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

Optimal. Leaf size=323 \[ -\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\left (1-\sqrt{3}\right ) d\right )} \]

[Out]

-(((c - (1 + Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTan[
(Sqrt[c^2 + c*d + d^2]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[c - d]*Sqrt[d]*S
qrt[(1 - x + 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 + Sq
rt[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 - (1 - Sqrt[3])*d)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[
-1 - x^3])

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Rubi [A]  time = 2.17888, antiderivative size = 323, 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 \[ -\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\left (1-\sqrt{3}\right ) d\right )} \]

Antiderivative was successfully verified.

[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]*ArcTan[
(Sqrt[c^2 + c*d + d^2]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[c - d]*Sqrt[d]*S
qrt[(1 - x + 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 + Sq
rt[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 - (1 - 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 = 1.02937, size = 216, normalized size = 0.67 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left (c-\left (1+\sqrt{3}\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{-x^3-1}} \]

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 + Sq
rt[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 = 0.06, size = 266, normalized size = 0.8 \[{\frac{-{\frac{2\,i}{3}}\sqrt{3}}{d}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{{\frac{2\,i}{3}} \left ( d\sqrt{3}-c+d \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}} \]

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]

-2/3*I/d*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/
2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3
*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))
^(1/2))-2/3*I*(d*3^(1/2)-c+d)/d^2*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2
)*((1+x)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x
^3-1)^(1/2)/(1/2+1/2*I*3^(1/2)+c/d)*EllipticPi(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/
2))*3^(1/2))^(1/2),I*3^(1/2)/(1/2+1/2*I*3^(1/2)+c/d),(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. \[ \int \frac{x + 1 + \sqrt{3}}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \]

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 + 1 + sqrt(3))/(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)