3.538 \(\int (a+b x^3)^{2/3} \, dx\)

Optimal. Leaf size=91 \[ \frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0165869, antiderivative size = 91, normalized size of antiderivative = 1., 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 = {195, 239} \[ \frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3),x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.169412, size = 196, normalized size = 2.15 \[ \frac{3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a+b x^3\right )^{2/3} F_1\left (\frac{5}{3};-\frac{2}{3},-\frac{2}{3};\frac{8}{3};-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{5\ 2^{2/3} \sqrt [3]{b} \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3),x]

[Out]

(3*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^(2/3)*AppellF1[5/3, -2/3, -2/3, 8/3, ((-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])])/(5*2^(2/3)*b^(1/
3)*((a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(2/3)*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + S
qrt[3]))^(2/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.09983, size = 944, normalized size = 10.37 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} b x^{3} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 2 \, a\right ) + 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x - 2 \, a \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + a \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{9 \, b}, -\frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} x - 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}{x}\right ) - 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x + 2 \, a \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - a \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{9 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/9*(3*sqrt(1/3)*a*b*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^
(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 3*(b*x^3
+ a)^(2/3)*b*x - 2*a*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + a*(-b)^(2/3)*log(((-b)^(2/3)*x^2
- (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2))/b, -1/9*(6*sqrt(1/3)*a*b*sqrt(-(-b)^(1/3)/b)*arcta
n(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) - 3*(b*x^3 + a)^(2/3)*b*x + 2*a*(-b)^
(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - a*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1
/3)*x + (b*x^3 + a)^(2/3))/x^2))/b]

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Sympy [C]  time = 1.6529, size = 37, normalized size = 0.41 \begin{align*} \frac{a^{\frac{2}{3}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{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)**(2/3),x)

[Out]

a**(2/3)*x*gamma(1/3)*hyper((-2/3, 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{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3), x)