3.829 \(\int \frac{x^2 \sqrt{1+x}}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-\sqrt{1-x} \sqrt{x+1}-\frac{4 \sqrt{x+1}}{\sqrt{1-x}}+3 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0136305, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {89, 21, 47, 50, 41, 216} \[ -\frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+\frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}-3 \sqrt{1-x} \sqrt{x+1}+3 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 47

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 50

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 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 216

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

Mathematica [A]  time = 0.0272696, size = 49, normalized size = 0.8 \[ -\frac{\sqrt{x+1} \left (3 x^2-19 x+14\right )}{3 (1-x)^{3/2}}-6 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + x]*(14 - 19*x + 3*x^2))/(3*(1 - x)^(3/2)) - 6*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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Maple [A]  time = 0.01, size = 83, normalized size = 1.4 \begin{align*}{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 9\,\arcsin \left ( x \right ){x}^{2}-3\,{x}^{2}\sqrt{-{x}^{2}+1}-18\,\arcsin \left ( x \right ) x+19\,x\sqrt{-{x}^{2}+1}+9\,\arcsin \left ( x \right ) -14\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(1+x)^(1/2)/(1-x)^(5/2),x)

[Out]

1/3*(9*arcsin(x)*x^2-3*x^2*(-x^2+1)^(1/2)-18*arcsin(x)*x+19*x*(-x^2+1)^(1/2)+9*arcsin(x)-14*(-x^2+1)^(1/2))*(1
-x)^(1/2)*(1+x)^(1/2)/(-1+x)^2/(-x^2+1)^(1/2)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 2.0842, size = 205, normalized size = 3.36 \begin{align*} -\frac{14 \, x^{2} +{\left (3 \, x^{2} - 19 \, x + 14\right )} \sqrt{x + 1} \sqrt{-x + 1} + 18 \,{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 28 \, x + 14}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(14*x^2 + (3*x^2 - 19*x + 14)*sqrt(x + 1)*sqrt(-x + 1) + 18*(x^2 - 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x +
1) - 1)/x) - 28*x + 14)/(x^2 - 2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{x + 1}}{\left (1 - x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(x + 1)/(1 - x)**(5/2), x)

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Giac [A]  time = 3.19669, size = 59, normalized size = 0.97 \begin{align*} -\frac{{\left ({\left (3 \, x - 22\right )}{\left (x + 1\right )} + 36\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((3*x - 22)*(x + 1) + 36)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 6*arcsin(1/2*sqrt(2)*sqrt(x + 1))