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

Optimal. Leaf size=325 \[ \frac{18 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}+\frac{2}{13} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^2+\frac{18}{91} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{54 \sqrt{x+1} \sqrt{x^2-x+1}}{91 \left (x+\sqrt{3}+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

[Out]

(18*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/91 + (54*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(91*(1 + Sqrt[3] + x)) + (2*x^2
*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/13 - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*S
qrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])
/(91*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (18*Sqrt[2]*3^(3/4)*(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]])/(91*
Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

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Rubi [A]  time = 0.106791, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {809, 279, 303, 218, 1877} \[ \frac{2}{13} \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right ) x^2+\frac{18}{91} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{54 \sqrt{x+1} \sqrt{x^2-x+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{18 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (x^3+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/91 + (54*Sqrt[1 + x]*Sqrt[1 - x + x^2])/(91*(1 + Sqrt[3] + x)) + (2*x^2
*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3))/13 - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)^(3/2)*Sqrt[1 - x + x^2]*S
qrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])
/(91*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3)) + (18*Sqrt[2]*3^(3/4)*(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]])/(91*
Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*(1 + x^3))

Rule 809

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((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[(f + g*x)*(a*d + c*e*x^
3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[m, p] && 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 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
 b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

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]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int x (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 \left (1+x^3\right )^{3/2} \, dx}{\sqrt{1+x^3}}\\ &=\frac{2}{13} x^2 \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 \sqrt{1+x^3} \, dx}{13 \sqrt{1+x^3}}\\ &=\frac{18}{91} x^2 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{2}{13} x^2 \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}{\sqrt{1+x^3}} \, dx}{91 \sqrt{1+x^3}}\\ &=\frac{18}{91} x^2 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{2}{13} x^2 \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{1-\sqrt{3}+x}{\sqrt{1+x^3}} \, dx}{91 \sqrt{1+x^3}}+\frac{\left (27 \sqrt{2 \left (2-\sqrt{3}\right )} \sqrt{1+x} \sqrt{1-x+x^2}\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx}{91 \sqrt{1+x^3}}\\ &=\frac{18}{91} x^2 \sqrt{1+x} \sqrt{1-x+x^2}+\frac{54 \sqrt{1+x} \sqrt{1-x+x^2}}{91 \left (1+\sqrt{3}+x\right )}+\frac{2}{13} x^2 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )-\frac{27 \sqrt [4]{3} \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}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \left (1+x^3\right )}+\frac{18 \sqrt{2} 3^{3/4} (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 )}{91 \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \left (1+x^3\right )}\\ \end{align*}

Mathematica [C]  time = 0.488259, size = 244, normalized size = 0.75 \[ \frac{\sqrt{x+1} \left (4 x^2 \left (x^2-x+1\right ) \left (7 x^3+16\right )-\frac{27 \sqrt{2} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \left (\left (\sqrt{3}-3 i\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )-\left (\sqrt{3}-i\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}\right ),\frac{\sqrt{3}+3 i}{-\sqrt{3}+3 i}\right )\right )}{\sqrt{-\frac{i (x+1)}{-2 i x+\sqrt{3}+i}}}\right )}{182 \sqrt{x^2-x+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + x]*(4*x^2*(1 - x + x^2)*(16 + 7*x^3) - (27*Sqrt[2]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*(
(-3*I + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqr
t[3])] - (-I + Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*
I - Sqrt[3])]))/Sqrt[((-I)*(1 + x))/(I + Sqrt[3] - (2*I)*x)]))/(182*Sqrt[1 - x + x^2])

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Maple [A]  time = 0.577, size = 366, normalized size = 1.1 \begin{align*}{\frac{1}{91\,{x}^{3}+91}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( 14\,{x}^{8}+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}+46\,{x}^{5}+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 ) -162\,\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 EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{i\sqrt{3}+3}}} \right ) +32\,{x}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/91*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(14*x^8+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)+46*x^5+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))-162*(-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)*EllipticE((-2*(1+x)/(I*3^(1/2)-3))^(1/2),(-(I*3^(1/2)-3)/(I*3^(1/2)+3))^(1/2))+32*x^2)/(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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{4} + x\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*(1+x)^(3/2)*(x^2-x+1)^(3/2),x, algorithm="fricas")

[Out]

integral((x^4 + x)*sqrt(x^2 - x + 1)*sqrt(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)