3.26.21 \(\int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\)

Optimal. Leaf size=120 \[ \frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \]

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Rubi [A]  time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {123} \begin {gather*} \frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (d - 3*e*x)^(2/3)/(Sqrt[3]*d^(1/3)*(d + 3*e*x)^(1/3))])/(4*d^(2/3)*e) + Log[d + e*
x]/(4*d^(2/3)*e) - (3*Log[-(d - 3*e*x)^(2/3)/(2*d^(1/3)) - (d + 3*e*x)^(1/3)])/(8*d^(2/3)*e)

Rule 123

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 100, normalized size = 0.83 \begin {gather*} -\frac {3 \sqrt [3]{1-\frac {4 d}{3 (d+e x)}} \sqrt [3]{1-\frac {2 d}{3 (d+e x)}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {4 d}{3 (d+e x)},\frac {2 d}{3 (d+e x)}\right )}{2 e \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 - (4*d)/(3*(d + e*x)))^(1/3)*(1 - (2*d)/(3*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (4*d)/(3*(d +
e*x)), (2*d)/(3*(d + e*x))])/(2*e*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3))

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IntegrateAlgebraic [B]  time = 1.67, size = 495, normalized size = 4.12 \begin {gather*} -\frac {\log \left (\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}+2^{2/3} \sqrt [3]{d}\right )}{2 d^{2/3} e}-\frac {\log \left (2 \sqrt [3]{d-3 e x}+\sqrt [3]{2} \sqrt [3]{d+3 e x}-2^{2/3} \sqrt [3]{d}\right )}{4 d^{2/3} e}+\frac {\log \left (2 \sqrt [3]{2} d^{2/3}+2\ 2^{2/3} \sqrt [3]{d} \sqrt [3]{d-3 e x}-4 \sqrt [3]{d} \sqrt [3]{d+3 e x}+4 (d-3 e x)^{2/3}+2^{2/3} (d+3 e x)^{2/3}-2 \sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}+\frac {\log \left (2 \sqrt [3]{2} d^{2/3}-2^{2/3} \sqrt [3]{d} \sqrt [3]{d-3 e x}-4 \sqrt [3]{d} \sqrt [3]{d+3 e x}+(d-3 e x)^{2/3}+2^{2/3} (d+3 e x)^{2/3}+\sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}\right )}{4 d^{2/3} e}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}+2^{2/3} \sqrt [3]{d}}\right )}{4 d^{2/3} e}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{\sqrt [3]{d-3 e x}+2 \sqrt [3]{2} \sqrt [3]{d+3 e x}-2\ 2^{2/3} \sqrt [3]{d}}\right )}{2 d^{2/3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d - 3*e*x)^(1/3)*(d + e*x)*(d + 3*e*x)^(1/3)),x]

[Out]

-1/4*(Sqrt[3]*ArcTan[(Sqrt[3]*(d - 3*e*x)^(1/3))/(2^(2/3)*d^(1/3) + (d - 3*e*x)^(1/3) - 2^(1/3)*(d + 3*e*x)^(1
/3))])/(d^(2/3)*e) - (Sqrt[3]*ArcTan[(Sqrt[3]*(d - 3*e*x)^(1/3))/(-2*2^(2/3)*d^(1/3) + (d - 3*e*x)^(1/3) + 2*2
^(1/3)*(d + 3*e*x)^(1/3))])/(2*d^(2/3)*e) - Log[2^(2/3)*d^(1/3) + (d - 3*e*x)^(1/3) - 2^(1/3)*(d + 3*e*x)^(1/3
)]/(2*d^(2/3)*e) - Log[-(2^(2/3)*d^(1/3)) + 2*(d - 3*e*x)^(1/3) + 2^(1/3)*(d + 3*e*x)^(1/3)]/(4*d^(2/3)*e) + L
og[2*2^(1/3)*d^(2/3) + 2*2^(2/3)*d^(1/3)*(d - 3*e*x)^(1/3) + 4*(d - 3*e*x)^(2/3) - 4*d^(1/3)*(d + 3*e*x)^(1/3)
- 2*2^(1/3)*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3) + 2^(2/3)*(d + 3*e*x)^(2/3)]/(8*d^(2/3)*e) + Log[2*2^(1/3)*d^
(2/3) - 2^(2/3)*d^(1/3)*(d - 3*e*x)^(1/3) + (d - 3*e*x)^(2/3) - 4*d^(1/3)*(d + 3*e*x)^(1/3) + 2^(1/3)*(d - 3*e
*x)^(1/3)*(d + 3*e*x)^(1/3) + 2^(2/3)*(d + 3*e*x)^(2/3)]/(4*d^(2/3)*e)

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fricas [F(-1)]  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(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-3 e x +d \right )^{\frac {1}{3}} \left (e x +d \right ) \left (3 e x +d \right )^{\frac {1}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (d+e\,x\right )\,{\left (d-3\,e\,x\right )}^{1/3}\,{\left (d+3\,e\,x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{d - 3 e x} \left (d + e x\right ) \sqrt [3]{d + 3 e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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