3.279 \(\int \frac{1}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=64 \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{4 c \sqrt{c+d x^3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.10081, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{4 c \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.1706, size = 51, normalized size = 0.8 \[ \frac{x \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},1,\frac{4}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{4 c^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

x*sqrt(c + d*x**3)*appellf1(1/3, 1/2, 1, 4/3, -d*x**3/c, -d*x**3/(4*c))/(4*c**2*
sqrt(1 + d*x**3/c))

_______________________________________________________________________________________

Mathematica [B]  time = 0.0695813, size = 165, normalized size = 2.58 \[ \frac{16 c x F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\sqrt{c+d x^3} \left (4 c+d x^3\right ) \left (16 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(16*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/(Sqrt[c + d*x^
3]*(4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(d*x^3)/(4*c)]
- 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*AppellF1
[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)])))

_______________________________________________________________________________________

Maple [C]  time = 0.008, size = 416, normalized size = 6.5 \[{\frac{-{\frac{i}{9}}\sqrt{2}}{{d}^{3}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d+4\,c \right ) }{\frac{1}{{{\it \_alpha}}^{2}}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)

[Out]

-1/9*I/d^3/c*2^(1/2)*sum(1/_alpha^2*(-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),1/6/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*_alph
a*(-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+4*c))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

integral(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

Integral(1/(sqrt(c + d*x**3)*(4*c + d*x**3)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)), x)