3.6.84 \(\int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx\)

Optimal. Leaf size=206 \[ \frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log \left (-\frac {1}{2} \left (1-\frac {3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac {3 e x}{d}+1}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac {\sqrt {3} \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\left (1-\frac {3 e x}{d}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 e x}{d}+1}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}} \]

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Rubi [A]  time = 0.07, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {754, 753, 123} \begin {gather*} \frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log \left (-\frac {1}{2} \left (1-\frac {3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac {3 e x}{d}+1}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac {\sqrt {3} \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\left (1-\frac {3 e x}{d}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 e x}{d}+1}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*(1 - (9*e^2*x^2)/d^2)^(1/3)*ArcTan[1/Sqrt[3] - (1 - (3*e*x)/d)^(2/3)/(Sqrt[3]*(1 + (3*e*x)/d)^(1/3))]
)/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) + ((1 - (9*e^2*x^2)/d^2)^(1/3)*Log[d + e*x])/(4*e*(d^2 - 9*e^2*x^2)^(1/3)) - (
3*(1 - (9*e^2*x^2)/d^2)^(1/3)*Log[-(1 - (3*e*x)/d)^(2/3)/2 - (1 + (3*e*x)/d)^(1/3)])/(8*e*(d^2 - 9*e^2*x^2)^(1
/3))

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]

Rule 753

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[a^(1/3), Int[1/((d + e*x)*(1 - (3*e
*x)/d)^(1/3)*(1 + (3*e*x)/d)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]

Rule 754

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[(1 + (c*x^2)/a)^(1/3)/(a + c*x^2)^(
1/3), Int[1/((d + e*x)*(1 + (c*x^2)/a)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] &&
!GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx &=\frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \int \frac {1}{(d+e x) \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}}} \, dx}{\sqrt [3]{d^2-9 e^2 x^2}}\\ &=\frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \int \frac {1}{(d+e x) \sqrt [3]{1-\frac {3 e x}{d}} \sqrt [3]{1+\frac {3 e x}{d}}} \, dx}{\sqrt [3]{d^2-9 e^2 x^2}}\\ &=\frac {\sqrt {3} \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\left (1-\frac {3 e x}{d}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1+\frac {3 e x}{d}}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log \left (-\frac {1}{2} \left (1-\frac {3 e x}{d}\right )^{2/3}-\sqrt [3]{1+\frac {3 e x}{d}}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 155, normalized size = 0.75 \begin {gather*} -\frac {\sqrt [3]{3} \sqrt [3]{-\frac {e \left (\sqrt {\frac {d^2}{e^2}}-3 x\right )}{d+e x}} \sqrt [3]{\frac {e \left (\sqrt {\frac {d^2}{e^2}}+3 x\right )}{d+e x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3 d-\sqrt {\frac {d^2}{e^2}} e}{3 d+3 e x},\frac {3 d+\sqrt {\frac {d^2}{e^2}} e}{3 d+3 e x}\right )}{2 e \sqrt [3]{d^2-9 e^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(3^(1/3)*(-((e*(Sqrt[d^2/e^2] - 3*x))/(d + e*x)))^(1/3)*((e*(Sqrt[d^2/e^2] + 3*x))/(d + e*x))^(1/3)*Appel
lF1[2/3, 1/3, 1/3, 5/3, (3*d - Sqrt[d^2/e^2]*e)/(3*d + 3*e*x), (3*d + Sqrt[d^2/e^2]*e)/(3*d + 3*e*x)])/(e*(d^2
- 9*e^2*x^2)^(1/3))

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IntegrateAlgebraic [C]  time = 1.10, size = 411, normalized size = 2.00 \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 \sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}}{2 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}-i \sqrt {3} d+d+3 i \sqrt {3} e x-3 e x}\right )}{4 d^{2/3} e}+\frac {\left (1+i \sqrt {3}\right ) \log \left (\sqrt {e} \left (4 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}+i \sqrt {3} d-d\right )+e^{3/2} \left (3 x-3 i \sqrt {3} x\right )\right )}{8 d^{2/3} e}-\frac {i \left (\sqrt {3}-i\right ) \log \left (-6 \sqrt [3]{d} e^2 x \sqrt [3]{d^2-9 e^2 x^2}-d^2 e+8 d^{2/3} e \left (d^2-9 e^2 x^2\right )^{2/3}+2 d^{4/3} e \sqrt [3]{d^2-9 e^2 x^2}+\sqrt {3} \left (6 i \sqrt [3]{d} e^2 x \sqrt [3]{d^2-9 e^2 x^2}-i d^2 e-2 i d^{4/3} e \sqrt [3]{d^2-9 e^2 x^2}+6 i d e^2 x-9 i e^3 x^2\right )+6 d e^2 x-9 e^3 x^2\right )}{16 d^{2/3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(-3 + (3*I)*Sqrt[3])/2]*ArcTan[(2*Sqrt[3]*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))/(d - I*Sqrt[3]*d - 3*e*x + (3
*I)*Sqrt[3]*e*x + 2*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))])/(4*d^(2/3)*e) + ((1 + I*Sqrt[3])*Log[e^(3/2)*(3*x - (3*
I)*Sqrt[3]*x) + Sqrt[e]*(-d + I*Sqrt[3]*d + 4*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))])/(8*d^(2/3)*e) - ((I/16)*(-I +
Sqrt[3])*Log[-(d^2*e) + 6*d*e^2*x - 9*e^3*x^2 + 2*d^(4/3)*e*(d^2 - 9*e^2*x^2)^(1/3) - 6*d^(1/3)*e^2*x*(d^2 -
9*e^2*x^2)^(1/3) + 8*d^(2/3)*e*(d^2 - 9*e^2*x^2)^(2/3) + Sqrt[3]*((-I)*d^2*e + (6*I)*d*e^2*x - (9*I)*e^3*x^2 -
(2*I)*d^(4/3)*e*(d^2 - 9*e^2*x^2)^(1/3) + (6*I)*d^(1/3)*e^2*x*(d^2 - 9*e^2*x^2)^(1/3))])/(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/(e*x+d)/(-9*e^2*x^2+d^2)^(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 (-9 \, 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)/(-9*e^2*x^2+d^2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((-9*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)

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maple [F]  time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e x +d \right ) \left (-9 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)/(-9*e^2*x^2+d^2)^(1/3),x)

[Out]

int(1/(e*x+d)/(-9*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 (-9 \, 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)/(-9*e^2*x^2+d^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((-9*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 = -0.00 \begin {gather*} \int \frac {1}{{\left (d^2-9\,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 - 9*e^2*x^2)^(1/3)*(d + e*x)),x)

[Out]

int(1/((d^2 - 9*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}{\sqrt [3]{- \left (- d + 3 e x\right ) \left (d + 3 e x\right )} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(-9*e**2*x**2+d**2)**(1/3),x)

[Out]

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

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