3.6.52 \(\int \frac {x}{\sqrt [3]{a+b x^3}} \, dx\) [552]

Optimal. Leaf size=38 \[ \frac {x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (1,\frac {4}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 a} \]

[Out]

1/2*x^2*(b*x^3+a)^(2/3)*hypergeom([1, 4/3],[5/3],-b*x^3/a)/a

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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {372, 371} \begin {gather*} \frac {x^2 \sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^3)^(1/3),x]

[Out]

(x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(a + b*x^3)^(1/3))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {x}{\sqrt [3]{a+b x^3}} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {x}{\sqrt [3]{1+\frac {b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}}\\ &=\frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\\ \end {align*}

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Mathematica [A]
time = 5.20, size = 51, normalized size = 1.34 \begin {gather*} \frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^3)^(1/3),x]

[Out]

(x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(a + b*x^3)^(1/3))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x/(b*x^3 + a)^(1/3), x)

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Fricas [F]
time = 0.39, size = 13, normalized size = 0.34 \begin {gather*} {\rm integral}\left (\frac {x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/(b*x^3 + a)^(1/3), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.37, size = 37, normalized size = 0.97 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*gamma(2/3)*hyper((1/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(5/3))

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

[Out]

integrate(x/(b*x^3 + a)^(1/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{{\left (b\,x^3+a\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^3)^(1/3),x)

[Out]

int(x/(a + b*x^3)^(1/3), x)

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