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

Optimal. Leaf size=246 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{2 d \sqrt{x^3+1}}{x+\sqrt{3}+1}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.208461, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{2 d \sqrt{x^3+1}}{x+\sqrt{3}+1}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/Sqrt[1 + x^3],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.4651, size = 221, normalized size = 0.9 \[ \frac{2 d \sqrt{x^{3} + 1}}{x + 1 + \sqrt{3}} - \frac{\sqrt [4]{3} d \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \left (c - d + \sqrt{3} d\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*sqrt(x**3 + 1)/(x + 1 + sqrt(3)) - 3**(1/4)*d*sqrt((x**2 - x + 1)/(x + 1 + s
qrt(3))**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_e(asin((x - sqrt(3) + 1)/(x + 1
+ sqrt(3))), -7 - 4*sqrt(3))/(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)*sqrt(sqrt(3) + 2)*(x + 1
)*(c - d + sqrt(3)*d)*elliptic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 -
4*sqrt(3))/(3*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.176978, size = 136, normalized size = 0.55 \[ -\frac{2 \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} \left (\sqrt [6]{-1} \sqrt{3} \left ((-1)^{2/3} d-c\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 d E\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*Sqrt[-((-1)^(1/6)*((-1)^(2/3) + x))]*Sqrt[1 + (-1)^(1/3)*x + (-1)^(2/3)*x^2]
*(3*d*EllipticE[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x))]/3^(1/4)], (-1)^(1/3)] + (-1)^
(1/6)*Sqrt[3]*(-c + (-1)^(2/3)*d)*EllipticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x))]/3
^(1/4)], (-1)^(1/3)]))/(3^(3/4)*Sqrt[1 + x^3])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 291, normalized size = 1.2 \[ 2\,{\frac{c \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{d \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}} \left ( \left ( -3/2-i/2\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) + \left ( 1/2+i/2\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(x^3+1)^(1/2),x)

[Out]

2*c*(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))+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)*((-3/2-1/2*I*3^(1/2))*EllipticE(((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))+
(1/2+1/2*I*3^(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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)/sqrt(x^3 + 1), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)/sqrt(x^3 + 1), x)

_______________________________________________________________________________________

Sympy [A]  time = 1.73082, size = 61, normalized size = 0.25 \[ \frac{c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4/3)) +
d*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), x**3*exp_polar(I*pi))/(3*gamma(5/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)/sqrt(x^3 + 1), x)