\(\int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} (e+f x^3)} \, dx\) [3]

Optimal result

Integrand size = 38, antiderivative size = 467 \[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=-\frac {(3 b D e-3 b C f+2 a D f) x \left (a+b x^3\right )^{2/3}}{9 b^2 f^2}+\frac {D x^4 \left (a+b x^3\right )^{2/3}}{6 b f}+\frac {\left (2 a^2 D f^2+3 a b f (D e-C f)+9 b^2 \left (D e^2-f (C e-B f)\right )\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} f^3}-\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b e-a f} x}{\sqrt [3]{e} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} e^{2/3} f^3 \sqrt [3]{b e-a f}}-\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \log \left (e+f x^3\right )}{6 e^{2/3} f^3 \sqrt [3]{b e-a f}}+\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \log \left (\frac {\sqrt [3]{b e-a f} x}{\sqrt [3]{e}}-\sqrt [3]{a+b x^3}\right )}{2 e^{2/3} f^3 \sqrt [3]{b e-a f}}-\frac {\left (2 a^2 D f^2+3 a b f (D e-C f)+9 b^2 \left (D e^2-f (C e-B f)\right )\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{7/3} f^3} \] Output:

-1/9*(-3*C*b*f+2*D*a*f+3*D*b*e)*x*(b*x^3+a)^(2/3)/b^2/f^2+1/6*D*x^4*(b*x^3 
+a)^(2/3)/b/f+1/27*(2*a^2*D*f^2+3*a*b*f*(-C*f+D*e)+9*b^2*(D*e^2-f*(-B*f+C* 
e)))*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(7/3)/f 
^3-1/3*(D*e^3-f*(C*e^2-f*(-A*f+B*e)))*arctan(1/3*(1+2*(-a*f+b*e)^(1/3)*x/e 
^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/e^(2/3)/f^3/(-a*f+b*e)^(1/3)-1/6* 
(D*e^3-f*(C*e^2-f*(-A*f+B*e)))*ln(f*x^3+e)/e^(2/3)/f^3/(-a*f+b*e)^(1/3)+1/ 
2*(D*e^3-f*(C*e^2-f*(-A*f+B*e)))*ln((-a*f+b*e)^(1/3)*x/e^(1/3)-(b*x^3+a)^( 
1/3))/e^(2/3)/f^3/(-a*f+b*e)^(1/3)-1/18*(2*a^2*D*f^2+3*a*b*f*(-C*f+D*e)+9* 
b^2*(D*e^2-f*(-B*f+C*e)))*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(7/3)/f^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.59 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.72 \[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\frac {-\frac {12 a D f^2 x \left (a+b x^3\right )^{2/3}}{b^2}+\frac {18 f (-D e+C f) x \left (a+b x^3\right )^{2/3}}{b}+\frac {9 D f^2 x^4 \left (a+b x^3\right )^{2/3}}{b}+\frac {4 \sqrt {3} a^2 D f^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{b^{7/3}}+\frac {6 \sqrt {3} a f (D e-C f) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{b^{4/3}}+\frac {18 \sqrt {3} \left (D e^2+f (-C e+B f)\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {18 \sqrt {3} \left (D e^3-f \left (C e^2+f (-B e+A f)\right )\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b e-a f} x}{\sqrt [3]{e} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{e^{2/3} \sqrt [3]{b e-a f}}-\frac {4 a^2 D f^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{7/3}}+\frac {6 a f (-D e+C f) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{4/3}}-\frac {18 \left (D e^2+f (-C e+B f)\right ) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}+\frac {2 a^2 D f^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{7/3}}+\frac {3 a f (D e-C f) \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{4/3}}+\frac {9 \left (D e^2+f (-C e+B f)\right ) \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}+\frac {18 \left (D e^3-f \left (C e^2+f (-B e+A f)\right )\right ) \log \left (\sqrt [3]{e}-\frac {\sqrt [3]{b e-a f} x}{\sqrt [3]{a+b x^3}}\right )}{e^{2/3} \sqrt [3]{b e-a f}}-\frac {9 \left (D e^3-f \left (C e^2+f (-B e+A f)\right )\right ) \log \left (e^{2/3}+\frac {(b e-a f)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{e} \sqrt [3]{b e-a f} x}{\sqrt [3]{a+b x^3}}\right )}{e^{2/3} \sqrt [3]{b e-a f}}}{54 f^3} \] Input:

Integrate[(A + B*x^3 + C*x^6 + D*x^9)/((a + b*x^3)^(1/3)*(e + f*x^3)),x]
 

Output:

((-12*a*D*f^2*x*(a + b*x^3)^(2/3))/b^2 + (18*f*(-(D*e) + C*f)*x*(a + b*x^3 
)^(2/3))/b + (9*D*f^2*x^4*(a + b*x^3)^(2/3))/b + (4*Sqrt[3]*a^2*D*f^2*ArcT 
an[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/b^(7/3) + (6*Sqrt[3]*a* 
f*(D*e - C*f)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/b^(4/ 
3) + (18*Sqrt[3]*(D*e^2 + f*(-(C*e) + B*f))*ArcTan[(1 + (2*b^(1/3)*x)/(a + 
 b*x^3)^(1/3))/Sqrt[3]])/b^(1/3) - (18*Sqrt[3]*(D*e^3 - f*(C*e^2 + f*(-(B* 
e) + A*f)))*ArcTan[(1 + (2*(b*e - a*f)^(1/3)*x)/(e^(1/3)*(a + b*x^3)^(1/3) 
))/Sqrt[3]])/(e^(2/3)*(b*e - a*f)^(1/3)) - (4*a^2*D*f^2*Log[1 - (b^(1/3)*x 
)/(a + b*x^3)^(1/3)])/b^(7/3) + (6*a*f*(-(D*e) + C*f)*Log[1 - (b^(1/3)*x)/ 
(a + b*x^3)^(1/3)])/b^(4/3) - (18*(D*e^2 + f*(-(C*e) + B*f))*Log[1 - (b^(1 
/3)*x)/(a + b*x^3)^(1/3)])/b^(1/3) + (2*a^2*D*f^2*Log[1 + (b^(2/3)*x^2)/(a 
 + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/b^(7/3) + (3*a*f*(D*e - 
C*f)*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/ 
3)])/b^(4/3) + (9*(D*e^2 + f*(-(C*e) + B*f))*Log[1 + (b^(2/3)*x^2)/(a + b* 
x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/b^(1/3) + (18*(D*e^3 - f*(C*e 
^2 + f*(-(B*e) + A*f)))*Log[e^(1/3) - ((b*e - a*f)^(1/3)*x)/(a + b*x^3)^(1 
/3)])/(e^(2/3)*(b*e - a*f)^(1/3)) - (9*(D*e^3 - f*(C*e^2 + f*(-(B*e) + A*f 
)))*Log[e^(2/3) + ((b*e - a*f)^(2/3)*x^2)/(a + b*x^3)^(2/3) + (e^(1/3)*(b* 
e - a*f)^(1/3)*x)/(a + b*x^3)^(1/3)])/(e^(2/3)*(b*e - a*f)^(1/3)))/(54*f^3 
)
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {A f^3-B e f^2+C e^2 f-D e^3}{f^3 \sqrt [3]{a+b x^3} \left (e+f x^3\right )}+\frac {B f^2-C e f+D e^2}{f^3 \sqrt [3]{a+b x^3}}-\frac {x^3 (D e-C f)}{f^2 \sqrt [3]{a+b x^3}}+\frac {D x^6}{f \sqrt [3]{a+b x^3}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^2 D \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} f}-\frac {a^2 D \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{9 b^{7/3} f}-\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b e-a f}}{\sqrt [3]{e} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right )}{\sqrt {3} e^{2/3} f^3 \sqrt [3]{b e-a f}}-\frac {\log \left (e+f x^3\right ) \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right )}{6 e^{2/3} f^3 \sqrt [3]{b e-a f}}+\frac {\log \left (\frac {x \sqrt [3]{b e-a f}}{\sqrt [3]{e}}-\sqrt [3]{a+b x^3}\right ) \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right )}{2 e^{2/3} f^3 \sqrt [3]{b e-a f}}+\frac {a \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (D e-C f)}{3 \sqrt {3} b^{4/3} f^2}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (B f^2-C e f+D e^2\right )}{\sqrt {3} \sqrt [3]{b} f^3}-\frac {a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right ) (D e-C f)}{6 b^{4/3} f^2}-\frac {2 a D x \left (a+b x^3\right )^{2/3}}{9 b^2 f}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right ) \left (B f^2-C e f+D e^2\right )}{2 \sqrt [3]{b} f^3}-\frac {x \left (a+b x^3\right )^{2/3} (D e-C f)}{3 b f^2}+\frac {D x^4 \left (a+b x^3\right )^{2/3}}{6 b f}\)

Input:

Int[(A + B*x^3 + C*x^6 + D*x^9)/((a + b*x^3)^(1/3)*(e + f*x^3)),x]
 

Output:

(-2*a*D*x*(a + b*x^3)^(2/3))/(9*b^2*f) - ((D*e - C*f)*x*(a + b*x^3)^(2/3)) 
/(3*b*f^2) + (D*x^4*(a + b*x^3)^(2/3))/(6*b*f) + (2*a^2*D*ArcTan[(1 + (2*b 
^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(7/3)*f) + (a*(D*e - C 
*f)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(4 
/3)*f^2) + ((D*e^2 - C*e*f + B*f^2)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^ 
(1/3))/Sqrt[3]])/(Sqrt[3]*b^(1/3)*f^3) - ((D*e^3 - f*(C*e^2 - f*(B*e - A*f 
)))*ArcTan[(1 + (2*(b*e - a*f)^(1/3)*x)/(e^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[ 
3]])/(Sqrt[3]*e^(2/3)*f^3*(b*e - a*f)^(1/3)) - ((D*e^3 - f*(C*e^2 - f*(B*e 
 - A*f)))*Log[e + f*x^3])/(6*e^(2/3)*f^3*(b*e - a*f)^(1/3)) + ((D*e^3 - f* 
(C*e^2 - f*(B*e - A*f)))*Log[((b*e - a*f)^(1/3)*x)/e^(1/3) - (a + b*x^3)^( 
1/3)])/(2*e^(2/3)*f^3*(b*e - a*f)^(1/3)) - (a^2*D*Log[-(b^(1/3)*x) + (a + 
b*x^3)^(1/3)])/(9*b^(7/3)*f) - (a*(D*e - C*f)*Log[-(b^(1/3)*x) + (a + b*x^ 
3)^(1/3)])/(6*b^(4/3)*f^2) - ((D*e^2 - C*e*f + B*f^2)*Log[-(b^(1/3)*x) + ( 
a + b*x^3)^(1/3)])/(2*b^(1/3)*f^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 3.43 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {-e \left (-\left (b \,x^{3}+a \right )^{\frac {2}{3}} f \left (-\frac {2 D f \,b^{\frac {7}{3}} a}{3}+b^{\frac {10}{3}} \left (-D e +f \left (\frac {D x^{3}}{2}+C \right )\right )\right ) x +\left (\arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) b^{2} \left (\left (B \,b^{2}-\frac {1}{3} C a b +\frac {2}{9} D a^{2}\right ) f^{2}-b e \left (C b -\frac {D a}{3}\right ) f +D b^{2} e^{2}\right )\right ) \left (\frac {a f -b e}{e}\right )^{\frac {1}{3}}+\left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{e}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right )+\ln \left (\frac {\left (\frac {a f -b e}{e}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {\left (\frac {a f -b e}{e}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a f -b e}{e}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (A \,f^{3}-B e \,f^{2}+C \,e^{2} f -D e^{3}\right ) b^{\frac {13}{3}}}{3 b^{\frac {13}{3}} \left (\frac {a f -b e}{e}\right )^{\frac {1}{3}} f^{3} e}\)	\(384\)

Input:

int((D*x^9+C*x^6+B*x^3+A)/(b*x^3+a)^(1/3)/(f*x^3+e),x,method=_RETURNVERBOS 
E)
 

Output:

1/3*(-e*(-(b*x^3+a)^(2/3)*f*(-2/3*D*f*b^(7/3)*a+b^(10/3)*(-D*e+f*(1/2*D*x^ 
3+C)))*x+(arctan(1/3*3^(1/2)*(2/b^(1/3)*(b*x^3+a)^(1/3)+x)/x)*3^(1/2)+ln(( 
-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-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))*b^2*((B*b^2-1/3*C*a*b+2/9*D*a^2)*f^2-b*e*(C*b-1/ 
3*D*a)*f+D*b^2*e^2))*((a*f-b*e)/e)^(1/3)+(3^(1/2)*arctan(1/3*3^(1/2)*(-2/( 
(a*f-b*e)/e)^(1/3)*(b*x^3+a)^(1/3)+x)/x)+ln((((a*f-b*e)/e)^(1/3)*x+(b*x^3+ 
a)^(1/3))/x)-1/2*ln((((a*f-b*e)/e)^(2/3)*x^2-((a*f-b*e)/e)^(1/3)*(b*x^3+a) 
^(1/3)*x+(b*x^3+a)^(2/3))/x^2))*(A*f^3-B*e*f^2+C*e^2*f-D*e^3)*b^(13/3))/b^ 
(13/3)/((a*f-b*e)/e)^(1/3)/f^3/e
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((D*x^9+C*x^6+B*x^3+A)/(b*x^3+a)^(1/3)/(f*x^3+e),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\int \frac {A + B x^{3} + C x^{6} + D x^{9}}{\sqrt [3]{a + b x^{3}} \left (e + f x^{3}\right )}\, dx \] Input:

integrate((D*x**9+C*x**6+B*x**3+A)/(b*x**3+a)**(1/3)/(f*x**3+e),x)
 

Output:

Integral((A + B*x**3 + C*x**6 + D*x**9)/((a + b*x**3)**(1/3)*(e + f*x**3)) 
, x)
 

Maxima [F]

\[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\int { \frac {D x^{9} + C x^{6} + B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (f x^{3} + e\right )}} \,d x } \] Input:

integrate((D*x^9+C*x^6+B*x^3+A)/(b*x^3+a)^(1/3)/(f*x^3+e),x, algorithm="ma 
xima")
 

Output:

integrate((D*x^9 + C*x^6 + B*x^3 + A)/((b*x^3 + a)^(1/3)*(f*x^3 + e)), x)
 

Giac [F]

\[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\int { \frac {D x^{9} + C x^{6} + B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (f x^{3} + e\right )}} \,d x } \] Input:

integrate((D*x^9+C*x^6+B*x^3+A)/(b*x^3+a)^(1/3)/(f*x^3+e),x, algorithm="gi 
ac")
 

Output:

integrate((D*x^9 + C*x^6 + B*x^3 + A)/((b*x^3 + a)^(1/3)*(f*x^3 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\int \frac {A+B\,x^3+C\,x^6+x^9\,D}{{\left (b\,x^3+a\right )}^{1/3}\,\left (f\,x^3+e\right )} \,d x \] Input:

int((A + B*x^3 + C*x^6 + x^9*D)/((a + b*x^3)^(1/3)*(e + f*x^3)),x)
 

Output:

int((A + B*x^3 + C*x^6 + x^9*D)/((a + b*x^3)^(1/3)*(e + f*x^3)), x)
 

Reduce [F]

\[ \int \frac {A+B x^3+C x^6+D x^9}{\sqrt [3]{a+b x^3} \left (e+f x^3\right )} \, dx=\left (\int \frac {x^{9}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} e +\left (b \,x^{3}+a \right )^{\frac {1}{3}} f \,x^{3}}d x \right ) d +\left (\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} e +\left (b \,x^{3}+a \right )^{\frac {1}{3}} f \,x^{3}}d x \right ) c +\left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} e +\left (b \,x^{3}+a \right )^{\frac {1}{3}} f \,x^{3}}d x \right ) b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} e +\left (b \,x^{3}+a \right )^{\frac {1}{3}} f \,x^{3}}d x \right ) a \] Input:

int((D*x^9+C*x^6+B*x^3+A)/(b*x^3+a)^(1/3)/(f*x^3+e),x)
 

Output:

int(x**9/((a + b*x**3)**(1/3)*e + (a + b*x**3)**(1/3)*f*x**3),x)*d + int(x 
**6/((a + b*x**3)**(1/3)*e + (a + b*x**3)**(1/3)*f*x**3),x)*c + int(x**3/( 
(a + b*x**3)**(1/3)*e + (a + b*x**3)**(1/3)*f*x**3),x)*b + int(1/((a + b*x 
**3)**(1/3)*e + (a + b*x**3)**(1/3)*f*x**3),x)*a