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

Optimal. Leaf size=33 \[ \frac{x \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac{5}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]

[Out]

(x*(a + b*x^3)^(4/3)*Hypergeometric2F1[1, 5/3, 4/3, -((b*x^3)/a)])/a

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Rubi [A]  time = 0.0085533, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3),x]

[Out]

(x*(a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(1/3)

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \sqrt [3]{a+b x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \sqrt [3]{1+\frac{b x^3}{a}} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [C]  time = 0.204643, size = 196, normalized size = 5.94 \[ \frac{3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} F_1\left (\frac{4}{3};-\frac{1}{3},-\frac{1}{3};\frac{7}{3};-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(1/3),x]

[Out]

(3*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^(1/3)*AppellF1[4/3, -1/3, -1/3, 7/3, ((-I)*((-1)^(2/3)*a^(1/3)
+ b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(4*2^(1/3)*b^(1/
3)*((a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(1/3)*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + S
qrt[3]))^(1/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b{x}^{3}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3 + a)^(1/3), x)

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Sympy [C]  time = 0.76344, size = 37, normalized size = 1.12 \begin{align*} \frac{\sqrt [3]{a} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(1/3)*x*gamma(1/3)*hyper((-1/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3), x)