3.12.28 \(\int \frac {(1-x)^{5/2}}{(1+x)^{5/2}} \, dx\) [1128]

Optimal. Leaf size=63 \[ -\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \sin ^{-1}(x) \]

[Out]

-2/3*(1-x)^(5/2)/(1+x)^(3/2)+5*arcsin(x)+10/3*(1-x)^(3/2)/(1+x)^(1/2)+5*(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 = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 52, 41, 222} \begin {gather*} -\frac {2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {x+1}}+5 \sqrt {x+1} \sqrt {1-x}+5 \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(5/2))/(3*(1 + x)^(3/2)) + (10*(1 - x)^(3/2))/(3*Sqrt[1 + x]) + 5*Sqrt[1 - x]*Sqrt[1 + x] + 5*ArcS
in[x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1-x)^{5/2}}{(1+x)^{5/2}} \, dx &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}-\frac {5}{3} \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{3 (1+x)^{3/2}}+\frac {10 (1-x)^{3/2}}{3 \sqrt {1+x}}+5 \sqrt {1-x} \sqrt {1+x}+5 \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 51, normalized size = 0.81 \begin {gather*} \frac {\sqrt {1-x} \left (23+34 x+3 x^2\right )}{3 (1+x)^{3/2}}+10 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x]*(23 + 34*x + 3*x^2))/(3*(1 + x)^(3/2)) + 10*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 6.47, size = 171, normalized size = 2.71 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-8 \sqrt {\frac {1-x}{1+x}}+\left (1+x\right ) \left (15 I \text {Log}\left [1+x\right ]+15 I \text {Log}\left [\frac {1}{1+x}\right ]+3 \left (1+x\right ) \sqrt {\frac {1-x}{1+x}}+28 \sqrt {\frac {1-x}{1+x}}+30 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]\right )}{3 \left (1+x\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-10 I \text {Log}\left [1+\sqrt {1-\frac {2}{1+x}}\right ]-\frac {8 I \sqrt {1-\frac {2}{1+x}}}{3 \left (1+x\right )}+I 5 \text {Log}\left [\frac {1}{1+x}\right ]+\frac {I 28 \sqrt {1-\frac {2}{1+x}}}{3}+I \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(5/2)/(1 + x)^(5/2),x]')

[Out]

Piecewise[{{(-8 Sqrt[(1 - x) / (1 + x)] + (1 + x) (15 I Log[1 + x] + 15 I Log[1 / (1 + x)] + 3 (1 + x) Sqrt[(1
- x) / (1 + x)] + 28 Sqrt[(1 - x) / (1 + x)] + 30 ArcSin[Sqrt[2] Sqrt[1 + x] / 2])) / (3 (1 + x)), 1 / Abs[1
+ x] > 1 / 2}}, -10 I Log[1 + Sqrt[1 - 2 / (1 + x)]] - 8 I Sqrt[1 - 2 / (1 + x)] / (3 (1 + x)) + I 5 Log[1 / (
1 + x)] + I 28 Sqrt[1 - 2 / (1 + x)] / 3 + I (1 + x) Sqrt[1 - 2 / (1 + x)]]

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Maple [A]
time = 0.17, size = 79, normalized size = 1.25

method	result	size
risch	\(-\frac {\left (3 x^{3}+31 x^{2}-11 x -23\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^3+31*x^2-11*x-23)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+5*((1+x)*(1-x))^
(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (47) = 94\).
time = 0.35, size = 98, normalized size = 1.56 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} + 5 \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x^2 + 1)^(5/2)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 5/3*(-x^2 + 1)^(3/2)/(x^3 + 3*x^2 + 3*x + 1) - 10/3*sqrt(-x
^2 + 1)/(x^2 + 2*x + 1) + 35/3*sqrt(-x^2 + 1)/(x + 1) + 5*arcsin(x)

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Fricas [A]
time = 0.29, size = 75, normalized size = 1.19 \begin {gather*} \frac {23 \, x^{2} + {\left (3 \, x^{2} + 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} - 30 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 46 \, x + 23}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(23*x^2 + (3*x^2 + 34*x + 23)*sqrt(x + 1)*sqrt(-x + 1) - 30*(x^2 + 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x +
1) - 1)/x) + 46*x + 23)/(x^2 + 2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 4.83, size = 162, normalized size = 2.57 \begin {gather*} \begin {cases} \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) + \frac {28 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log {\left (\frac {1}{x + 1} \right )} + 5 i \log {\left (x + 1 \right )} + 10 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + \frac {28 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log {\left (\frac {1}{x + 1} \right )} - 10 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(5/2)/(1+x)**(5/2),x)

[Out]

Piecewise((sqrt(-1 + 2/(x + 1))*(x + 1) + 28*sqrt(-1 + 2/(x + 1))/3 - 8*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + 5*I
*log(1/(x + 1)) + 5*I*log(x + 1) + 10*asin(sqrt(2)*sqrt(x + 1)/2), 1/Abs(x + 1) > 1/2), (I*sqrt(1 - 2/(x + 1))
*(x + 1) + 28*I*sqrt(1 - 2/(x + 1))/3 - 8*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)) + 5*I*log(1/(x + 1)) - 10*I*log(sq
rt(1 - 2/(x + 1)) + 1), True))

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Giac [A]
time = 0.01, size = 86, normalized size = 1.37 \begin {gather*} \frac {2 \left (\left (\frac {1}{2} \sqrt {-x+1} \sqrt {-x+1}-\frac {20}{3}\right ) \sqrt {-x+1} \sqrt {-x+1}+10\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}-10 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(5/2)/(1+x)^(5/2),x)

[Out]

1/3*((3*x + 37)*(x - 1) + 60)*sqrt(-x + 1)/(x + 1)^(3/2) - 10*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (1-x\right )}^{5/2}}{{\left (x+1\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(5/2)/(x + 1)^(5/2),x)

[Out]

int((1 - x)^(5/2)/(x + 1)^(5/2), x)

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