3.77 \(\int \frac{x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx\)

Optimal. Leaf size=202 \[ \frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 \sqrt{c} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}} \]

[Out]

(2*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(9*Sqrt[c]*d^2) - (
Sqrt[2 + Sqrt[3]]*(c - 2*d*x)*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + Sqrt[3])*c
- 2*d*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c - 2*d*x)/((1 + Sqrt[3])*c - 2*d*x)
], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d^2*Sqrt[(c*(c - 2*d*x))/((1 + Sqrt[3])*c - 2*d*x
)^2]*Sqrt[c^3 - 8*d^3*x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.517249, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 \sqrt{c} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]

[Out]

(2*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(9*Sqrt[c]*d^2) - (
Sqrt[2 + Sqrt[3]]*(c - 2*d*x)*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + Sqrt[3])*c
- 2*d*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c - 2*d*x)/((1 + Sqrt[3])*c - 2*d*x)
], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d^2*Sqrt[(c*(c - 2*d*x))/((1 + Sqrt[3])*c - 2*d*x
)^2]*Sqrt[c^3 - 8*d^3*x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 132.163, size = 561, normalized size = 2.78 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{2} + 2 c d x + 4 d^{2} x^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (c - 2 d x\right ) F\left (\operatorname{asin}{\left (- \frac{- c \left (-1 + \sqrt{3}\right ) - 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} + \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{3 + 2 \sqrt{3}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \operatorname{atanh}{\left (\frac{\sqrt{- \frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} + 1}}{\sqrt{3 + 2 \sqrt{3}} \sqrt{\frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{9 d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{c^{3} - 8 d^{3} x^{3}}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{c \left (-1 + \sqrt{3}\right ) + 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)

[Out]

3**(3/4)*sqrt((c**2 + 2*c*d*x + 4*d**2*x**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*(1 +
sqrt(3))*sqrt(sqrt(3) + 2)*(c - 2*d*x)*elliptic_f(asin(-(-c*(-1 + sqrt(3)) - 2*d
*x)/(c*(1 + sqrt(3)) - 2*d*x)), -7 - 4*sqrt(3))/(3*d**2*sqrt(c*(c - 2*d*x)/(c*(1
+ sqrt(3)) - 2*d*x)**2)*(sqrt(3) + 3)*sqrt(c**3 - 8*d**3*x**3)) + 3**(3/4)*sqrt
(c**2*(1 + 2*d*x/c + 4*d**2*x**2/c**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(3 + 2*
sqrt(3))*sqrt(-sqrt(3) + 2)*(c - 2*d*x)*atanh(sqrt(-(c*(-1 + sqrt(3)) + 2*d*x)**
2/(c*(1 + sqrt(3)) - 2*d*x)**2 + 1)/(sqrt(3 + 2*sqrt(3))*sqrt((c*(-1 + sqrt(3))
+ 2*d*x)**2/(c*(1 + sqrt(3)) - 2*d*x)**2 - 4*sqrt(3) + 7)))/(9*d**2*sqrt(c*(c -
2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(c**3 - 8*d**3*x**3)) - 4*3**(1/4)*sqrt
(c**2*(1 + 2*d*x/c + 4*d**2*x**2/c**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(-sqrt(
3) + 2)*(c - 2*d*x)*elliptic_pi(4*sqrt(3) + 7, asin((c*(-1 + sqrt(3)) + 2*d*x)/(
c*(1 + sqrt(3)) - 2*d*x)), -7 - 4*sqrt(3))/(d**2*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt
(3)) - 2*d*x)**2)*sqrt(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(c**3 -
8*d**3*x**3))

_______________________________________________________________________________________

Mathematica [C]  time = 1.1701, size = 295, normalized size = 1.46 \[ \frac{\sqrt{\frac{c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\left (\sqrt [3]{-1}-2\right ) \left (\sqrt [3]{-1} c+2 d x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (c+2 \sqrt [3]{-1} d x\right )}{\left (1+\sqrt [3]{-1}\right ) c}} F\left (\sin ^{-1}\left (\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )+\frac{2 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) c \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{c^2}} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}\right )}{\left (\sqrt [3]{-1}-2\right ) d^2 \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3-8 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]

[Out]

(Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*((-2 + (-1)^(1/3))*((-1)^(1/3)*c + 2*d*x
)*Sqrt[((-1)^(1/3)*(c + 2*(-1)^(1/3)*d*x))/((1 + (-1)^(1/3))*c)]*EllipticF[ArcSi
n[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)] + (2*(-1)^(1/3
)*(1 + (-1)^(1/3))*c*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[(c^2
+ 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt
[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-
1)^(1/3))*d^2*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3
*x^3])

_______________________________________________________________________________________

Maple [B]  time = 0.011, size = 653, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x)

[Out]

2/d*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1/2*(-1/2-1/
2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(
1/2)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/
2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(
1/2)/(-8*d^3*x^3+c^3)^(1/2)*EllipticF(((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1
/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2),((1/2*(-1/2-1/2*I*3^(
1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^
(1/2))-2*c/d^2*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1
/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/
2))*c/d))^(1/2)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2)*((x-1
/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/
2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d)*Ellipt
icPi(((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1
/2*I*3^(1/2))*c/d))^(1/2),(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))
*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d),((1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/
2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="fricas")

[Out]

integral(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- \left (- c + 2 d x\right ) \left (c^{2} + 2 c d x + 4 d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)

[Out]

Integral(x/(sqrt(-(-c + 2*d*x)*(c**2 + 2*c*d*x + 4*d**2*x**2))*(c + d*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="giac")

[Out]

integrate(x/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)