\(\int \frac {x^2 (A+B x)}{(a+b x^3)^2} \, dx\) [92]

Optimal result

Integrand size = 18, antiderivative size = 141 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=-\frac {A+B x}{3 b \left (a+b x^3\right )}-\frac {B \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{4/3}}+\frac {B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{4/3}}-\frac {B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}} \] Output:

-1/3*(B*x+A)/b/(b*x^3+a)-1/9*B*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^ 
(1/3))*3^(1/2)/a^(2/3)/b^(4/3)+1/9*B*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(4/3) 
-1/18*B*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(4/3)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 \sqrt [3]{b} (A+B x)}{a+b x^3}-\frac {2 \sqrt {3} B \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{18 b^{4/3}} \] Input:

Integrate[(x^2*(A + B*x))/(a + b*x^3)^2,x]
 

Output:

((-6*b^(1/3)*(A + B*x))/(a + b*x^3) - (2*Sqrt[3]*B*ArcTan[(1 - (2*b^(1/3)* 
x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*B*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - ( 
B*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^(4/3))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2363, 27, 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.

\(\displaystyle \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx\)

\(\Big \downarrow \) 2363

\(\displaystyle \frac {\int \frac {B}{b x^3+a}dx}{3 b}-\frac {A+B x}{3 b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B \int \frac {1}{b x^3+a}dx}{3 b}-\frac {A+B x}{3 b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {B \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{3 b}-\frac {A+B x}{3 b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {B \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 b}-\frac {A+B x}{3 b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

Int[(x^2*(A + B*x))/(a + b*x^3)^2,x]
 

Output:

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

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 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 2363

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Pq*(( 
a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[D[Pq, x] 
*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && E 
qQ[m - n + 1, 0] && LtQ[p, -1]
 

Maple [C] (verified)

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

Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.38

method	result	size
risch	\(\frac {-\frac {B x}{3 b}-\frac {A}{3 b}}{b \,x^{3}+a}+\frac {B \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{2}}\)	\(53\)
default	\(\frac {-\frac {B x}{3 b}-\frac {A}{3 b}}{b \,x^{3}+a}+\frac {B \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 b}\)	\(122\)

Input:

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

Output:

(-1/3*B*x/b-1/3*A/b)/(b*x^3+a)+1/9*B/b^2*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^ 
3*b+a))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.00 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=\left [-\frac {6 \, B a^{2} b x + 6 \, A a^{2} b - 3 \, \sqrt {\frac {1}{3}} {\left (B a b^{2} x^{3} + B a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) + {\left (B b x^{3} + B a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left (B b x^{3} + B a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}, -\frac {6 \, B a^{2} b x + 6 \, A a^{2} b - 6 \, \sqrt {\frac {1}{3}} {\left (B a b^{2} x^{3} + B a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + {\left (B b x^{3} + B a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left (B b x^{3} + B a\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}\right ] \] Input:

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

Output:

[-1/18*(6*B*a^2*b*x + 6*A*a^2*b - 3*sqrt(1/3)*(B*a*b^2*x^3 + B*a^2*b)*sqrt 
(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3 
)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/ 
(b*x^3 + a)) + (B*b*x^3 + B*a)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x 
 + (a^2*b)^(1/3)*a) - 2*(B*b*x^3 + B*a)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^ 
(2/3)))/(a^2*b^3*x^3 + a^3*b^2), -1/18*(6*B*a^2*b*x + 6*A*a^2*b - 6*sqrt(1 
/3)*(B*a*b^2*x^3 + B*a^2*b)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2 
*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + (B*b*x^3 + B*a 
)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 2*(B*b* 
x^3 + B*a)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^2*b^3*x^3 + a^3*b^ 
2)]
 

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.33 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=B \operatorname {RootSum} {\left (729 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log {\left (9 t a b + x \right )} \right )\right )} + \frac {- A - B x}{3 a b + 3 b^{2} x^{3}} \] Input:

integrate(x**2*(B*x+A)/(b*x**3+a)**2,x)
 

Output:

B*RootSum(729*_t**3*a**2*b**4 - 1, Lambda(_t, _t*log(9*_t*a*b + x))) + (-A 
 - B*x)/(3*a*b + 3*b**2*x**3)
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=-\frac {B x + A}{3 \, {\left (b^{2} x^{3} + a b\right )}} + \frac {\sqrt {3} B \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {B \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {B \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:

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

Output:

-1/3*(B*x + A)/(b^2*x^3 + a*b) + 1/9*sqrt(3)*B*arctan(1/3*sqrt(3)*(2*x - ( 
a/b)^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(2/3)) - 1/18*B*log(x^2 - x*(a/b)^(1/3 
) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) + 1/9*B*log(x + (a/b)^(1/3))/(b^2*(a/b) 
^(2/3))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} B \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {B \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {B \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b} - \frac {B x + A}{3 \, {\left (b x^{3} + a\right )} b} \] Input:

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

Output:

-1/9*sqrt(3)*B*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(-a*b 
^2)^(2/3) - 1/18*B*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(-a*b^2)^(2/3) 
 - 1/9*B*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) - 1/3*(B*x + A)/((b 
*x^3 + a)*b)
 

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {B\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{9\,a^{2/3}\,b^{4/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (B+\sqrt {3}\,B\,1{}\mathrm {i}\right )}{18\,a^{2/3}\,b^{4/3}}-\frac {\frac {A}{3\,b}+\frac {B\,x}{3\,b}}{b\,x^3+a}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (B-\sqrt {3}\,B\,1{}\mathrm {i}\right )}{18\,a^{2/3}\,b^{4/3}} \] Input:

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

Output:

(B*log(b^(1/3)*x + a^(1/3)))/(9*a^(2/3)*b^(4/3)) - (log(3^(1/2)*a^(1/3)*1i 
 - 2*b^(1/3)*x + a^(1/3))*(B + 3^(1/2)*B*1i))/(18*a^(2/3)*b^(4/3)) - (A/(3 
*b) + (B*x)/(3*b))/(a + b*x^3) - (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a 
^(1/3))*(B - 3^(1/2)*B*1i))/(18*a^(2/3)*b^(4/3))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.28 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {-2 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )-2 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{3}-a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )-a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{3}+2 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{3}+6 b^{\frac {1}{3}} a \,x^{3}-6 b^{\frac {1}{3}} a x}{18 b^{\frac {1}{3}} a \left (b \,x^{3}+a \right )} \] Input:

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

Output:

( - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* 
a - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* 
b*x**3 - a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a - 
a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b*x**3 + 2*a* 
*(1/3)*log(a**(1/3) + b**(1/3)*x)*a + 2*a**(1/3)*log(a**(1/3) + b**(1/3)*x 
)*b*x**3 + 6*b**(1/3)*a*x**3 - 6*b**(1/3)*a*x)/(18*b**(1/3)*a*(a + b*x**3) 
)