3.2.42 \(\int \frac {g+h x}{\sqrt [3]{-\frac {c g^2}{h^2}+9 c x^2} (g^2+3 h^2 x^2)} \, dx\) [142]

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

[Out]

1/12*(1-9*h^2*x^2/g^2)^(1/3)*ln(3*h^2*x^2+g^2)*2^(1/3)/h/(-c*g^2/h^2+9*c*x^2)^(1/3)-1/4*(1-9*h^2*x^2/g^2)^(1/3
)*ln((1-3*h*x/g)^(2/3)+2^(1/3)*(1+3*h*x/g)^(1/3))*2^(1/3)/h/(-c*g^2/h^2+9*c*x^2)^(1/3)-1/6*(1-9*h^2*x^2/g^2)^(
1/3)*arctan(-1/3*3^(1/2)+1/3*2^(2/3)*(1-3*h*x/g)^(2/3)/(1+3*h*x/g)^(1/3)*3^(1/2))*2^(1/3)/h/(-c*g^2/h^2+9*c*x^
2)^(1/3)*3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1023, 1022} \begin {gather*} \frac {\sqrt [3]{1-\frac {9 h^2 x^2}{g^2}} \text {ArcTan}\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}}}+\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}}} \end {gather*}

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 1022

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]

Rule 1023

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]

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*}

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Mathematica [A]
time = 0.72, size = 381, normalized size = 1.57 \begin {gather*} \frac {\sqrt [3]{-\frac {2 g^2}{h^2}+18 x^2} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{g} h^{2/3} \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}}{2^{2/3} g-3\ 2^{2/3} h x+\sqrt [3]{g} h^{2/3} \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}}\right )-2 \log \left (\sqrt [3]{g} \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}\right )+\log \left (g^{2/3} \left (-\frac {g^2}{h^2}+9 x^2\right )^{2/3}\right )+2 \log \left (2^{2/3} g-3\ 2^{2/3} h x-2 \sqrt [3]{g} h^{2/3} \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}\right )-\log \left (\sqrt [3]{2} g^2-6 \sqrt [3]{2} g h x+9 \sqrt [3]{2} h^2 x^2+2^{2/3} g^{4/3} h^{2/3} \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}-3\ 2^{2/3} \sqrt [3]{g} h^{5/3} x \sqrt [3]{-\frac {g^2}{h^2}+9 x^2}+2 g^{2/3} h^{4/3} \left (-\frac {g^2}{h^2}+9 x^2\right )^{2/3}\right )\right )}{12 g^{2/3} \sqrt [3]{h} \sqrt [3]{c \left (-\frac {g^2}{h^2}+9 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[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]

(((-2*g^2)/h^2 + 18*x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*g^(1/3)*h^(2/3)*(-(g^2/h^2) + 9*x^2)^(1/3))/(2^(2/3)
*g - 3*2^(2/3)*h*x + g^(1/3)*h^(2/3)*(-(g^2/h^2) + 9*x^2)^(1/3))] - 2*Log[g^(1/3)*(-(g^2/h^2) + 9*x^2)^(1/3)]
+ Log[g^(2/3)*(-(g^2/h^2) + 9*x^2)^(2/3)] + 2*Log[2^(2/3)*g - 3*2^(2/3)*h*x - 2*g^(1/3)*h^(2/3)*(-(g^2/h^2) +
9*x^2)^(1/3)] - Log[2^(1/3)*g^2 - 6*2^(1/3)*g*h*x + 9*2^(1/3)*h^2*x^2 + 2^(2/3)*g^(4/3)*h^(2/3)*(-(g^2/h^2) +
9*x^2)^(1/3) - 3*2^(2/3)*g^(1/3)*h^(5/3)*x*(-(g^2/h^2) + 9*x^2)^(1/3) + 2*g^(2/3)*h^(4/3)*(-(g^2/h^2) + 9*x^2)
^(2/3)]))/(12*g^(2/3)*h^(1/3)*(c*(-(g^2/h^2) + 9*x^2))^(1/3))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {h x +g}{\left (-\frac {c \,g^{2}}{h^{2}}+9 c \,x^{2}\right )^{\frac {1}{3}} \left (3 h^{2} x^{2}+g^{2}\right )}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {g+h\,x}{\left (g^2+3\,h^2\,x^2\right )\,{\left (9\,c\,x^2-\frac {c\,g^2}{h^2}\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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