\(\int (a+b x^3)^{2/3} \, dx\) [538]

Optimal result

Integrand size = 11, antiderivative size = 91 \[ \int \left (a+b x^3\right )^{2/3} \, dx=\frac {1}{3} x \left (a+b x^3\right )^{2/3}+\frac {2 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}-\frac {a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \]

[Out]

1/3*x*(b*x^3+a)^(2/3)-1/3*a*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)+2/9*a*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(
1/3))*3^(1/2))/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 245} \[ \int \left (a+b x^3\right )^{2/3} \, dx=\frac {2 a \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {1}{3} x \left (a+b x^3\right )^{2/3}-\frac {a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}} \]

[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 201

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 245

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*} \text {integral}& = \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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.16 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.23 \[ \int \left (a+b x^3\right )^{2/3} \, dx=\frac {3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a+b x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},-\frac {2}{3},\frac {8}{3},-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}},\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}}{3 i+\sqrt {3}}\right )}{5\ 2^{2/3} \sqrt [3]{b} \left (\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \left (\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}\right )^{2/3}} \]

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

Maple [A] (verified)

Time = 4.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {2 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a -\frac {3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} x \,b^{\frac {1}{3}}}{2}+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a -\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a}{2}\right )}{9 b^{\frac {1}{3}}}\)	\(118\)

[In]

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

[Out]

-2/9/b^(1/3)*(3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)*a-3/2*(b*x^3+a)^(2/3)*x*b^(1
/3)+ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a-1/2*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a
)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (68) = 136\).

Time = 0.26 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.98 \[ \int \left (a+b x^3\right )^{2/3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x - 2 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b}, -\frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 2 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b}\right ] \]

[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]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.41 \[ \int \left (a+b x^3\right )^{2/3} \, dx=\frac {a^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int \left (a+b x^3\right )^{2/3} \, dx=-\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{9 \, b^{\frac {1}{3}}} + \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b^{\frac {1}{3}}} - \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{9 \, b^{\frac {1}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{3 \, {\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{2}} \]

[In]

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

[Out]

-2/9*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(1/3) + 1/9*a*log(b^(2/3) + (b*
x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(1/3) - 2/9*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(1/3)
- 1/3*(b*x^3 + a)^(2/3)*a/((b - (b*x^3 + a)/x^3)*x^2)

Giac [F]

\[ \int \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} \,d x } \]

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3), x)

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.41 \[ \int \left (a+b x^3\right )^{2/3} \, dx=\frac {x\,{\left (b\,x^3+a\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (\frac {b\,x^3}{a}+1\right )}^{2/3}} \]

[In]

int((a + b*x^3)^(2/3),x)

[Out]

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