3.78 \(\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{(2 \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {a+b x^3}} \, dx\)

Optimal. Leaf size=50 \[ \frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

[Out]

2/3*arctanh(1/3*(a^(1/3)+b^(1/3)*x)^2/a^(1/6)/(b*x^3+a)^(1/2))/a^(1/6)/b^(1/3)

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Rubi [A]  time = 0.13, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2138, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]

[Out]

(2*ArcTanh[(a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[a + b*x^3])])/(3*a^(1/6)*b^(1/3))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx &=\frac {\left (2 \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{9-a x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 1.02 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \left (\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )^2}{3 \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]

[Out]

(2*ArcTanh[(Sqrt[a]*(1 + (b^(1/3)*x)/a^(1/3))^2)/(3*Sqrt[a + b*x^3])])/(3*a^(1/6)*b^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}}}{\left (-b^{\frac {1}{3}} x +2 a^{\frac {1}{3}}\right ) \sqrt {b \,x^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(1/3)+b^(1/3)*x)/(2*a^(1/3)-b^(1/3)*x)/(b*x^3+a)^(1/2),x)

[Out]

int((a^(1/3)+b^(1/3)*x)/(2*a^(1/3)-b^(1/3)*x)/(b*x^3+a)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {b^{\frac {1}{3}} x + a^{\frac {1}{3}}}{\sqrt {b x^{3} + a} {\left (b^{\frac {1}{3}} x - 2 \, a^{\frac {1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((b^(1/3)*x + a^(1/3))/(sqrt(b*x^3 + a)*(b^(1/3)*x - 2*a^(1/3))), x)

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mupad [B]  time = 3.44, size = 65, normalized size = 1.30 \[ \frac {\ln \left (\frac {\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}-\sqrt {a}+2\,a^{1/6}\,b^{1/3}\,x\right )}^3}{x^3\,{\left (b^{1/3}\,x-2\,a^{1/3}\right )}^3}\right )}{3\,a^{1/6}\,b^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^(1/3)*x + a^(1/3))/((b^(1/3)*x - 2*a^(1/3))*(a + b*x^3)^(1/2)),x)

[Out]

log((((a + b*x^3)^(1/2) + a^(1/2))*((a + b*x^3)^(1/2) - a^(1/2) + 2*a^(1/6)*b^(1/3)*x)^3)/(x^3*(b^(1/3)*x - 2*
a^(1/3))^3))/(3*a^(1/6)*b^(1/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt {a + b x^{3}} + \sqrt [3]{b} x \sqrt {a + b x^{3}}}\, dx - \int \frac {\sqrt [3]{b} x}{- 2 \sqrt [3]{a} \sqrt {a + b x^{3}} + \sqrt [3]{b} x \sqrt {a + b x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**(1/3)+b**(1/3)*x)/(2*a**(1/3)-b**(1/3)*x)/(b*x**3+a)**(1/2),x)

[Out]

-Integral(a**(1/3)/(-2*a**(1/3)*sqrt(a + b*x**3) + b**(1/3)*x*sqrt(a + b*x**3)), x) - Integral(b**(1/3)*x/(-2*
a**(1/3)*sqrt(a + b*x**3) + b**(1/3)*x*sqrt(a + b*x**3)), x)

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