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

Optimal. Leaf size=476 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c+\sqrt{3} d+d\right )} \]

[Out]

-(((d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[c^2
- c*d + d^2]*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2])/(Sqrt[d]*Sqrt[c + d]*Sqrt[(1 +
x + x^2)/(1 + Sqrt[3] - x)^2])])/(Sqrt[d]*Sqrt[c + d]*Sqrt[c^2 - c*d + d^2]*Sqrt
[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])) - (2*Sqrt[2 + Sqrt[3]]*(e + f + Sq
rt[3]*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - S
qrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sq
rt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) + (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(d
*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(c + d + Sq
rt[3]*d)^2/(c + d - Sqrt[3]*d)^2, -ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)],
-7 - 4*Sqrt[3]])/((c^2 + 2*c*d - 2*d^2)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1
- x^3])

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Rubi [A]  time = 2.83603, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c+\sqrt{3} d+d\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[c^2
- c*d + d^2]*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2])/(Sqrt[d]*Sqrt[c + d]*Sqrt[(1 +
x + x^2)/(1 + Sqrt[3] - x)^2])])/(Sqrt[d]*Sqrt[c + d]*Sqrt[c^2 - c*d + d^2]*Sqrt
[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])) - (2*Sqrt[2 + Sqrt[3]]*(e + f + Sq
rt[3]*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - S
qrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sq
rt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3]) + (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*(d
*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(c + d + Sq
rt[3]*d)^2/(c + d - Sqrt[3]*d)^2, -ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)],
-7 - 4*Sqrt[3]])/((c^2 + 2*c*d - 2*d^2)*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((f*x+e)/(d*x+c)/(-x**3+1)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.22881, size = 233, normalized size = 0.49 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{3 f \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} (c f-d e) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1} d-c}\right )}{3 d \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-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))] + ((-1)^(1/3)*Sq
rt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*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)))/(3*d*Sqrt[1 - x^3])

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Maple [A]  time = 0.011, size = 265, normalized size = 0.6 \[{\frac{-{\frac{2\,i}{3}}f\sqrt{3}}{d}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}-{\frac{{\frac{2\,i}{3}} \left ( -cf+de \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}} \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f*x + e)/(sqrt(-x^3 + 1)*(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((e + f*x)/(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. \[ \int \frac{f x + e}{\sqrt{-x^{3} + 1}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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