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

Optimal. Leaf size=332 \[ \frac {(-1)^{5/6} \sqrt {3} \tanh ^{-1}\left (\frac {\frac {i \sqrt [3]{a^3 x^3-b^3}}{\sqrt {3}}+\frac {\left (-\sqrt {3} a-i a\right ) x}{2^{2/3} \sqrt {3}}+\frac {\sqrt {3} b+i b}{2^{2/3} \sqrt {3}}}{\sqrt [3]{a^3 x^3-b^3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\sqrt [3]{-\frac {1}{2}} \log \left ((-1)^{2/3} a^{3/2} \sqrt {b} x-2^{2/3} \sqrt {a} \sqrt {b} \sqrt [3]{a^3 x^3-b^3}-(-1)^{2/3} \sqrt {a} b^{3/2}\right )}{2 a b}-\frac {\sqrt [3]{-\frac {1}{2}} \log \left (-2 \sqrt [3]{2} a b \left (a^3 x^3-b^3\right )^{2/3}+\sqrt [3]{-1} a^3 b x^2-2 \sqrt [3]{-1} a^2 b^2 x+\left ((-2)^{2/3} a b^2-(-2)^{2/3} a^2 b x\right ) \sqrt [3]{a^3 x^3-b^3}+\sqrt [3]{-1} a b^3\right )}{4 a b} \]

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Rubi [A]  time = 0.08, antiderivative size = 139, normalized size of antiderivative = 0.42, 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 = {2148} \begin {gather*} -\frac {3 \log \left (2^{2/3} a \sqrt [3]{a^3 x^3-b^3}+a (b-a x)\right )}{4 \sqrt [3]{2} a b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (b-a x)}{\sqrt [3]{a^3 x^3-b^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\log \left ((b-a x) (a x+b)^2\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]

(Sqrt[3]*ArcTan[(1 - (2^(1/3)*(b - a*x))/(-b^3 + a^3*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*a*b) + Log[(b - a*x)*(b
+ a*x)^2]/(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 2148

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) (b+a x)^2\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 [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(b+a x) \sqrt [3]{-b^3+a^3 x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.34, size = 385, normalized size = 1.16 \begin {gather*} \frac {(-1)^{5/6} \sqrt {3} \tanh ^{-1}\left (\frac {\frac {i b+\sqrt {3} b}{2^{2/3} \sqrt {3}}+\frac {\left (-i a-\sqrt {3} a\right ) x}{2^{2/3} \sqrt {3}}+\frac {i \sqrt [3]{-b^3+a^3 x^3}}{\sqrt {3}}}{\sqrt [3]{-b^3+a^3 x^3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\sqrt [3]{-\frac {1}{2}} \log \left (\left (-1+i \sqrt {3}\right ) \sqrt {a} b^{3/2}+a^{3/2} \sqrt {b} x-i \sqrt {3} a^{3/2} \sqrt {b} x+2\ 2^{2/3} \sqrt {a} \sqrt {b} \sqrt [3]{-b^3+a^3 x^3}\right )}{2 a b}-\frac {\sqrt [3]{-\frac {1}{2}} \log \left (-a b^3-i \sqrt {3} a b^3+2 a^2 b^2 x+2 i \sqrt {3} a^2 b^2 x-a^3 b x^2-i \sqrt {3} a^3 b x^2+\left (-2 (-2)^{2/3} a b^2+2 (-2)^{2/3} a^2 b x\right ) \sqrt [3]{-b^3+a^3 x^3}+4 \sqrt [3]{2} a b \left (-b^3+a^3 x^3\right )^{2/3}\right )}{4 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(5/6)*Sqrt[3]*ArcTanh[((I*b + Sqrt[3]*b)/(2^(2/3)*Sqrt[3]) + (((-I)*a - Sqrt[3]*a)*x)/(2^(2/3)*Sqrt[3])
+ (I*(-b^3 + a^3*x^3)^(1/3))/Sqrt[3])/(-b^3 + a^3*x^3)^(1/3)])/(2*2^(1/3)*a*b) + ((-1/2)^(1/3)*Log[(-1 + I*Sqr
t[3])*Sqrt[a]*b^(3/2) + a^(3/2)*Sqrt[b]*x - I*Sqrt[3]*a^(3/2)*Sqrt[b]*x + 2*2^(2/3)*Sqrt[a]*Sqrt[b]*(-b^3 + a^
3*x^3)^(1/3)])/(2*a*b) - ((-1/2)^(1/3)*Log[-(a*b^3) - I*Sqrt[3]*a*b^3 + 2*a^2*b^2*x + (2*I)*Sqrt[3]*a^2*b^2*x
- a^3*b*x^2 - I*Sqrt[3]*a^3*b*x^2 + (-2*(-2)^(2/3)*a*b^2 + 2*(-2)^(2/3)*a^2*b*x)*(-b^3 + a^3*x^3)^(1/3) + 4*2^
(1/3)*a*b*(-b^3 + a^3*x^3)^(2/3)])/(4*a*b)

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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/(a*x+b)/(a^3*x^3-b^3)^(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 (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} {\left (a x + b\right )}}\,{d x} \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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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*} \int \frac {1}{{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} {\left (a x + b\right )}}\,{d x} \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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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/((a^3*x^3 - b^3)^(1/3)*(b + a*x)),x)

[Out]

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