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

Optimal. Leaf size=141 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{6 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18 c^{5/6} d^{2/3}} \]

[Out]

-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]]/(6*Sqrt[3]*c^(5
/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])]/(18*
c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(18*c^(5/6)*d^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.755241, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{6 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18 c^{5/6} d^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]]/(6*Sqrt[3]*c^(5
/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])]/(18*
c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(18*c^(5/6)*d^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.1782, size = 51, normalized size = 0.36 \[ \frac{x^{2} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{d x^{3}}{c},\frac{d x^{3}}{8 c} \right )}}{16 c^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(c + d*x**3)*appellf1(2/3, 1/2, 1, 5/3, -d*x**3/c, d*x**3/(8*c))/(16*c*
*2*sqrt(1 + d*x**3/c))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0803912, size = 168, normalized size = 1.19 \[ \frac{20 c x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3} \left (3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+40 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(20*c*x^2*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3
)*Sqrt[c + d*x^3]*(40*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)]
+ 3*d*x^3*(AppellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[
5/3, 3/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])))

_______________________________________________________________________________________

Maple [C]  time = 0.009, size = 416, normalized size = 3. \[{\frac{-{\frac{i}{27}}\sqrt{2}}{{d}^{3}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*I/d^3/c*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*
(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/
(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2
)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d
^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3
)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2
*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha
^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alp
ha*(-c*d^2)^(2/3)-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/
2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.679283, size = 3586, normalized size = 25.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(1/(c^5*d^4))^(1/6)*arctan(3*(3*sqrt(3)*(5*c^4*d^5*x^7 + 64*c^5*d^4
*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4))^(2/3) - 3*sqrt(3)*(c^2*d^4*x^8 + 38*c^3*d^3*x
^5 + 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3) + sqrt(d*x^3 + c)*(6*sqrt(3)*(5*c^5*d^5
*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) - sqrt(3)*(c*d^3*x^7 + 80*c^2*d^2*x^4
+ 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)))/(d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3
+ 640*c^3 - 9*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4))^(2/3
) + 3*sqrt(d*x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) -
2*(7*c^3*d^4*x^6 + 152*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*x^7
+ 80*c^2*d^2*x^4 + 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + (d^3*x^9 - 24*c*d^2*x^6 +
192*c^2*d*x^3 - 512*c^3)*sqrt((d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1088*
c^3 - 18*(c^4*d^5*x^7 - 52*c^5*d^4*x^4 - 80*c^6*d^3*x)*(1/(c^5*d^4))^(2/3) - 6*s
qrt(d*x^3 + c)*(24*(c^5*d^5*x^5 + c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) - 4*(c^3*d^4*
x^6 + 41*c^4*d^3*x^3 + 40*c^5*d^2)*sqrt(1/(c^5*d^4)) - (c*d^3*x^7 - 28*c^2*d^2*x
^4 - 272*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^4*x^8 + 20*c^3*d^3*x^5 - 8*c^
4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^
3)) - 9*(c^2*d^4*x^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))) +
1/54*sqrt(3)*(1/(c^5*d^4))^(1/6)*arctan(-3*(3*sqrt(3)*(5*c^4*d^5*x^7 + 64*c^5*d^
4*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4))^(2/3) - 3*sqrt(3)*(c^2*d^4*x^8 + 38*c^3*d^3*
x^5 + 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3) - sqrt(d*x^3 + c)*(6*sqrt(3)*(5*c^5*d^
5*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) - sqrt(3)*(c*d^3*x^7 + 80*c^2*d^2*x^
4 + 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)))/(d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3
+ 640*c^3 - 9*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4))^(2/
3) - 3*sqrt(d*x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) -
2*(7*c^3*d^4*x^6 + 152*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*x^7
+ 80*c^2*d^2*x^4 + 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + (d^3*x^9 - 24*c*d^2*x^6
+ 192*c^2*d*x^3 - 512*c^3)*sqrt((d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1088
*c^3 - 18*(c^4*d^5*x^7 - 52*c^5*d^4*x^4 - 80*c^6*d^3*x)*(1/(c^5*d^4))^(2/3) + 6*
sqrt(d*x^3 + c)*(24*(c^5*d^5*x^5 + c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) - 4*(c^3*d^4
*x^6 + 41*c^4*d^3*x^3 + 40*c^5*d^2)*sqrt(1/(c^5*d^4)) - (c*d^3*x^7 - 28*c^2*d^2*
x^4 - 272*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^4*x^8 + 20*c^3*d^3*x^5 - 8*c
^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c
^3)) - 9*(c^2*d^4*x^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))) +
1/108*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3 + 640*c
^3 + 18*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4))^(2/3) + 6*
sqrt(d*x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5/6) + (7*c^3
*d^4*x^6 + 152*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*x^7 + 80*c^2
*d^2*x^4 + 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^4*x^8 + 38*c^3*d^3*x^5
+ 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 -
512*c^3)) - 1/108*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d
*x^3 + 640*c^3 + 18*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x)*(1/(c^5*d^4)
)^(2/3) - 6*sqrt(d*x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2)*(1/(c^5*d^4))^(5
/6) + (7*c^3*d^4*x^6 + 152*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*
x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^4*x^8 + 38*
c^3*d^3*x^5 + 64*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192
*c^2*d*x^3 - 512*c^3)) - 1/216*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 - 276*c*d^2*x^6
- 1608*c^2*d*x^3 - 1088*c^3 - 18*(c^4*d^5*x^7 - 52*c^5*d^4*x^4 - 80*c^6*d^3*x)*(
1/(c^5*d^4))^(2/3) + 6*sqrt(d*x^3 + c)*(24*(c^5*d^5*x^5 + c^6*d^4*x^2)*(1/(c^5*d
^4))^(5/6) - 4*(c^3*d^4*x^6 + 41*c^4*d^3*x^3 + 40*c^5*d^2)*sqrt(1/(c^5*d^4)) - (
c*d^3*x^7 - 28*c^2*d^2*x^4 - 272*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^4*x^8
+ 20*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6
+ 192*c^2*d*x^3 - 512*c^3)) + 1/216*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 - 276*c*d^2
*x^6 - 1608*c^2*d*x^3 - 1088*c^3 - 18*(c^4*d^5*x^7 - 52*c^5*d^4*x^4 - 80*c^6*d^3
*x)*(1/(c^5*d^4))^(2/3) - 6*sqrt(d*x^3 + c)*(24*(c^5*d^5*x^5 + c^6*d^4*x^2)*(1/(
c^5*d^4))^(5/6) - 4*(c^3*d^4*x^6 + 41*c^4*d^3*x^3 + 40*c^5*d^2)*sqrt(1/(c^5*d^4)
) - (c*d^3*x^7 - 28*c^2*d^2*x^4 - 272*c^3*d*x)*(1/(c^5*d^4))^(1/6)) + 18*(c^2*d^
4*x^8 + 20*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24*c*d^2
*x^6 + 192*c^2*d*x^3 - 512*c^3))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{- 8 c \sqrt{c + d x^{3}} + d x^{3} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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