3.2.73 \(\int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx\) [173]

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

[Out]

1/8*(-c*f+d*e)*ln((-d*x+c)*(d*x+c)^2)*2^(2/3)/c/d^2-1/2*f*ln(-d*x+(d^3*x^3-c^3)^(1/3))/d^2-3/8*(-c*f+d*e)*ln(d
*(-d*x+c)+2^(2/3)*d*(d^3*x^3-c^3)^(1/3))*2^(2/3)/c/d^2+1/3*f*arctan(1/3*(1+2*d*x/(d^3*x^3-c^3)^(1/3))*3^(1/2))
/d^2*3^(1/2)+1/4*(-c*f+d*e)*arctan(1/3*(1-2^(1/3)*(-d*x+c)/(d^3*x^3-c^3)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)/c/d^2

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Rubi [A]
time = 0.16, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2177, 245, 2174} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right ) (d e-c f)}{2 \sqrt [3]{2} c d^2}+\frac {f \text {ArcTan}\left (\frac {\frac {2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}-\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 {f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \end {gather*}

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 245

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 2174

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]

Rule 2177

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]

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*}

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Mathematica [F]
time = 10.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {f x +e}{\left (d x +c \right ) \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}\, dx\]

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 = 0.00, 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((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)] 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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