3.6.77 \(\int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx\)

Optimal. Leaf size=151 \[ -\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}+\frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e} \]

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Rubi [A]  time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {751} \begin {gather*} \frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}-\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]

[Out]

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

Rule 751

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps

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

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Mathematica [C]  time = 0.15, size = 176, normalized size = 1.17 \begin {gather*} -\frac {\sqrt [3]{\frac {e \left (\sqrt {3} \sqrt {-\frac {d^2}{e^2}}+3 x\right )}{d+e x}} \sqrt [3]{\frac {e \left (9 x-3 \sqrt {3} \sqrt {-\frac {d^2}{e^2}}\right )}{d+e x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3 d-\sqrt {3} \sqrt {-\frac {d^2}{e^2}} e}{3 d+3 e x},\frac {3 d+\sqrt {3} \sqrt {-\frac {d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2+3 e^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]

[Out]

-1/2*(((e*(Sqrt[3]*Sqrt[-(d^2/e^2)] + 3*x))/(d + e*x))^(1/3)*((e*(-3*Sqrt[3]*Sqrt[-(d^2/e^2)] + 9*x))/(d + e*x
))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3*d - Sqrt[3]*Sqrt[-(d^2/e^2)]*e)/(3*d + 3*e*x), (3*d + Sqrt[3]*Sqrt[-(
d^2/e^2)]*e)/(3*d + 3*e*x)])/(e*(d^2 + 3*e^2*x^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.41, size = 298, normalized size = 1.97 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}{\sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}+2^{2/3} d-2^{2/3} e x}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}+\frac {\log \left (2 \sqrt [3]{d} \sqrt {e} \sqrt [3]{d^2+3 e^2 x^2}-2^{2/3} d \sqrt {e}+2^{2/3} e^{3/2} x\right )}{3\ 2^{2/3} d^{2/3} e}-\frac {\log \left (-2^{2/3} \sqrt [3]{d} e^2 x \sqrt [3]{d^2+3 e^2 x^2}+\sqrt [3]{2} d^2 e+2 d^{2/3} e \left (d^2+3 e^2 x^2\right )^{2/3}+2^{2/3} d^{4/3} e \sqrt [3]{d^2+3 e^2 x^2}-2 \sqrt [3]{2} d e^2 x+\sqrt [3]{2} e^3 x^2\right )}{6\ 2^{2/3} d^{2/3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]

[Out]

ArcTan[(Sqrt[3]*d^(1/3)*(d^2 + 3*e^2*x^2)^(1/3))/(2^(2/3)*d - 2^(2/3)*e*x + d^(1/3)*(d^2 + 3*e^2*x^2)^(1/3))]/
(2^(2/3)*Sqrt[3]*d^(2/3)*e) + Log[-(2^(2/3)*d*Sqrt[e]) + 2^(2/3)*e^(3/2)*x + 2*d^(1/3)*Sqrt[e]*(d^2 + 3*e^2*x^
2)^(1/3)]/(3*2^(2/3)*d^(2/3)*e) - Log[2^(1/3)*d^2*e - 2*2^(1/3)*d*e^2*x + 2^(1/3)*e^3*x^2 + 2^(2/3)*d^(4/3)*e*
(d^2 + 3*e^2*x^2)^(1/3) - 2^(2/3)*d^(1/3)*e^2*x*(d^2 + 3*e^2*x^2)^(1/3) + 2*d^(2/3)*e*(d^2 + 3*e^2*x^2)^(2/3)]
/(6*2^(2/3)*d^(2/3)*e)

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fricas [B]  time = 40.92, size = 337, normalized size = 2.23 \begin {gather*} -\frac {4 \, \sqrt {3} d \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}} \arctan \left (\frac {\sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (e x - d\right )} + 4^{\frac {1}{3}} {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 4 \, {\left (d e^{2} x^{2} - 2 \, d^{2} e x + d^{3}\right )} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}}}{6 \, {\left (d e^{3} x^{3} - 9 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - 3 \, d^{4}\right )}}\right ) + 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (d e x - d^{2}\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \cdot 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} {\left (e x - d\right )} + 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} d}{e x + d}\right )}{24 \, d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="fricas")

[Out]

-1/24*(4*sqrt(3)*d*sqrt(4^(1/3)*(d^2)^(1/3))*arctan(1/6*sqrt(3)*(2*4^(2/3)*(3*e^2*x^2 + d^2)^(2/3)*(d^2)^(2/3)
*(e*x - d) + 4^(1/3)*(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(d^2)^(1/3) + 4*(d*e^2*x^2 - 2*d^2*e*x + d^3)*(
3*e^2*x^2 + d^2)^(1/3))*sqrt(4^(1/3)*(d^2)^(1/3))/(d*e^3*x^3 - 9*d^2*e^2*x^2 + 3*d^3*e*x - 3*d^4)) + 4^(2/3)*(
d^2)^(2/3)*log((4^(2/3)*(3*e^2*x^2 + d^2)^(2/3)*(d^2)^(2/3) + 4^(1/3)*(e^2*x^2 - 2*d*e*x + d^2)*(d^2)^(1/3) -
2*(3*e^2*x^2 + d^2)^(1/3)*(d*e*x - d^2))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*4^(2/3)*(d^2)^(2/3)*log((4^(1/3)*(d^2)
^(1/3)*(e*x - d) + 2*(3*e^2*x^2 + d^2)^(1/3)*d)/(e*x + d)))/(d^2*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)

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maple [F]  time = 0.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e x +d \right ) \left (3 e^{2} x^{2}+d^{2}\right )^{\frac {1}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x)

[Out]

int(1/(e*x+d)/(3*e^2*x^2+d^2)^(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^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (d^2+3\,e^2\,x^2\right )}^{1/3}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d^2 + 3*e^2*x^2)^(1/3)*(d + e*x)),x)

[Out]

int(1/((d^2 + 3*e^2*x^2)^(1/3)*(d + e*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(3*e**2*x**2+d**2)**(1/3),x)

[Out]

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

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