3.173 \(\int \frac{e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx\)

Optimal. Leaf size=234 \[ -\frac{3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left (\frac{1-\frac{\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac{f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac{f \tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]

[Out]

(f*ArcTan[(1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (Sqrt[3]*(d*e - c*f)*ArcTan[(1 - (2^(
1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*c*d^2) + ((d*e - c*f)*Log[(c - d*x)*(c + d*x)^2])
/(4*2^(1/3)*c*d^2) - (f*Log[-(d*x) + (-c^3 + d^3*x^3)^(1/3)])/(2*d^2) - (3*(d*e - c*f)*Log[d*(c - d*x) + 2^(2/
3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d^2)

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Rubi [A]  time = 0.220086, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2152, 239, 2148} \[ -\frac{3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left (\frac{1-\frac{\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac{f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac{f \tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]

[Out]

(f*ArcTan[(1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (Sqrt[3]*(d*e - c*f)*ArcTan[(1 - (2^(
1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*c*d^2) + ((d*e - c*f)*Log[(c - d*x)*(c + d*x)^2])
/(4*2^(1/3)*c*d^2) - (f*Log[-(d*x) + (-c^3 + d^3*x^3)^(1/3)])/(2*d^2) - (3*(d*e - c*f)*Log[d*(c - d*x) + 2^(2/
3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d^2)

Rule 2152

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[f/d, Int[1/(a +
b*x^3)^(1/3), x], x] + Dist[(d*e - c*f)/d, Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d,
e, f}, x]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx &=\frac{f \int \frac{1}{\sqrt [3]{-c^3+d^3 x^3}} \, dx}{d}+\frac{(d e-c f) \int \frac{1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx}{d}\\ &=\frac{f \tan ^{-1}\left (\frac{1+\frac{2 d x}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left (\frac{1-\frac{\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} c d^2}+\frac{(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2}-\frac{f \log \left (-d x+\sqrt [3]{-c^3+d^3 x^3}\right )}{2 d^2}-\frac{3 (d e-c f) \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d^2}\\ \end{align*}

Mathematica [F]  time = 0.183859, size = 0, normalized size = 0. \[ \int \frac{e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]

[Out]

Integrate[(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)), x]

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{fx+e}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}-{c}^{3}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)

[Out]

int((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="maxima")

[Out]

integrate((f*x + e)/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)

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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((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e + f x}{\sqrt [3]{\left (- c + d x\right ) \left (c^{2} + c d x + d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(d**3*x**3-c**3)**(1/3),x)

[Out]

Integral((e + f*x)/(((-c + d*x)*(c**2 + c*d*x + d**2*x**2))**(1/3)*(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x, algorithm="giac")

[Out]

integrate((f*x + e)/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)