3.142 \(\int \frac{g+h x}{\sqrt [3]{-\frac{c g^2}{h^2}+9 c x^2} \left (g^2+3 h^2 x^2\right )} \, dx\)

Optimal. Leaf size=242 \[ \frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}-\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (\left (1-\frac{3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h x}{g}+1}\right )}{2\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}+\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h x}{g}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h x}{g}+1}}\right )}{2^{2/3} \sqrt{3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}} \]

[Out]

((1 - (9*h^2*x^2)/g^2)^(1/3)*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/
(Sqrt[3]*(1 + (3*h*x)/g)^(1/3))])/(2^(2/3)*Sqrt[3]*h*(-((c*g^2)/h^2) + 9*c*x^2)^
(1/3)) + ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[g^2 + 3*h^2*x^2])/(6*2^(2/3)*h*(-((c*g
^2)/h^2) + 9*c*x^2)^(1/3)) - ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[(1 - (3*h*x)/g)^(2
/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1
/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.344257, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}-\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (\left (1-\frac{3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h x}{g}+1}\right )}{2\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}+\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h x}{g}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h x}{g}+1}}\right )}{2^{2/3} \sqrt{3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(g + h*x)/((-((c*g^2)/h^2) + 9*c*x^2)^(1/3)*(g^2 + 3*h^2*x^2)),x]

[Out]

((1 - (9*h^2*x^2)/g^2)^(1/3)*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/
(Sqrt[3]*(1 + (3*h*x)/g)^(1/3))])/(2^(2/3)*Sqrt[3]*h*(-((c*g^2)/h^2) + 9*c*x^2)^
(1/3)) + ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[g^2 + 3*h^2*x^2])/(6*2^(2/3)*h*(-((c*g
^2)/h^2) + 9*c*x^2)^(1/3)) - ((1 - (9*h^2*x^2)/g^2)^(1/3)*Log[(1 - (3*h*x)/g)^(2
/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1
/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.7888, size = 224, normalized size = 0.93 \[ \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{9 h^{2} x^{2}}{g^{2}}} \log{\left (g^{2} + 3 h^{2} x^{2} \right )}}{12 h \sqrt [3]{- \frac{c g^{2}}{h^{2}} + 9 c x^{2}}} - \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{9 h^{2} x^{2}}{g^{2}}} \log{\left (\left (1 - \frac{3 h x}{g}\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{1 + \frac{3 h x}{g}} \right )}}{4 h \sqrt [3]{- \frac{c g^{2}}{h^{2}} + 9 c x^{2}}} - \frac{\sqrt [3]{2} \sqrt{3} \sqrt [3]{1 - \frac{9 h^{2} x^{2}}{g^{2}}} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (1 - \frac{3 h x}{g}\right )^{\frac{2}{3}}}{3 \sqrt [3]{1 + \frac{3 h x}{g}}} - \frac{\sqrt{3}}{3} \right )}}{6 h \sqrt [3]{- \frac{c g^{2}}{h^{2}} + 9 c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x+g)/(-c*g**2/h**2+9*c*x**2)**(1/3)/(3*h**2*x**2+g**2),x)

[Out]

2**(1/3)*(1 - 9*h**2*x**2/g**2)**(1/3)*log(g**2 + 3*h**2*x**2)/(12*h*(-c*g**2/h*
*2 + 9*c*x**2)**(1/3)) - 2**(1/3)*(1 - 9*h**2*x**2/g**2)**(1/3)*log((1 - 3*h*x/g
)**(2/3) + 2**(1/3)*(1 + 3*h*x/g)**(1/3))/(4*h*(-c*g**2/h**2 + 9*c*x**2)**(1/3))
- 2**(1/3)*sqrt(3)*(1 - 9*h**2*x**2/g**2)**(1/3)*atan(2**(2/3)*sqrt(3)*(1 - 3*h
*x/g)**(2/3)/(3*(1 + 3*h*x/g)**(1/3)) - sqrt(3)/3)/(6*h*(-c*g**2/h**2 + 9*c*x**2
)**(1/3))

_______________________________________________________________________________________

Mathematica [C]  time = 1.0381, size = 331, normalized size = 1.37 \[ \frac{g^2 x \left (\frac{g F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )}{g^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )+2 h^2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )\right )}-\frac{h x F_1\left (1;\frac{1}{3},1;2;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )}{3 h^2 x^2 \left (F_1\left (2;\frac{1}{3},2;3;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )-F_1\left (2;\frac{4}{3},1;3;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )\right )-2 g^2 F_1\left (1;\frac{1}{3},1;2;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )}\right )}{\left (g^2+3 h^2 x^2\right ) \sqrt [3]{c \left (9 x^2-\frac{g^2}{h^2}\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(g + h*x)/((-((c*g^2)/h^2) + 9*c*x^2)^(1/3)*(g^2 + 3*h^2*x^2)),x]

[Out]

(g^2*x*((g*AppellF1[1/2, 1/3, 1, 3/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2])/(g^2*A
ppellF1[1/2, 1/3, 1, 3/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2] + 2*h^2*x^2*(-Appel
lF1[3/2, 1/3, 2, 5/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2] + AppellF1[3/2, 4/3, 1,
5/2, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2])) - (h*x*AppellF1[1, 1/3, 1, 2, (9*h^2*
x^2)/g^2, (-3*h^2*x^2)/g^2])/(-2*g^2*AppellF1[1, 1/3, 1, 2, (9*h^2*x^2)/g^2, (-3
*h^2*x^2)/g^2] + 3*h^2*x^2*(AppellF1[2, 1/3, 2, 3, (9*h^2*x^2)/g^2, (-3*h^2*x^2)
/g^2] - AppellF1[2, 4/3, 1, 3, (9*h^2*x^2)/g^2, (-3*h^2*x^2)/g^2]))))/((c*(-(g^2
/h^2) + 9*x^2))^(1/3)*(g^2 + 3*h^2*x^2))

_______________________________________________________________________________________

Maple [F]  time = 0.314, size = 0, normalized size = 0. \[ \int{\frac{hx+g}{3\,{h}^{2}{x}^{2}+{g}^{2}}{\frac{1}{\sqrt [3]{-{\frac{c{g}^{2}}{{h}^{2}}}+9\,c{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x+g)/(-c*g^2/h^2+9*c*x^2)^(1/3)/(3*h^2*x^2+g^2),x)

[Out]

int((h*x+g)/(-c*g^2/h^2+9*c*x^2)^(1/3)/(3*h^2*x^2+g^2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{h x + g}{{\left (3 \, h^{2} x^{2} + g^{2}\right )}{\left (9 \, c x^{2} - \frac{c g^{2}}{h^{2}}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)/((3*h^2*x^2 + g^2)*(9*c*x^2 - c*g^2/h^2)^(1/3)),x, algorithm="maxima")

[Out]

integrate((h*x + g)/((3*h^2*x^2 + g^2)*(9*c*x^2 - c*g^2/h^2)^(1/3)), x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)/((3*h^2*x^2 + g^2)*(9*c*x^2 - c*g^2/h^2)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x+g)/(-c*g**2/h**2+9*c*x**2)**(1/3)/(3*h**2*x**2+g**2),x)

[Out]

Integral((g + h*x)/((c*(-g/h + 3*x)*(g/h + 3*x))**(1/3)*(g**2 + 3*h**2*x**2)), x
)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{h x + g}{{\left (3 \, h^{2} x^{2} + g^{2}\right )}{\left (9 \, c x^{2} - \frac{c g^{2}}{h^{2}}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x + g)/((3*h^2*x^2 + g^2)*(9*c*x^2 - c*g^2/h^2)^(1/3)),x, algorithm="giac")

[Out]

integrate((h*x + g)/((3*h^2*x^2 + g^2)*(9*c*x^2 - c*g^2/h^2)^(1/3)), x)