3.828 \(\int \frac{x^3 \sqrt{1+x}}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 \sqrt{x+1} x^3}{3 (1-x)^{3/2}}-\frac{13 \sqrt{x+1} x^2}{3 \sqrt{1-x}}-\frac{1}{6} \sqrt{1-x} \sqrt{x+1} (33 x+52)+\frac{11}{2} \sin ^{-1}(x) \]

[Out]

(-13*x^2*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*x^3*Sqrt[1 + x])/(3*(1 - x)^(3/2)) - (Sqrt[1 - x]*Sqrt[1 + x]*(52 +
33*x))/6 + (11*ArcSin[x])/2

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Rubi [A]  time = 0.0149358, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 150, 147, 41, 216} \[ \frac{2 \sqrt{x+1} x^3}{3 (1-x)^{3/2}}-\frac{13 \sqrt{x+1} x^2}{3 \sqrt{1-x}}-\frac{1}{6} \sqrt{1-x} \sqrt{x+1} (33 x+52)+\frac{11}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*x^2*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*x^3*Sqrt[1 + x])/(3*(1 - x)^(3/2)) - (Sqrt[1 - x]*Sqrt[1 + x]*(52 +
33*x))/6 + (11*ArcSin[x])/2

Rule 97

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

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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^3 \sqrt{1+x}}{(1-x)^{5/2}} \, dx &=\frac{2 x^3 \sqrt{1+x}}{3 (1-x)^{3/2}}-\frac{2}{3} \int \frac{x^2 \left (3+\frac{7 x}{2}\right )}{(1-x)^{3/2} \sqrt{1+x}} \, dx\\ &=-\frac{13 x^2 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 x^3 \sqrt{1+x}}{3 (1-x)^{3/2}}-\frac{2}{3} \int \frac{\left (-13-\frac{33 x}{2}\right ) x}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{13 x^2 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 x^3 \sqrt{1+x}}{3 (1-x)^{3/2}}-\frac{1}{6} \sqrt{1-x} \sqrt{1+x} (52+33 x)+\frac{11}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{13 x^2 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 x^3 \sqrt{1+x}}{3 (1-x)^{3/2}}-\frac{1}{6} \sqrt{1-x} \sqrt{1+x} (52+33 x)+\frac{11}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{13 x^2 \sqrt{1+x}}{3 \sqrt{1-x}}+\frac{2 x^3 \sqrt{1+x}}{3 (1-x)^{3/2}}-\frac{1}{6} \sqrt{1-x} \sqrt{1+x} (52+33 x)+\frac{11}{2} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.034439, size = 54, normalized size = 0.69 \[ -\frac{\sqrt{x+1} \left (3 x^3+12 x^2-71 x+52\right )}{6 (1-x)^{3/2}}-11 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + x]*(52 - 71*x + 12*x^2 + 3*x^3))/(6*(1 - x)^(3/2)) - 11*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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Maple [A]  time = 0.013, size = 97, normalized size = 1.2 \begin{align*}{\frac{1}{6\, \left ( -1+x \right ) ^{2}} \left ( -3\,{x}^{3}\sqrt{-{x}^{2}+1}+33\,\arcsin \left ( x \right ){x}^{2}-12\,{x}^{2}\sqrt{-{x}^{2}+1}-66\,\arcsin \left ( x \right ) x+71\,x\sqrt{-{x}^{2}+1}+33\,\arcsin \left ( x \right ) -52\,\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^3*(1+x)^(1/2)/(1-x)^(5/2),x)

[Out]

1/6*(-3*x^3*(-x^2+1)^(1/2)+33*arcsin(x)*x^2-12*x^2*(-x^2+1)^(1/2)-66*arcsin(x)*x+71*x*(-x^2+1)^(1/2)+33*arcsin
(x)-52*(-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^3*(1+x)^(1/2)/(1-x)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.05301, size = 219, normalized size = 2.81 \begin{align*} -\frac{52 \, x^{2} +{\left (3 \, x^{3} + 12 \, x^{2} - 71 \, x + 52\right )} \sqrt{x + 1} \sqrt{-x + 1} + 66 \,{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 104 \, x + 52}{6 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(1+x)^(1/2)/(1-x)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(52*x^2 + (3*x^3 + 12*x^2 - 71*x + 52)*sqrt(x + 1)*sqrt(-x + 1) + 66*(x^2 - 2*x + 1)*arctan((sqrt(x + 1)*
sqrt(-x + 1) - 1)/x) - 104*x + 52)/(x^2 - 2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \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**3*(1+x)**(1/2)/(1-x)**(5/2),x)

[Out]

Integral(x**3*sqrt(x + 1)/(1 - x)**(5/2), x)

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Giac [A]  time = 1.2245, size = 66, normalized size = 0.85 \begin{align*} -\frac{{\left ({\left (3 \,{\left (x + 2\right )}{\left (x + 1\right )} - 86\right )}{\left (x + 1\right )} + 132\right )} \sqrt{x + 1} \sqrt{-x + 1}}{6 \,{\left (x - 1\right )}^{2}} + 11 \, \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^3*(1+x)^(1/2)/(1-x)^(5/2),x, algorithm="giac")

[Out]

-1/6*((3*(x + 2)*(x + 1) - 86)*(x + 1) + 132)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 11*arcsin(1/2*sqrt(2)*sqrt(
x + 1))