3.28.76 \(\int \frac {1}{(-b+a x) \sqrt [3]{b^3+a^3 x^3}} \, dx\) [2776]

Optimal. Leaf size=266 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{b^3+a^3 x^3}}{\sqrt [3]{2} b+\sqrt [3]{2} a x+\sqrt [3]{b^3+a^3 x^3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\log \left (\sqrt [3]{2} \sqrt {a} b^{3/2}+\sqrt [3]{2} a^{3/2} \sqrt {b} x-2 \sqrt {a} \sqrt {b} \sqrt [3]{b^3+a^3 x^3}\right )}{2 \sqrt [3]{2} a b}-\frac {\log \left (2^{2/3} a b^3+2\ 2^{2/3} a^2 b^2 x+2^{2/3} a^3 b x^2+\left (2 \sqrt [3]{2} a b^2+2 \sqrt [3]{2} a^2 b x\right ) \sqrt [3]{b^3+a^3 x^3}+4 a b \left (b^3+a^3 x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} a b} \]

[Out]

1/4*3^(1/2)*arctan(3^(1/2)*(a^3*x^3+b^3)^(1/3)/(2^(1/3)*b+2^(1/3)*a*x+(a^3*x^3+b^3)^(1/3)))*2^(2/3)/a/b+1/4*ln
(2^(1/3)*a^(1/2)*b^(3/2)+2^(1/3)*a^(3/2)*b^(1/2)*x-2*a^(1/2)*b^(1/2)*(a^3*x^3+b^3)^(1/3))*2^(2/3)/a/b-1/8*ln(2
^(2/3)*a*b^3+2*2^(2/3)*a^2*b^2*x+2^(2/3)*a^3*b*x^2+(2*2^(1/3)*a*b^2+2*2^(1/3)*a^2*b*x)*(a^3*x^3+b^3)^(1/3)+4*a
*b*(a^3*x^3+b^3)^(2/3))*2^(2/3)/a/b

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Rubi [A]
time = 0.05, antiderivative size = 134, normalized size of antiderivative = 0.50, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2174} \begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (a x+b)}{\sqrt [3]{a^3 x^3+b^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a b}+\frac {3 \log \left (2^{2/3} a \sqrt [3]{a^3 x^3+b^3}-a (a x+b)\right )}{4 \sqrt [3]{2} a b}-\frac {\log \left (-(b-a x)^2 (a x+b)\right )}{4 \sqrt [3]{2} a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-b + a*x)*(b^3 + a^3*x^3)^(1/3)),x]

[Out]

-1/2*(Sqrt[3]*ArcTan[(1 + (2^(1/3)*(b + a*x))/(b^3 + a^3*x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*a*b) - Log[-((b - a*x)
^2*(b + a*x))]/(4*2^(1/3)*a*b) + (3*Log[-(a*(b + a*x)) + 2^(2/3)*a*(b^3 + a^3*x^3)^(1/3)])/(4*2^(1/3)*a*b)

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]

Rubi steps

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

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Mathematica [A]
time = 1.23, size = 207, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{b^3+a^3 x^3}}{\sqrt [3]{2} b+\sqrt [3]{2} a x+\sqrt [3]{b^3+a^3 x^3}}\right )+2 \log \left (\sqrt {a} \sqrt {b} \left (\sqrt [3]{2} b+\sqrt [3]{2} a x-2 \sqrt [3]{b^3+a^3 x^3}\right )\right )-\log \left (a b \left (2^{2/3} b^2+2\ 2^{2/3} a b x+2^{2/3} a^2 x^2+2 \sqrt [3]{2} (b+a x) \sqrt [3]{b^3+a^3 x^3}+4 \left (b^3+a^3 x^3\right )^{2/3}\right )\right )}{4 \sqrt [3]{2} a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-b + a*x)*(b^3 + a^3*x^3)^(1/3)),x]

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(b^3 + a^3*x^3)^(1/3))/(2^(1/3)*b + 2^(1/3)*a*x + (b^3 + a^3*x^3)^(1/3))] + 2*Log[S
qrt[a]*Sqrt[b]*(2^(1/3)*b + 2^(1/3)*a*x - 2*(b^3 + a^3*x^3)^(1/3))] - Log[a*b*(2^(2/3)*b^2 + 2*2^(2/3)*a*b*x +
2^(2/3)*a^2*x^2 + 2*2^(1/3)*(b + a*x)*(b^3 + a^3*x^3)^(1/3) + 4*(b^3 + a^3*x^3)^(2/3))])/(4*2^(1/3)*a*b)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x -b \right ) \left (a^{3} x^{3}+b^{3}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-b)/(a^3*x^3+b^3)^(1/3),x)

[Out]

int(1/(a*x-b)/(a^3*x^3+b^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(1/(a*x-b)/(a^3*x^3+b^3)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((a^3*x^3 + b^3)^(1/3)*(a*x - b)), 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(1/(a*x-b)/(a^3*x^3+b^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 {1}{\sqrt [3]{\left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )} \left (a x - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-b)/(a**3*x**3+b**3)**(1/3),x)

[Out]

Integral(1/(((a*x + b)*(a**2*x**2 - a*b*x + b**2))**(1/3)*(a*x - b)), 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(1/(a*x-b)/(a^3*x^3+b^3)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((a^3*x^3 + b^3)^(1/3)*(a*x - b)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((b^3 + a^3*x^3)^(1/3)*(b - a*x)),x)

[Out]

-int(1/((b^3 + a^3*x^3)^(1/3)*(b - a*x)), x)

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