\(\int \frac {a+b x^3+c x^6}{(d+e x^3)^3} \, dx\) [8]

Optimal result

Integrand size = 22, antiderivative size = 242 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=\frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{18 d^2 e^2 \left (d+e x^3\right )}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{9 \sqrt {3} d^{8/3} e^{7/3}}+\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{27 d^{8/3} e^{7/3}}-\frac {\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}} \] Output:

1/6*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)^2-1/18*(7*c*d^2-e*(5*a*e+b*d))*x 
/d^2/e^2/(e*x^3+d)-1/27*(2*c*d^2+e*(5*a*e+b*d))*arctan(1/3*(d^(1/3)-2*e^(1 
/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(8/3)/e^(7/3)+1/27*(2*c*d^2+e*(5*a*e+b*d 
))*ln(d^(1/3)+e^(1/3)*x)/d^(8/3)/e^(7/3)-1/54*(2*c*d^2+e*(5*a*e+b*d))*ln(d 
^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(8/3)/e^(7/3)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=\frac {-\frac {3 d^{2/3} \sqrt [3]{e} x \left (c d^2 \left (4 d+7 e x^3\right )-e \left (b d \left (-2 d+e x^3\right )+a e \left (8 d+5 e x^3\right )\right )\right )}{\left (d+e x^3\right )^2}-2 \sqrt {3} \left (2 c d^2+e (b d+5 a e)\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )+2 \left (2 c d^2+e (b d+5 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-\left (2 c d^2+e (b d+5 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{54 d^{8/3} e^{7/3}} \] Input:

Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^3,x]
 

Output:

((-3*d^(2/3)*e^(1/3)*x*(c*d^2*(4*d + 7*e*x^3) - e*(b*d*(-2*d + e*x^3) + a* 
e*(8*d + 5*e*x^3))))/(d + e*x^3)^2 - 2*Sqrt[3]*(2*c*d^2 + e*(b*d + 5*a*e)) 
*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(2*c*d^2 + e*(b*d + 5*a*e 
))*Log[d^(1/3) + e^(1/3)*x] - (2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(2/3) - d^ 
(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(54*d^(8/3)*e^(7/3))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1739, 910, 750, 16, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx\)

\(\Big \downarrow \) 1739

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\int \frac {-6 c d e x^3+c d^2-e (b d+5 a e)}{\left (e x^3+d\right )^2}dx}{6 d e^2}\)

\(\Big \downarrow \) 910

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \int \frac {1}{e x^3+d}dx}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{e} x+\sqrt [3]{d}}dx}{3 d^{2/3}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {\int \frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac {\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{3 d \left (d+e x^3\right )}-\frac {2 \left (e (5 a e+b d)+2 c d^2\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{3 d}}{6 d e^2}\)

Input:

Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^3,x]
 

Output:

((c*d^2 - b*d*e + a*e^2)*x)/(6*d*e^2*(d + e*x^3)^2) - (((7*c*d^2 - e*(b*d 
+ 5*a*e))*x)/(3*d*(d + e*x^3)) - (2*(2*c*d^2 + e*(b*d + 5*a*e))*(Log[d^(1/ 
3) + e^(1/3)*x]/(3*d^(2/3)*e^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*e^(1/3)*x 
)/d^(1/3))/Sqrt[3]])/e^(1/3)) - Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)* 
x^2]/(2*e^(1/3)))/(3*d^(2/3))))/(3*d))/(6*d*e^2)
 

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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 750

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

rule 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 0])
 

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 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1739

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Simp[(-(c*d^2 - b*d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d* 
e^2*n*(q + 1))), x] + Simp[1/(n*(q + 1)*d*e^2)   Int[(d + e*x^n)^(q + 1)*Si 
mp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] 
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && N 
eQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.47

method	result	size
risch	\(\frac {\frac {\left (5 a \,e^{2}+b d e -7 c \,d^{2}\right ) x^{4}}{18 d^{2} e}+\frac {\left (4 a \,e^{2}-b d e -2 c \,d^{2}\right ) x}{9 d \,e^{2}}}{\left (e \,x^{3}+d \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (5 a \,e^{2}+b d e +2 c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 d^{2} e^{3}}\)	\(114\)
default	\(\frac {\frac {\left (5 a \,e^{2}+b d e -7 c \,d^{2}\right ) x^{4}}{18 d^{2} e}+\frac {\left (4 a \,e^{2}-b d e -2 c \,d^{2}\right ) x}{9 d \,e^{2}}}{\left (e \,x^{3}+d \right )^{2}}+\frac {\left (5 a \,e^{2}+b d e +2 c \,d^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )}{9 d^{2} e^{2}}\)	\(183\)

Input:

int((c*x^6+b*x^3+a)/(e*x^3+d)^3,x,method=_RETURNVERBOSE)
 

Output:

(1/18*(5*a*e^2+b*d*e-7*c*d^2)/d^2/e*x^4+1/9*(4*a*e^2-b*d*e-2*c*d^2)/d/e^2* 
x)/(e*x^3+d)^2+1/27/d^2/e^3*sum((5*a*e^2+b*d*e+2*c*d^2)/_R^2*ln(x-_R),_R=R 
ootOf(_Z^3*e+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (201) = 402\).

Time = 0.09 (sec) , antiderivative size = 941, normalized size of antiderivative = 3.89 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/54*(3*(7*c*d^4*e^2 - b*d^3*e^3 - 5*a*d^2*e^4)*x^4 - 3*sqrt(1/3)*(2*c*d 
^5*e + b*d^4*e^2 + 5*a*d^3*e^3 + (2*c*d^3*e^3 + b*d^2*e^4 + 5*a*d*e^5)*x^6 
 + 2*(2*c*d^4*e^2 + b*d^3*e^3 + 5*a*d^2*e^4)*x^3)*sqrt(-(d^2*e)^(1/3)/e)*l 
og((2*d*e*x^3 - 3*(d^2*e)^(1/3)*d*x - d^2 + 3*sqrt(1/3)*(2*d*e*x^2 + (d^2* 
e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt(-(d^2*e)^(1/3)/e))/(e*x^3 + d)) + ((2*c 
*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2 
*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*(d^2*e)^(2/3)*log(d*e*x^2 - (d^2*e) 
^(2/3)*x + (d^2*e)^(1/3)*d) - 2*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2 
*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3 
)*(d^2*e)^(2/3)*log(d*e*x + (d^2*e)^(2/3)) + 6*(2*c*d^5*e + b*d^4*e^2 - 4* 
a*d^3*e^3)*x)/(d^4*e^5*x^6 + 2*d^5*e^4*x^3 + d^6*e^3), -1/54*(3*(7*c*d^4*e 
^2 - b*d^3*e^3 - 5*a*d^2*e^4)*x^4 - 6*sqrt(1/3)*(2*c*d^5*e + b*d^4*e^2 + 5 
*a*d^3*e^3 + (2*c*d^3*e^3 + b*d^2*e^4 + 5*a*d*e^5)*x^6 + 2*(2*c*d^4*e^2 + 
b*d^3*e^3 + 5*a*d^2*e^4)*x^3)*sqrt((d^2*e)^(1/3)/e)*arctan(sqrt(1/3)*(2*(d 
^2*e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt((d^2*e)^(1/3)/e)/d^2) + ((2*c*d^2*e^ 
2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3* 
e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*(d^2*e)^(2/3)*log(d*e*x^2 - (d^2*e)^(2/3)* 
x + (d^2*e)^(1/3)*d) - 2*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 
+ b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*(d^2* 
e)^(2/3)*log(d*e*x + (d^2*e)^(2/3)) + 6*(2*c*d^5*e + b*d^4*e^2 - 4*a*d^...
 

Sympy [A] (verification not implemented)

Time = 2.37 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=\frac {x^{4} \cdot \left (5 a e^{3} + b d e^{2} - 7 c d^{2} e\right ) + x \left (8 a d e^{2} - 2 b d^{2} e - 4 c d^{3}\right )}{18 d^{4} e^{2} + 36 d^{3} e^{3} x^{3} + 18 d^{2} e^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} d^{8} e^{7} - 125 a^{3} e^{6} - 75 a^{2} b d e^{5} - 150 a^{2} c d^{2} e^{4} - 15 a b^{2} d^{2} e^{4} - 60 a b c d^{3} e^{3} - 60 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 12 b c^{2} d^{5} e - 8 c^{3} d^{6}, \left ( t \mapsto t \log {\left (\frac {27 t d^{3} e^{2}}{5 a e^{2} + b d e + 2 c d^{2}} + x \right )} \right )\right )} \] Input:

integrate((c*x**6+b*x**3+a)/(e*x**3+d)**3,x)
                                                                                    
                                                                                    
 

Output:

(x**4*(5*a*e**3 + b*d*e**2 - 7*c*d**2*e) + x*(8*a*d*e**2 - 2*b*d**2*e - 4* 
c*d**3))/(18*d**4*e**2 + 36*d**3*e**3*x**3 + 18*d**2*e**4*x**6) + RootSum( 
19683*_t**3*d**8*e**7 - 125*a**3*e**6 - 75*a**2*b*d*e**5 - 150*a**2*c*d**2 
*e**4 - 15*a*b**2*d**2*e**4 - 60*a*b*c*d**3*e**3 - 60*a*c**2*d**4*e**2 - b 
**3*d**3*e**3 - 6*b**2*c*d**4*e**2 - 12*b*c**2*d**5*e - 8*c**3*d**6, Lambd 
a(_t, _t*log(27*_t*d**3*e**2/(5*a*e**2 + b*d*e + 2*c*d**2) + x)))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2} e} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{54 \, \left (-d e^{2}\right )^{\frac {2}{3}} d^{2} e} - \frac {{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{27 \, d^{3} e^{2}} - \frac {7 \, c d^{2} e x^{4} - b d e^{2} x^{4} - 5 \, a e^{3} x^{4} + 4 \, c d^{3} x + 2 \, b d^{2} e x - 8 \, a d e^{2} x}{18 \, {\left (e x^{3} + d\right )}^{2} d^{2} e^{2}} \] Input:

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

Output:

-1/27*sqrt(3)*(2*c*d^2 + b*d*e + 5*a*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d/e) 
^(1/3))/(-d/e)^(1/3))/((-d*e^2)^(2/3)*d^2*e) - 1/54*(2*c*d^2 + b*d*e + 5*a 
*e^2)*log(x^2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/((-d*e^2)^(2/3)*d^2*e) - 1/ 
27*(2*c*d^2 + b*d*e + 5*a*e^2)*(-d/e)^(1/3)*log(abs(x - (-d/e)^(1/3)))/(d^ 
3*e^2) - 1/18*(7*c*d^2*e*x^4 - b*d*e^2*x^4 - 5*a*e^3*x^4 + 4*c*d^3*x + 2*b 
*d^2*e*x - 8*a*d*e^2*x)/((e*x^3 + d)^2*d^2*e^2)
 

Mupad [B] (verification not implemented)

Time = 11.41 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx=\frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\frac {x\,\left (2\,c\,d^2+b\,d\,e-4\,a\,e^2\right )}{9\,d\,e^2}-\frac {x^4\,\left (-7\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{18\,d^2\,e}}{d^2+2\,d\,e\,x^3+e^2\,x^6}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,c\,d^2+b\,d\,e+5\,a\,e^2\right )}{27\,d^{8/3}\,e^{7/3}} \] Input:

int((a + b*x^3 + c*x^6)/(d + e*x^3)^3,x)
 

Output:

(log(e^(1/3)*x + d^(1/3))*(5*a*e^2 + 2*c*d^2 + b*d*e))/(27*d^(8/3)*e^(7/3) 
) - ((x*(2*c*d^2 - 4*a*e^2 + b*d*e))/(9*d*e^2) - (x^4*(5*a*e^2 - 7*c*d^2 + 
 b*d*e))/(18*d^2*e))/(d^2 + e^2*x^6 + 2*d*e*x^3) + (log(3^(1/2)*d^(1/3)*1i 
 + 2*e^(1/3)*x - d^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(5*a*e^2 + 2*c*d^2 + b*d* 
e))/(27*d^(8/3)*e^(7/3)) - (log(3^(1/2)*d^(1/3)*1i - 2*e^(1/3)*x + d^(1/3) 
)*((3^(1/2)*1i)/2 + 1/2)*(5*a*e^2 + 2*c*d^2 + b*d*e))/(27*d^(8/3)*e^(7/3))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.46 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx =\text {Too large to display} \] Input:

int((c*x^6+b*x^3+a)/(e*x^3+d)^3,x)
 

Output:

( - 10*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3))) 
*a*d**2*e**2 - 20*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3 
)*sqrt(3)))*a*d*e**3*x**3 - 10*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3 
)*x)/(d**(1/3)*sqrt(3)))*a*e**4*x**6 - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 
 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*b*d**3*e - 4*d**(1/3)*sqrt(3)*atan((d** 
(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*b*d**2*e**2*x**3 - 2*d**(1/3)*sq 
rt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*b*d*e**3*x**6 - 4 
*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*c*d** 
4 - 8*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))* 
c*d**3*e*x**3 - 4*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3 
)*sqrt(3)))*c*d**2*e**2*x**6 - 5*d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3) 
*x + e**(2/3)*x**2)*a*d**2*e**2 - 10*d**(1/3)*log(d**(2/3) - e**(1/3)*d**( 
1/3)*x + e**(2/3)*x**2)*a*d*e**3*x**3 - 5*d**(1/3)*log(d**(2/3) - e**(1/3) 
*d**(1/3)*x + e**(2/3)*x**2)*a*e**4*x**6 - d**(1/3)*log(d**(2/3) - e**(1/3 
)*d**(1/3)*x + e**(2/3)*x**2)*b*d**3*e - 2*d**(1/3)*log(d**(2/3) - e**(1/3 
)*d**(1/3)*x + e**(2/3)*x**2)*b*d**2*e**2*x**3 - d**(1/3)*log(d**(2/3) - e 
**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*b*d*e**3*x**6 - 2*d**(1/3)*log(d**(2/3 
) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*c*d**4 - 4*d**(1/3)*log(d**(2/3) 
- e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*c*d**3*e*x**3 - 2*d**(1/3)*log(d**( 
2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*c*d**2*e**2*x**6 + 10*d**(1...