1}} + \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 59, normalized size = 0.95 \begin {gather*} \frac {2 \, x^{4} - 4 \, x^{3} - {\left (2 \, x^{3} - 4 \, x^{2} + x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x - 2}{5 \, {\left (x^{4} - 2 \, x^{3} + 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 13.06, size = 284, normalized size = 4.58 \begin {gather*} \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {10 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac {15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {5 \sqrt {-1 + \frac {2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {10 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac {15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac {5 i \sqrt {1 - \frac {2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \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)**(3/2),x)

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) + 10*sqrt(-1 + 2/(x +
1))*(x + 1)**2/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) - 15*sqrt(-1 + 2/(x + 1))*(x + 1)/(60*x + 5*(x + 1)
**3 - 30*(x + 1)**2 + 20) + 5*sqrt(-1 + 2/(x + 1))/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20), 1/Abs(x + 1) >
1/2), (-2*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) + 10*I*sqrt(1 - 2/(x + 1
))*(x + 1)**2/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) - 15*I*sqrt(1 - 2/(x + 1))*(x + 1)/(60*x + 5*(x + 1)*
*3 - 30*(x + 1)**2 + 20) + 5*I*sqrt(1 - 2/(x + 1))/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (44) = 88\).
time = 0.01, size = 243, normalized size = 3.92 \begin {gather*} 2 \left (\frac {\frac {1}{5}\cdot 68719476736 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}[phone] \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {1305670057984 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{35184372088832}+\frac {-190 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-15 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-1}{2560 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}}-\frac {\sqrt {-x+1} \sqrt {x+1}}{16 \left (x+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x)

[Out]

-1/1280*(sqrt(2) - sqrt(x + 1))^5/(-x + 1)^(5/2) - 3/256*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 19/128*(sq
rt(2) - sqrt(x + 1))/sqrt(-x + 1) - 1/8*sqrt(-x + 1)/sqrt(x + 1) + 1/1280*(190*(sqrt(2) - sqrt(x + 1))^4/(x -
1)^2 - 15*(sqrt(2) - sqrt(x + 1))^2/(x - 1) + 1)*(-x + 1)^(5/2)/(sqrt(2) - sqrt(x + 1))^5

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Mupad [B]
time = 0.34, size = 55, normalized size = 0.89 \begin {gather*} -\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-4\,x^2\,\sqrt {1-x}+2\,x^3\,\sqrt {1-x}}{5\,{\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)^(3/2)),x)

[Out]

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

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