3.496 \(\int x^3 (1+x)^{3/2} (1-x+x^2)^{3/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{935 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{2}{17} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^4+\frac{18}{187} \sqrt{x+1} \sqrt{x^2-x+1} x^4+\frac{54}{935} \sqrt{x+1} \sqrt{x^2-x+1} x \]

[Out]

(54*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/935 + (18*x^4*Sqrt[1 + x]*Sqrt[1 - x + x^2])/187 + (2*x^4*Sqrt[1 + x]*Sqr
t[1 - x + x^2]*(1 + x^3))/17 - (36*3^(3/4)*Sqrt[2 + Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2
)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(935*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.0761565, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {915, 279, 321, 218} \[ \frac{2}{17} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^4+\frac{18}{187} \sqrt{x+1} \sqrt{x^2-x+1} x^4+\frac{54}{935} \sqrt{x+1} \sqrt{x^2-x+1} x-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{935 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54*x*Sqrt[1 + x]*Sqrt[1 - x + x^2])/935 + (18*x^4*Sqrt[1 + x]*Sqrt[1 - x + x^2])/187 + (2*x^4*Sqrt[1 + x]*Sqr
t[1 - x + x^2]*(1 + x^3))/17 - (36*3^(3/4)*Sqrt[2 + Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*Sqrt[(1 - x + x^2
)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(935*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

Rule 915

Int[((g_.)*(x_))^(n_)*((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((d
+ e*x)^FracPart[p]*(a + b*x + c*x^2)^FracPart[p])/(a*d + c*e*x^3)^FracPart[p], Int[(g*x)^n*(a*d + c*e*x^3)^p,
x], x] /; FreeQ[{a, b, c, d, e, g, m, n, p}, x] && EqQ[m - p, 0] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps

\begin{align*} \int x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \, dx &=\frac{\left (\sqrt{1+x} \sqrt{1-x+x^2}\right ) \int x^3 \left (1+x^3\right )^{3/2} \, dx}{\sqrt{1+x^3}}\\ &=\frac{2}{17} x^4 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )+\frac{\left (9 \sqrt{1+x} \sqrt{1-x+x^2}\right ) \int x^3 \sqrt{1+x^3} \, dx}{17 \sqrt{1+x^3}}\\ &=\frac{18}{187} x^4 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{2}{17} x^4 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )+\frac{\left (27 \sqrt{1+x} \sqrt{1-x+x^2}\right ) \int \frac{x^3}{\sqrt{1+x^3}} \, dx}{187 \sqrt{1+x^3}}\\ &=\frac{54}{935} x \sqrt{1+x} \sqrt{1-x+x^2}+\frac{18}{187} x^4 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{2}{17} x^4 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )-\frac{\left (54 \sqrt{1+x} \sqrt{1-x+x^2}\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx}{935 \sqrt{1+x^3}}\\ &=\frac{54}{935} x \sqrt{1+x} \sqrt{1-x+x^2}+\frac{18}{187} x^4 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{2}{17} x^4 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} (1+x)^{3/2} \sqrt{1-x+x^2} \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{935 \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \left (1+x^3\right )}\\ \end{align*}

Mathematica [C]  time = 0.890009, size = 235, normalized size = 1.17 \[ \frac{2 \left (x \sqrt{x+1} \left (55 x^8-55 x^7+55 x^6+100 x^5-100 x^4+100 x^3+27 x^2-27 x+27\right )-\frac{9 i \sqrt{6} (x+1) \sqrt{\frac{\left (\sqrt{3}-3 i\right ) x+\sqrt{3}+3 i}{\left (\sqrt{3}-3 i\right ) (x+1)}} \sqrt{\frac{\left (\sqrt{3}+3 i\right ) x+\sqrt{3}-3 i}{\left (\sqrt{3}+3 i\right ) (x+1)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{\sqrt{3}+3 i}}}{\sqrt{x+1}}\right ),\frac{\sqrt{3}+3 i}{-\sqrt{3}+3 i}\right )}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}\right )}{935 \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*Sqrt[1 + x]*(27 - 27*x + 27*x^2 + 100*x^3 - 100*x^4 + 100*x^5 + 55*x^6 - 55*x^7 + 55*x^8) - ((9*I)*Sqrt[
6]*(1 + x)*Sqrt[(3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[(-3*I + Sqrt[3] + (3*I +
Sqrt[3])*x)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I +
Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])]))/(935*Sqrt[1 - x + x^2])

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Maple [A]  time = 1.13, size = 262, normalized size = 1.3 \begin{align*}{\frac{2}{935\,{x}^{3}+935}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 55\,{x}^{10}+155\,{x}^{7}+27\,i\sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{2\,x-1+i\sqrt{3}}{i\sqrt{3}-3}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{i\sqrt{3}+3}}} \right ) \sqrt{3}-81\,\sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{2\,x-1+i\sqrt{3}}{i\sqrt{3}-3}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{i\sqrt{3}+3}}} \right ) +127\,{x}^{4}+27\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(1+x)^(3/2)*(x^2-x+1)^(3/2),x)

[Out]

2/935*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(55*x^10+155*x^7+27*I*(-2*(1+x)/(I*3^(1/2)-3))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3
^(1/2)+3))^(1/2)*((2*x-1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2)*EllipticF((-2*(1+x)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)
-3)/(I*3^(1/2)+3))^(1/2))*3^(1/2)-81*(-2*(1+x)/(I*3^(1/2)-3))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((
2*x-1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2)*EllipticF((-2*(1+x)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3))^
(1/2))+127*x^4+27*x)/(x^3+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{6} + x^{3}\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((x^6 + x^3)*sqrt(x^2 - x + 1)*sqrt(x + 1), x)

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Sympy [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)**(3/2)*(x**2-x+1)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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