\(\int \frac {A+B x^3}{x^2 (a+b x^3)^3} \, dx\) [106]

Optimal result

Integrand size = 20, antiderivative size = 208 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=-\frac {A}{a^3 x}-\frac {(A b-a B) x^2}{6 a^2 \left (a+b x^3\right )^2}-\frac {(5 A b-2 a B) x^2}{9 a^3 \left (a+b x^3\right )}+\frac {2 (7 A b-a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}-\frac {(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}} \] Output:

-A/a^3/x-1/6*(A*b-B*a)*x^2/a^2/(b*x^3+a)^2-1/9*(5*A*b-2*B*a)*x^2/a^3/(b*x^ 
3+a)+2/27*(7*A*b-B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^ 
(1/2)/a^(10/3)/b^(2/3)+2/27*(7*A*b-B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^( 
2/3)-1/27*(7*A*b-B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/b 
^(2/3)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {54 \sqrt [3]{a} A}{x}+\frac {9 a^{4/3} (-A b+a B) x^2}{\left (a+b x^3\right )^2}+\frac {6 \sqrt [3]{a} (-5 A b+2 a B) x^2}{a+b x^3}+\frac {4 \sqrt {3} (7 A b-a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {4 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {2 (-7 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{54 a^{10/3}} \] Input:

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

Output:

((-54*a^(1/3)*A)/x + (9*a^(4/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (6*a^( 
1/3)*(-5*A*b + 2*a*B)*x^2)/(a + b*x^3) + (4*Sqrt[3]*(7*A*b - a*B)*ArcTan[( 
1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(7*A*b - a*B)*Log[a^(1/3 
) + b^(1/3)*x])/b^(2/3) + (2*(-7*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)* 
x + b^(2/3)*x^2])/b^(2/3))/(54*a^(10/3))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {957, 819, 847, 821, 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.

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

\(\Big \downarrow \) 957

\(\displaystyle \frac {(7 A b-a B) \int \frac {1}{x^2 \left (b x^3+a\right )^2}dx}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \int \frac {1}{x^2 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \int \frac {x}{b x^3+a}dx}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 821

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(7 A b-a B) \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a b}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(A*b - a*B)/(6*a*b*x*(a + b*x^3)^2) + ((7*A*b - a*B)*(1/(3*a*x*(a + b*x^3) 
) + (4*(-(1/(a*x)) - (b*(-1/3*Log[a^(1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + 
 (-((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^(1/3)*b^(1/3)) 
))/a))/(3*a)))/(6*a*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 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(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 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

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]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {A}{a^{3} x}-\frac {\frac {\left (\frac {5}{9} b^{2} A -\frac {2}{9} a b B \right ) x^{5}+\frac {a \left (13 A b -7 B a \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {14 A b}{9}-\frac {2 B a}{9}\right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}\)	\(159\)
risch	\(\frac {-\frac {2 b \left (7 A b -B a \right ) x^{6}}{9 a^{3}}-\frac {7 \left (7 A b -B a \right ) x^{3}}{18 a^{2}}-\frac {A}{a}}{x \left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} b^{2} \textit {\_Z}^{3}-343 A^{3} b^{3}+147 A^{2} B a \,b^{2}-21 A \,B^{2} a^{2} b +B^{3} a^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b^{2}+1029 A^{3} b^{3}-441 A^{2} B a \,b^{2}+63 A \,B^{2} a^{2} b -3 B^{3} a^{3}\right ) x +\left (-7 A \,a^{7} b^{2}+B \,a^{8} b \right ) \textit {\_R}^{2}\right )\right )}{27}\)	\(183\)

Input:

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

Output:

-A/a^3/x-1/a^3*(((5/9*b^2*A-2/9*a*b*B)*x^5+1/18*a*(13*A*b-7*B*a)*x^2)/(b*x 
^3+a)^2+(14/9*A*b-2/9*B*a)*(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/ 
b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arcta 
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (164) = 328\).

Time = 0.17 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.73 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/54*(12*(B*a^2*b^3 - 7*A*a*b^4)*x^6 - 54*A*a^3*b^2 + 21*(B*a^3*b^2 - 7*A 
*a^2*b^3)*x^3 - 6*sqrt(1/3)*((B*a^2*b^3 - 7*A*a*b^4)*x^7 + 2*(B*a^3*b^2 - 
7*A*a^2*b^3)*x^4 + (B*a^4*b - 7*A*a^3*b^2)*x)*sqrt(-(a*b^2)^(1/3)/a)*log(( 
2*b^2*x^3 - a*b - 3*sqrt(1/3)*(a*b*x + 2*(a*b^2)^(2/3)*x^2 - (a*b^2)^(1/3) 
*a)*sqrt(-(a*b^2)^(1/3)/a) - 3*(a*b^2)^(2/3)*x)/(b*x^3 + a)) + 2*((B*a*b^2 
 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*b)*x)*(a* 
b^2)^(2/3)*log(b^2*x^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) - 4*((B*a*b^2 
- 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*b)*x)*(a*b 
^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^4*b^4*x^7 + 2*a^5*b^3*x^4 + a^6*b^2 
*x), 1/54*(12*(B*a^2*b^3 - 7*A*a*b^4)*x^6 - 54*A*a^3*b^2 + 21*(B*a^3*b^2 - 
 7*A*a^2*b^3)*x^3 - 12*sqrt(1/3)*((B*a^2*b^3 - 7*A*a*b^4)*x^7 + 2*(B*a^3*b 
^2 - 7*A*a^2*b^3)*x^4 + (B*a^4*b - 7*A*a^3*b^2)*x)*sqrt((a*b^2)^(1/3)/a)*a 
rctan(-sqrt(1/3)*(2*b*x - (a*b^2)^(1/3))*sqrt((a*b^2)^(1/3)/a)/b) + 2*((B* 
a*b^2 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*b)*x 
)*(a*b^2)^(2/3)*log(b^2*x^2 - (a*b^2)^(1/3)*b*x + (a*b^2)^(2/3)) - 4*((B*a 
*b^2 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*b)*x) 
*(a*b^2)^(2/3)*log(b*x + (a*b^2)^(1/3)))/(a^4*b^4*x^7 + 2*a^5*b^3*x^4 + a^ 
6*b^2*x)]
 

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {- 18 A a^{2} + x^{6} \left (- 28 A b^{2} + 4 B a b\right ) + x^{3} \left (- 49 A a b + 7 B a^{2}\right )}{18 a^{5} x + 36 a^{4} b x^{4} + 18 a^{3} b^{2} x^{7}} + \operatorname {RootSum} {\left (19683 t^{3} a^{10} b^{2} - 2744 A^{3} b^{3} + 1176 A^{2} B a b^{2} - 168 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a^{7} b}{196 A^{2} b^{2} - 56 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \] Input:

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

Output:

(-18*A*a**2 + x**6*(-28*A*b**2 + 4*B*a*b) + x**3*(-49*A*a*b + 7*B*a**2))/( 
18*a**5*x + 36*a**4*b*x**4 + 18*a**3*b**2*x**7) + RootSum(19683*_t**3*a**1 
0*b**2 - 2744*A**3*b**3 + 1176*A**2*B*a*b**2 - 168*A*B**2*a**2*b + 8*B**3* 
a**3, Lambda(_t, _t*log(729*_t**2*a**7*b/(196*A**2*b**2 - 56*A*B*a*b + 4*B 
**2*a**2) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {4 \, {\left (B a b - 7 \, A b^{2}\right )} x^{6} + 7 \, {\left (B a^{2} - 7 \, A a b\right )} x^{3} - 18 \, A a^{2}}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} + \frac {2 \, \sqrt {3} {\left (B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (B a - 7 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (B a - 7 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:

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

Output:

1/18*(4*(B*a*b - 7*A*b^2)*x^6 + 7*(B*a^2 - 7*A*a*b)*x^3 - 18*A*a^2)/(a^3*b 
^2*x^7 + 2*a^4*b*x^4 + a^5*x) + 2/27*sqrt(3)*(B*a - 7*A*b)*arctan(1/3*sqrt 
(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b*(a/b)^(1/3)) + 1/27*(B*a - 7*A 
*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(1/3)) - 2/27*(B*a 
 - 7*A*b)*log(x + (a/b)^(1/3))/(a^3*b*(a/b)^(1/3))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {2 \, \sqrt {3} {\left (B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} - \frac {{\left (B a - 7 \, A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} - \frac {2 \, {\left (B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, A b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4}} - \frac {A}{a^{3} x} + \frac {4 \, B a b x^{5} - 10 \, A b^{2} x^{5} + 7 \, B a^{2} x^{2} - 13 \, A a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3}} \] Input:

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

Output:

2/27*sqrt(3)*(B*a - 7*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^ 
(1/3))/((-a*b^2)^(1/3)*a^3) - 1/27*(B*a - 7*A*b)*log(x^2 + x*(-a/b)^(1/3) 
+ (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^3) - 2/27*(B*a*(-a/b)^(1/3) - 7*A*b*(-a/ 
b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 - A/(a^3*x) + 1/18*( 
4*B*a*b*x^5 - 10*A*b^2*x^5 + 7*B*a^2*x^2 - 13*A*a*b*x^2)/((b*x^3 + a)^2*a^ 
3)
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}}-\frac {\frac {A}{a}+\frac {7\,x^3\,\left (7\,A\,b-B\,a\right )}{18\,a^2}+\frac {2\,b\,x^6\,\left (7\,A\,b-B\,a\right )}{9\,a^3}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}}-\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}} \] Input:

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

Output:

(2*log(b^(1/3)*x + a^(1/3))*(7*A*b - B*a))/(27*a^(10/3)*b^(2/3)) - (A/a + 
(7*x^3*(7*A*b - B*a))/(18*a^2) + (2*b*x^6*(7*A*b - B*a))/(9*a^3))/(a^2*x + 
 b^2*x^7 + 2*a*b*x^4) + (2*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3)) 
*((3^(1/2)*1i)/2 - 1/2)*(7*A*b - B*a))/(27*a^(10/3)*b^(2/3)) - (2*log(3^(1 
/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(7*A*b - B* 
a))/(27*a^(10/3)*b^(2/3))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx=\frac {4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b x +4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{4}-9 b^{\frac {2}{3}} a^{\frac {4}{3}}-12 b^{\frac {5}{3}} a^{\frac {1}{3}} x^{3}-2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a b x -2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{4}+4 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a b x +4 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{4}}{9 b^{\frac {2}{3}} a^{\frac {7}{3}} x \left (b \,x^{3}+a \right )} \] Input:

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

Output:

(4*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b*x + 4*sq 
rt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**2*x**4 - 9*b** 
(2/3)*a**(1/3)*a - 12*b**(2/3)*a**(1/3)*b*x**3 - 2*log(a**(2/3) - b**(1/3) 
*a**(1/3)*x + b**(2/3)*x**2)*a*b*x - 2*log(a**(2/3) - b**(1/3)*a**(1/3)*x 
+ b**(2/3)*x**2)*b**2*x**4 + 4*log(a**(1/3) + b**(1/3)*x)*a*b*x + 4*log(a* 
*(1/3) + b**(1/3)*x)*b**2*x**4)/(9*b**(2/3)*a**(1/3)*a**2*x*(a + b*x**3))