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

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 = 0.0927643, 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, Rules used = {1009, 1008} \[ \frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (g^2+3 h^2 x^2\right )}{6\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}-\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \log \left (\left (1-\frac{3 h x}{g}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h x}{g}+1}\right )}{2\ 2^{2/3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}}+\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h x}{g}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h x}{g}+1}}\right )}{2^{2/3} \sqrt{3} h \sqrt [3]{9 c x^2-\frac{c g^2}{h^2}}} \]

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))

Rule 1009

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)^(1/3)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> Dist[(1 + (c*x^2)/a)
^(1/3)/(a + c*x^2)^(1/3), Int[(g + h*x)/((1 + (c*x^2)/a)^(1/3)*(d + f*x^2)), x], x] /; FreeQ[{a, c, d, f, g, h
}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^2, 0] &&  !GtQ[a, 0]

Rule 1008

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)^(1/3)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> Simp[(Sqrt[3]*h*ArcT
an[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)*a^(1/3)*f), x] + (-S
imp[(3*h*Log[(1 - (3*h*x)/g)^(2/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2^(5/3)*a^(1/3)*f), x] + Simp[(h*Log[d +
f*x^2])/(2^(5/3)*a^(1/3)*f), x]) /; FreeQ[{a, c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^
2, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \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 &=\frac{\sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} \int \frac{g+h x}{\left (g^2+3 h^2 x^2\right ) \sqrt [3]{1-\frac{9 h^2 x^2}{g^2}}} \, dx}{\sqrt [3]{-\frac{c g^2}{h^2}+9 c x^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]{1+\frac{3 h x}{g}}}\right )}{2^{2/3} \sqrt{3} h \sqrt [3]{-\frac{c g^2}{h^2}+9 c x^2}}+\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]{-\frac{c g^2}{h^2}+9 c x^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]{1+\frac{3 h x}{g}}\right )}{2\ 2^{2/3} h \sqrt [3]{-\frac{c g^2}{h^2}+9 c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.581694, size = 268, normalized size = 1.11 \[ \frac{h^2 x \left (-\frac{2 g^5 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )}{\left (g^2+3 h^2 x^2\right ) \left (g^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )+2 h^2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )\right )\right )}-h x \sqrt [3]{1-\frac{9 h^2 x^2}{g^2}} F_1\left (1;\frac{1}{3},1;2;\frac{9 h^2 x^2}{g^2},-\frac{3 h^2 x^2}{g^2}\right )\right ) \left (c \left (9 x^2-\frac{g^2}{h^2}\right )\right )^{2/3}}{2 c g^2 \left (g^2-9 h^2 x^2\right )} \]

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]

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

________________________________________________________________________________________

Maple [F]  time = 0.803, size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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. \begin{align*} \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} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)