\(\int \frac {A+B x^3}{x^3 (a+b x^3)^3} \, dx\) [107]

Optimal result

Integrand size = 20, antiderivative size = 206 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {A}{2 a^3 x^2}-\frac {(A b-a B) x}{6 a^2 \left (a+b x^3\right )^2}-\frac {(11 A b-5 a B) x}{18 a^3 \left (a+b x^3\right )}+\frac {5 (4 A b-a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}} \] Output:

-1/2*A/a^3/x^2-1/6*(A*b-B*a)*x/a^2/(b*x^3+a)^2-1/18*(11*A*b-5*B*a)*x/a^3/( 
b*x^3+a)+5/27*(4*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^(11/3)/b^(1/3)-5/27*(4*A*b-B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3) 
/b^(1/3)+5/54*(4*A*b-B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/ 
3)/b^(1/3)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {27 a^{2/3} A}{x^2}+\frac {9 a^{5/3} (-A b+a B) x}{\left (a+b x^3\right )^2}+\frac {3 a^{2/3} (-11 A b+5 a B) x}{a+b x^3}+\frac {10 \sqrt {3} (4 A b-a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 (-4 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{54 a^{11/3}} \] Input:

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

Output:

((-27*a^(2/3)*A)/x^2 + (9*a^(5/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (3*a^( 
2/3)*(-11*A*b + 5*a*B)*x)/(a + b*x^3) + (10*Sqrt[3]*(4*A*b - a*B)*ArcTan[( 
1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*(-4*A*b + a*B)*Log[a^(1 
/3) + b^(1/3)*x])/b^(1/3) + (5*(4*A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3) 
*x + b^(2/3)*x^2])/b^(1/3))/(54*a^(11/3))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 205, 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, 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.

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

\(\Big \downarrow \) 957

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

\(\Big \downarrow \) 819

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \int \frac {1}{x^3 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(4 A b-a B) \left (\frac {5 \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a b}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}\)

Input:

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

Output:

(A*b - a*B)/(6*a*b*x^2*(a + b*x^3)^2) + ((4*A*b - a*B)*(1/(3*a*x^2*(a + b* 
x^3)) + (5*(-1/2*1/(a*x^2) - (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))))/ 
a))/(3*a)))/(3*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 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 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 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 = 158, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {A}{2 a^{3} x^{2}}-\frac {\frac {\left (\frac {11}{18} b^{2} A -\frac {5}{18} a b B \right ) x^{4}+\frac {a \left (7 A b -4 B a \right ) x}{9}}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (4 A b -B a \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}}{a^{3}}\)	\(158\)
risch	\(\frac {-\frac {5 b \left (4 A b -B a \right ) x^{6}}{18 a^{3}}-\frac {4 \left (4 A b -B a \right ) x^{3}}{9 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b \,\textit {\_Z}^{3}+64 A^{3} b^{3}-48 A^{2} B a \,b^{2}+12 A \,B^{2} a^{2} b -B^{3} a^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b -192 A^{3} b^{3}+144 A^{2} B a \,b^{2}-36 A \,B^{2} a^{2} b +3 B^{3} a^{3}\right ) x +\left (-16 A^{2} a^{4} b^{2}+8 A B \,a^{5} b -B^{2} a^{6}\right ) \textit {\_R} \right )\right )}{27}\)	\(190\)

Input:

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

Output:

-1/2*A/a^3/x^2-1/a^3*(((11/18*b^2*A-5/18*a*b*B)*x^4+1/9*a*(7*A*b-4*B*a)*x) 
/(b*x^3+a)^2+5/9*(4*A*b-B*a)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a 
/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arct 
an(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. 387 vs. \(2 (160) = 320\).

Time = 0.12 (sec) , antiderivative size = 812, normalized size of antiderivative = 3.94 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/54*(15*(B*a^3*b^2 - 4*A*a^2*b^3)*x^6 - 27*A*a^4*b + 24*(B*a^4*b - 4*A*a 
^3*b^2)*x^3 - 15*sqrt(1/3)*((B*a^2*b^3 - 4*A*a*b^4)*x^8 + 2*(B*a^3*b^2 - 4 
*A*a^2*b^3)*x^5 + (B*a^4*b - 4*A*a^3*b^2)*x^2)*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)) - 5*((B 
*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*b - 4*A*a*b^2)*x^5 + (B*a^3 - 4*A*a^2*b)* 
x^2)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 1 
0*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*b - 4*A*a*b^2)*x^5 + (B*a^3 - 4*A*a^ 
2*b)*x^2)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)))/(a^5*b^3*x^8 + 2*a^6 
*b^2*x^5 + a^7*b*x^2), 1/54*(15*(B*a^3*b^2 - 4*A*a^2*b^3)*x^6 - 27*A*a^4*b 
 + 24*(B*a^4*b - 4*A*a^3*b^2)*x^3 + 30*sqrt(1/3)*((B*a^2*b^3 - 4*A*a*b^4)* 
x^8 + 2*(B*a^3*b^2 - 4*A*a^2*b^3)*x^5 + (B*a^4*b - 4*A*a^3*b^2)*x^2)*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) - 5*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*b - 
 4*A*a*b^2)*x^5 + (B*a^3 - 4*A*a^2*b)*x^2)*(-a^2*b)^(2/3)*log(a*b*x^2 - (- 
a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 10*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^ 
2*b - 4*A*a*b^2)*x^5 + (B*a^3 - 4*A*a^2*b)*x^2)*(-a^2*b)^(2/3)*log(a*b*x + 
 (-a^2*b)^(2/3)))/(a^5*b^3*x^8 + 2*a^6*b^2*x^5 + a^7*b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {- 9 A a^{2} + x^{6} \left (- 20 A b^{2} + 5 B a b\right ) + x^{3} \left (- 32 A a b + 8 B a^{2}\right )}{18 a^{5} x^{2} + 36 a^{4} b x^{5} + 18 a^{3} b^{2} x^{8}} + \operatorname {RootSum} {\left (19683 t^{3} a^{11} b + 8000 A^{3} b^{3} - 6000 A^{2} B a b^{2} + 1500 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left ( t \mapsto t \log {\left (\frac {27 t a^{4}}{- 20 A b + 5 B a} + x \right )} \right )\right )} \] Input:

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

Output:

(-9*A*a**2 + x**6*(-20*A*b**2 + 5*B*a*b) + x**3*(-32*A*a*b + 8*B*a**2))/(1 
8*a**5*x**2 + 36*a**4*b*x**5 + 18*a**3*b**2*x**8) + RootSum(19683*_t**3*a* 
*11*b + 8000*A**3*b**3 - 6000*A**2*B*a*b**2 + 1500*A*B**2*a**2*b - 125*B** 
3*a**3, Lambda(_t, _t*log(27*_t*a**4/(-20*A*b + 5*B*a) + x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {5 \, {\left (B a b - 4 \, A b^{2}\right )} x^{6} + 8 \, {\left (B a^{2} - 4 \, A a b\right )} x^{3} - 9 \, A a^{2}}{18 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} + \frac {5 \, \sqrt {3} {\left (B a - 4 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, {\left (B a - 4 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, {\left (B a - 4 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:

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

Output:

1/18*(5*(B*a*b - 4*A*b^2)*x^6 + 8*(B*a^2 - 4*A*a*b)*x^3 - 9*A*a^2)/(a^3*b^ 
2*x^8 + 2*a^4*b*x^5 + a^5*x^2) + 5/27*sqrt(3)*(B*a - 4*A*b)*arctan(1/3*sqr 
t(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b*(a/b)^(2/3)) - 5/54*(B*a - 4* 
A*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(2/3)) + 5/27*(B* 
a - 4*A*b)*log(x + (a/b)^(1/3))/(a^3*b*(a/b)^(2/3))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {5 \, {\left (B a - 4 \, A b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4}} + \frac {5 \, \sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} + \frac {5 \, {\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b} + \frac {5 \, B a b x^{6} - 20 \, A b^{2} x^{6} + 8 \, B a^{2} x^{3} - 32 \, A a b x^{3} - 9 \, A a^{2}}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3}} \] Input:

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

Output:

-5/27*(B*a - 4*A*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 + 5/27*sqr 
t(3)*((-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + 
 (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b) + 5/54*((-a*b^2)^(1/3)*B*a - 4*(-a*b^ 
2)^(1/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) + 1/18*(5*B 
*a*b*x^6 - 20*A*b^2*x^6 + 8*B*a^2*x^3 - 32*A*a*b*x^3 - 9*A*a^2)/((b*x^4 + 
a*x)^2*a^3)
 

Mupad [B] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {A}{2\,a}+\frac {4\,x^3\,\left (4\,A\,b-B\,a\right )}{9\,a^2}+\frac {5\,b\,x^6\,\left (4\,A\,b-B\,a\right )}{18\,a^3}}{a^2\,x^2+2\,a\,b\,x^5+b^2\,x^8}-\frac {5\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}}+\frac {5\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}}-\frac {5\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}} \] Input:

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

Output:

(5*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)* 
(4*A*b - B*a))/(27*a^(11/3)*b^(1/3)) - (5*log(b^(1/3)*x + a^(1/3))*(4*A*b 
- B*a))/(27*a^(11/3)*b^(1/3)) - (A/(2*a) + (4*x^3*(4*A*b - B*a))/(9*a^2) + 
 (5*b*x^6*(4*A*b - B*a))/(18*a^3))/(a^2*x^2 + b^2*x^8 + 2*a*b*x^5) - (5*lo 
g(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(4*A* 
b - B*a))/(27*a^(11/3)*b^(1/3))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx=\frac {10 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{2}+10 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{5}+5 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{2}+5 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{5}-10 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{2}-10 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{5}-9 b^{\frac {1}{3}} a^{2}-15 b^{\frac {4}{3}} a \,x^{3}}{18 b^{\frac {1}{3}} a^{3} x^{2} \left (b \,x^{3}+a \right )} \] Input:

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

Output:

(10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a* 
b*x**2 + 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt 
(3)))*b**2*x**5 + 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3) 
*x**2)*a*b*x**2 + 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3) 
*x**2)*b**2*x**5 - 10*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*x**2 - 10*a* 
*(1/3)*log(a**(1/3) + b**(1/3)*x)*b**2*x**5 - 9*b**(1/3)*a**2 - 15*b**(1/3 
)*a*b*x**3)/(18*b**(1/3)*a**3*x**2*(a + b*x**3))