\(\int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx\) [6]

Optimal result

Integrand size = 15, antiderivative size = 164 \[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\frac {x \sqrt [3]{a+b x^3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a x^3+b x^6}}-\frac {x \sqrt [3]{a+b x^3} \log (x)}{2 \sqrt [3]{a} \sqrt [3]{a x^3+b x^6}}+\frac {x \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a} \sqrt [3]{a x^3+b x^6}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\frac {x \sqrt [3]{a+b x^3} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )\right )}{6 \sqrt [3]{a} \sqrt [3]{x^3 \left (a+b x^3\right )}} \] Input:

Integrate[(a*x^3 + b*x^6)^(-1/3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1917, 798, 67, 16, 1082, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx\)

\(\Big \downarrow \) 1917

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \int \frac {1}{x \sqrt [3]{b x^3+a}}dx}{\sqrt [3]{a x^3+b x^6}}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3}{3 \sqrt [3]{a x^3+b x^6}}\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 \sqrt [3]{a x^3+b x^6}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 \sqrt [3]{a x^3+b x^6}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 \sqrt [3]{a x^3+b x^6}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x \sqrt [3]{a+b x^3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 \sqrt [3]{a x^3+b x^6}}\)

Input:

Int[(a*x^3 + b*x^6)^(-1/3),x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

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

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1917

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])   Int[ 
x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !Integ 
erQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

\[\int \frac {1}{\left (b \,x^{6}+a \,x^{3}\right )^{\frac {1}{3}}}d x\]

Input:

int(1/(b*x^6+a*x^3)^(1/3),x)
 

Output:

int(1/(b*x^6+a*x^3)^(1/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{5} + 3 \, a x^{2} - 3 \, {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}} a^{\frac {2}{3}} x - 3 \, \sqrt {\frac {1}{3}} {\left (a^{\frac {4}{3}} x^{2} + {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}} a x - 2 \, {\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}} a^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}}}{x^{5}}\right ) + 2 \, a^{\frac {2}{3}} \log \left (-\frac {a^{\frac {1}{3}} x - {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - a^{\frac {2}{3}} \log \left (\frac {a^{\frac {2}{3}} x^{2} + {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (a^{\frac {1}{3}} x + 2 \, {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}} x}\right ) + 2 \, a^{\frac {2}{3}} \log \left (-\frac {a^{\frac {1}{3}} x - {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - a^{\frac {2}{3}} \log \left (\frac {a^{\frac {2}{3}} x^{2} + {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}\right ] \] Input:

integrate(1/(b*x^6+a*x^3)^(1/3),x, algorithm="fricas")
 

Output:

[1/6*(3*sqrt(1/3)*a*sqrt(-1/a^(2/3))*log((2*b*x^5 + 3*a*x^2 - 3*(b*x^6 + a 
*x^3)^(1/3)*a^(2/3)*x - 3*sqrt(1/3)*(a^(4/3)*x^2 + (b*x^6 + a*x^3)^(1/3)*a 
*x - 2*(b*x^6 + a*x^3)^(2/3)*a^(2/3))*sqrt(-1/a^(2/3)))/x^5) + 2*a^(2/3)*l 
og(-(a^(1/3)*x - (b*x^6 + a*x^3)^(1/3))/x) - a^(2/3)*log((a^(2/3)*x^2 + (b 
*x^6 + a*x^3)^(1/3)*a^(1/3)*x + (b*x^6 + a*x^3)^(2/3))/x^2))/a, 1/6*(6*sqr 
t(1/3)*a^(2/3)*arctan(sqrt(1/3)*(a^(1/3)*x + 2*(b*x^6 + a*x^3)^(1/3))/(a^( 
1/3)*x)) + 2*a^(2/3)*log(-(a^(1/3)*x - (b*x^6 + a*x^3)^(1/3))/x) - a^(2/3) 
*log((a^(2/3)*x^2 + (b*x^6 + a*x^3)^(1/3)*a^(1/3)*x + (b*x^6 + a*x^3)^(2/3 
))/x^2))/a]
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\int \frac {1}{\sqrt [3]{a x^{3} + b x^{6}}}\, dx \] Input:

integrate(1/(b*x**6+a*x**3)**(1/3),x)
 

Output:

Integral((a*x**3 + b*x**6)**(-1/3), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(b*x^6+a*x^3)^(1/3),x, algorithm="maxima")
 

Output:

integrate((b*x^6 + a*x^3)^(-1/3), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(b*x^6+a*x^3)^(1/3),x, algorithm="giac")
 

Output:

integrate((b*x^6 + a*x^3)^(-1/3), x)
 

Mupad [B] (verification not implemented)

Time = 18.48 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=-\frac {x\,{\left (\frac {a}{b\,x^3}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -\frac {a}{b\,x^3}\right )}{{\left (b\,x^6+a\,x^3\right )}^{1/3}} \] Input:

int(1/(a*x^3 + b*x^6)^(1/3),x)
 

Output:

-(x*(a/(b*x^3) + 1)^(1/3)*hypergeom([1/3, 1/3], 4/3, -a/(b*x^3)))/(a*x^3 + 
 b*x^6)^(1/3)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{a x^3+b x^6}} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x}d x \] Input:

int(1/(b*x^6+a*x^3)^(1/3),x)
 

Output:

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