3.277 \(\int \frac{1}{x^2 \sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=697 \[ \frac{\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{2} \sqrt [4]{3} c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}}-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \sqrt{c+d x^3}}{4 c^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )} \]

[Out]

-Sqrt[c + d*x^3]/(4*c^2*x) + (d^(1/3)*Sqrt[c + d*x^3])/(4*c^2*((1 + Sqrt[3])*c^(
1/3) + d^(1/3)*x)) + (d^(1/3)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)
*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) - (d^(1/3)*ArcTan[Sqrt[c +
d*x^3]/(Sqrt[3]*Sqrt[c])])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) + (d^(1/3)*ArcTanh[(c^(
1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*c^(11/6)) - (d
^(1/3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(36*2^(2/3)*c^(11/6)) - (3^(1/4)*Sqrt[2
- Sqrt[3]]*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^
(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3]
)*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(8
*c^(5/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x
)^2]*Sqrt[c + d*x^3]) + (d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d
^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[
((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4
*Sqrt[3]])/(2*Sqrt[2]*3^(1/4)*c^(5/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 +
Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])

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Rubi [A]  time = 1.02005, antiderivative size = 697, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{2} \sqrt [4]{3} c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}}-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \sqrt{c+d x^3}}{4 c^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-Sqrt[c + d*x^3]/(4*c^2*x) + (d^(1/3)*Sqrt[c + d*x^3])/(4*c^2*((1 + Sqrt[3])*c^(
1/3) + d^(1/3)*x)) + (d^(1/3)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)
*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) - (d^(1/3)*ArcTan[Sqrt[c +
d*x^3]/(Sqrt[3]*Sqrt[c])])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) + (d^(1/3)*ArcTanh[(c^(
1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*c^(11/6)) - (d
^(1/3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(36*2^(2/3)*c^(11/6)) - (3^(1/4)*Sqrt[2
- Sqrt[3]]*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^
(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3]
)*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(8
*c^(5/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x
)^2]*Sqrt[c + d*x^3]) + (d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d
^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[
((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4
*Sqrt[3]])/(2*Sqrt[2]*3^(1/4)*c^(5/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 +
Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])

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Rubi in Sympy [A]  time = 63.6819, size = 706, normalized size = 1.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(1/3)*sqrt(c + d*x**3)/(4*c**2*(c**(1/3)*(1 + sqrt(3)) + d**(1/3)*x)) - sqrt(
c + d*x**3)/(4*c**2*x) - 3**(1/4)*d**(1/3)*sqrt((c**(2/3) - c**(1/3)*d**(1/3)*x
+ d**(2/3)*x**2)/(c**(1/3)*(1 + sqrt(3)) + d**(1/3)*x)**2)*sqrt(-sqrt(3) + 2)*(c
**(1/3) + d**(1/3)*x)*elliptic_e(asin((-c**(1/3)*(-1 + sqrt(3)) + d**(1/3)*x)/(c
**(1/3)*(1 + sqrt(3)) + d**(1/3)*x)), -7 - 4*sqrt(3))/(8*c**(5/3)*sqrt(c**(1/3)*
(c**(1/3) + d**(1/3)*x)/(c**(1/3)*(1 + sqrt(3)) + d**(1/3)*x)**2)*sqrt(c + d*x**
3)) + sqrt(2)*3**(3/4)*d**(1/3)*sqrt((c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*
x**2)/(c**(1/3)*(1 + sqrt(3)) + d**(1/3)*x)**2)*(c**(1/3) + d**(1/3)*x)*elliptic
_f(asin((-c**(1/3)*(-1 + sqrt(3)) + d**(1/3)*x)/(c**(1/3)*(1 + sqrt(3)) + d**(1/
3)*x)), -7 - 4*sqrt(3))/(12*c**(5/3)*sqrt(c**(1/3)*(c**(1/3) + d**(1/3)*x)/(c**(
1/3)*(1 + sqrt(3)) + d**(1/3)*x)**2)*sqrt(c + d*x**3)) - 2**(1/3)*d**(1/3)*log(1
- sqrt(c + d*x**3)/sqrt(c) - 2**(1/3)*d**(1/3)*x/c**(1/3))/(48*c**(11/6)) + 2**
(1/3)*d**(1/3)*log(1 + sqrt(c + d*x**3)/sqrt(c) - 2**(1/3)*d**(1/3)*x/c**(1/3))/
(48*c**(11/6)) + 2**(1/3)*sqrt(3)*d**(1/3)*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*(sq
rt(c) - sqrt(c + d*x**3))/(3*c**(1/6)*d**(1/3)*x))/(72*c**(11/6)) - 2**(1/3)*sqr
t(3)*d**(1/3)*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*(sqrt(c) + sqrt(c + d*x**3))/(3*
c**(1/6)*d**(1/3)*x))/(72*c**(11/6)) - 2**(1/3)*d**(1/3)*atanh(sqrt(c + d*x**3)/
sqrt(c))/(72*c**(11/6))

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Mathematica [C]  time = 0.420196, size = 348, normalized size = 0.5 \[ \frac{\frac{\frac{16 c d^2 x^6 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (32 c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}-5 \left (c+d x^3\right )}{c^2}+\frac{50 d x^3 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (20 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}}{20 x \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((50*d*x^3*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/((4*c + d*x
^3)*(20*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -(d*x^3)/(4*c)] - 3*d*x^3*(Ap
pellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*AppellF1[5/3, 3/2, 1,
8/3, -((d*x^3)/c), -(d*x^3)/(4*c)]))) + (-5*(c + d*x^3) + (16*c*d^2*x^6*AppellF
1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/((4*c + d*x^3)*(32*c*AppellF1
[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)] - 3*d*x^3*(AppellF1[8/3, 1/2, 2
, 11/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*AppellF1[8/3, 3/2, 1, 11/3, -((d*x^3)/
c), -(d*x^3)/(4*c)]))))/c^2)/(20*x*Sqrt[c + d*x^3])

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Maple [C]  time = 0.032, size = 874, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/c*(-(d*x^3+c)^(1/2)/c/x-1/3*I/c*3^(1/2)*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-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)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*
EllipticE(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),(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))+1/d*(-c*d^2)^(1/3)*EllipticF(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),(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))))+1/36*I/d^2/c^2*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),1/6/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*_alpha*(-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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d x^{5} + 4 \, c x^{2}\right )} \sqrt{d x^{3} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2),x, algorithm="fricas")

[Out]

integral(1/((d*x^5 + 4*c*x^2)*sqrt(d*x^3 + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2),x, algorithm="giac")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2), x)