3.137 \(\int \frac{x}{(3+x) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=334 \[ -\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2 \sqrt{2 \left (97+56 \sqrt{3}\right )} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{12 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

(-3*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(Sqrt[13/2]*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (2*Sqrt[2*(97 + 56*Sqrt[3])]*(1
+ x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)
/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]
*Sqrt[1 + x^3]) - (12*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*El
lipticPi[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*S
qrt[3]])/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

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Rubi [A]  time = 1.39459, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \tan ^{-1}\left (\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}}}{\sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}}}\right )}{\sqrt{26} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2 \sqrt{2 \left (97+56 \sqrt{3}\right )} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{12 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (97-56 \sqrt{3};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[x/((3 + x)*Sqrt[1 + x^3]),x]

[Out]

(-3*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[(Sqrt[13/2]*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2])/Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]])/(Sqrt[26]*Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (2*Sqrt[2*(97 + 56*Sqrt[3])]*(1
+ x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)
/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]
*Sqrt[1 + x^3]) - (12*3^(1/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*El
lipticPi[97 - 56*Sqrt[3], -ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*S
qrt[3]])/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

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Rubi in Sympy [A]  time = 89.3319, size = 379, normalized size = 1.13 \[ - \frac{3 \sqrt{26} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) \operatorname{atan}{\left (\frac{\sqrt{26} \cdot 3^{\frac{3}{4}} \sqrt{- \sqrt{3} + 2} \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{6 \sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{26 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (- \sqrt{3} + 2\right ) \sqrt{x^{3} + 1}} - \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) \Pi \left (\frac{\left (-2 + \sqrt{3}\right )^{2}}{\left (\sqrt{3} + 2\right )^{2}}; \operatorname{asin}{\left (\frac{- x - 1 + \sqrt{3}}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{- \sqrt{3} + 2} \left (\sqrt{3} + 2\right ) \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3+x)/(x**3+1)**(1/2),x)

[Out]

-3*sqrt(26)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(x + 1)*atan(sqrt(26)*3**(
3/4)*sqrt(-sqrt(3) + 2)*sqrt(-(-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 + 1)/(6
*sqrt((-x - 1 + sqrt(3))**2/(x + 1 + sqrt(3))**2 - 4*sqrt(3) + 7)))/(26*sqrt((x
+ 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1)) - 2*3**(3/4)*sqrt((x**2 - x + 1)/(x +
1 + sqrt(3))**2)*(1 + sqrt(3))*sqrt(sqrt(3) + 2)*(x + 1)*elliptic_f(asin((x - s
qrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(3*sqrt((x + 1)/(x + 1 + sqrt(3)
)**2)*(-sqrt(3) + 2)*sqrt(x**3 + 1)) - 12*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1 +
sqrt(3))**2)*(x + 1)*elliptic_pi((-2 + sqrt(3))**2/(sqrt(3) + 2)**2, asin((-x -
1 + sqrt(3))/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((x + 1)/(x + 1 + sqrt(3))
**2)*sqrt(-4*sqrt(3) + 7)*sqrt(-sqrt(3) + 2)*(sqrt(3) + 2)*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.382523, size = 194, normalized size = 0.58 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{3 i \sqrt{x^2-x+1} \Pi \left (\frac{i \sqrt{3}}{3+\sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3+\sqrt [3]{-1}}\right )}{\sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((3 + x)*Sqrt[1 + x^3]),x]

[Out]

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(
2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/
3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((3*I)*Sqrt[1 -
x + x^2]*EllipticPi[(I*Sqrt[3])/(3 + (-1)^(1/3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x
)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (-1)^(1/3))))/Sqrt[1 + x^3]

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Maple [A]  time = 0.009, size = 240, normalized size = 0.7 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }-3\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},-3/4+i/4\sqrt{3},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3+x)/(x^3+1)^(1/2),x)

[Out]

2*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(
-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x
^3+1)^(1/2)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-
3/2-1/2*I*3^(1/2)))^(1/2))-3*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/
2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3
/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1
/2),-3/4+1/4*I*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{3} + 1}{\left (x + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x^3 + 1)*(x + 3)),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(x^3 + 1)*(x + 3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{x^{3} + 1}{\left (x + 3\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x^3 + 1)*(x + 3)),x, algorithm="fricas")

[Out]

integral(x/(sqrt(x^3 + 1)*(x + 3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3+x)/(x**3+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{3} + 1}{\left (x + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x^3 + 1)*(x + 3)),x, algorithm="giac")

[Out]

integrate(x/(sqrt(x^3 + 1)*(x + 3)), x)