3.125 \(\int \frac{1-\sqrt{3}+x}{(c+d x) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=360 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{x^3+1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{x^3+1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh
[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]
)/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + x)^2/(1 - Sqrt[3] + x
)^2])])/(Sqrt[c - d]*Sqrt[d]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] +
x)^2)]*Sqrt[1 + x^3])) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2
)/(1 - Sqrt[3] + x)^2]*EllipticPi[(c - (1 - Sqrt[3])*d)^2/(c - (1 + Sqrt[3])*d)^
2, -ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/((c - d - Sqrt
[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[1 + x^3])

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Rubi [A]  time = 2.19756, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{x^3+1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{x^3+1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh
[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]
)/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + x)^2/(1 - Sqrt[3] + x
)^2])])/(Sqrt[c - d]*Sqrt[d]*Sqrt[c^2 + c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] +
x)^2)]*Sqrt[1 + x^3])) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2
)/(1 - Sqrt[3] + x)^2]*EllipticPi[(c - (1 - Sqrt[3])*d)^2/(c - (1 + Sqrt[3])*d)^
2, -ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/((c - d - Sqrt
[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[1 + x^3])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.98588, size = 213, normalized size = 0.59 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left (c+\left (\sqrt{3}-1\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 - Sqrt[3] + x)/((c + d*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))]) + (I*(c + (-1 + S
qrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin
[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(
d*Sqrt[1 + x^3])

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Maple [A]  time = 0.031, size = 275, normalized size = 0.8 \[ -2\,{\frac{ \left ( d\sqrt{3}+c-d \right ) \left ( 3/2-i/2\sqrt{3} \right ) }{{d}^{2}\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},{(-3/2+i/2\sqrt{3}) \left ( -1+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) \left ( -1+{\frac{c}{d}} \right ) ^{-1}}+2\,{\frac{3/2-i/2\sqrt{3}}{d\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x-3^(1/2))/(d*x+c)/(x^3+1)^(1/2),x)

[Out]

-2*(d*3^(1/2)+c-d)/d^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)/(-1+c/d)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))
^(1/2),(-3/2+1/2*I*3^(1/2))/(-1+c/d),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))
^(1/2))+2/d*(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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