\(\int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx\) [330]

Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {A \left (a+b x^3\right )^{2/3}}{3 a x^3}-\frac {(A b-3 a B) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {(A b-3 a B) \log (x)}{6 a^{4/3}}-\frac {(A b-3 a B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}} \] Output:

-1/3*A*(b*x^3+a)^(2/3)/a/x^3-1/9*(A*b-3*B*a)*arctan(1/3*(a^(1/3)+2*(b*x^3+ 
a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(4/3)+1/6*(A*b-3*B*a)*ln(x)/a^(4/3)-1 
/6*(A*b-3*B*a)*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(4/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=\frac {-\frac {6 \sqrt [3]{a} A \left (a+b x^3\right )^{2/3}}{x^3}-2 \sqrt {3} (A b-3 a B) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 (A b-3 a B) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+(A b-3 a B) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{18 a^{4/3}} \] Input:

Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(1/3)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {948, 87, 67, 16, 1082, 217}

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.

\(\displaystyle \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {B x^3+A}{x^6 \sqrt [3]{b x^3+a}}dx^3\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[(A + B*x^3)/(x^4*(a + b*x^3)^(1/3)),x]
 

Output:

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

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 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

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]
 

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {-6 A \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{\frac {1}{3}}+\left (-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right ) \left (A b -3 B a \right ) x^{3}}{18 a^{\frac {4}{3}} x^{3}}\)	\(112\)

Input:

int((B*x^3+A)/x^4/(b*x^3+a)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/18*(-6*A*(b*x^3+a)^(2/3)*a^(1/3)+(-2*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/ 
3))*3^(1/2)/a^(1/3))*3^(1/2)+ln((b*x^3+a)^(2/3)+a^(1/3)*(b*x^3+a)^(1/3)+a^ 
(2/3))-2*ln((b*x^3+a)^(1/3)-a^(1/3)))*(A*b-3*B*a)*x^3)/a^(4/3)/x^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.02 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=\left [-\frac {3 \, \sqrt {\frac {1}{3}} {\left (3 \, B a^{2} - A a b\right )} x^{3} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x^{3} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) + {\left (3 \, B a - A b\right )} \left (-a\right )^{\frac {2}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, {\left (3 \, B a - A b\right )} \left (-a\right )^{\frac {2}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} A a}{18 \, a^{2} x^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (3 \, B a^{2} - A a b\right )} x^{3} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - {\left (3 \, B a - A b\right )} \left (-a\right )^{\frac {2}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, {\left (3 \, B a - A b\right )} \left (-a\right )^{\frac {2}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} A a}{18 \, a^{2} x^{3}}\right ] \] Input:

integrate((B*x^3+A)/x^4/(b*x^3+a)^(1/3),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.63 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=- \frac {A \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x^{4} \Gamma \left (\frac {7}{3}\right )} - \frac {B \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x \Gamma \left (\frac {4}{3}\right )} \] Input:

integrate((B*x**3+A)/x**4/(b*x**3+a)**(1/3),x)
 

Output:

-A*gamma(4/3)*hyper((1/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b** 
(1/3)*x**4*gamma(7/3)) - B*gamma(1/3)*hyper((1/3, 1/3), (4/3,), a*exp_pola 
r(I*pi)/(b*x**3))/(3*b**(1/3)*x*gamma(4/3))
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.60 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} + \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{{\left (b x^{3} + a\right )} a - a^{2}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}}\right )} A + \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}\right )} B \] Input:

integrate((B*x^3+A)/x^4/(b*x^3+a)^(1/3),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=\frac {1}{18} \, b {\left (\frac {2 \, \sqrt {3} {\left (3 \, B a - A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}} b} - \frac {{\left (3 \, B a - A b\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}} b} + \frac {2 \, {\left (3 \, B a^{\frac {4}{3}} - A a^{\frac {1}{3}} b\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}} b} - \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} A}{a b x^{3}}\right )} \] Input:

integrate((B*x^3+A)/x^4/(b*x^3+a)^(1/3),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.10 \[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=\frac {\ln \left (\frac {{\left (A\,b-\sqrt {3}\,A\,b\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}-\frac {A^2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9\,a^2}\right )\,\left (A\,b-\sqrt {3}\,A\,b\,1{}\mathrm {i}\right )}{18\,a^{4/3}}+\frac {\ln \left (\frac {{\left (A\,b+\sqrt {3}\,A\,b\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}-\frac {A^2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9\,a^2}\right )\,\left (A\,b+\sqrt {3}\,A\,b\,1{}\mathrm {i}\right )}{18\,a^{4/3}}-\frac {\ln \left (B^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {a^{1/3}\,{\left (B-\sqrt {3}\,B\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (B-\sqrt {3}\,B\,1{}\mathrm {i}\right )}{6\,a^{1/3}}-\frac {\ln \left (B^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {a^{1/3}\,{\left (B+\sqrt {3}\,B\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (B+\sqrt {3}\,B\,1{}\mathrm {i}\right )}{6\,a^{1/3}}+\frac {B\,\ln \left (B^2\,{\left (b\,x^3+a\right )}^{1/3}-B^2\,a^{1/3}\right )}{3\,a^{1/3}}-\frac {A\,b\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a^{1/3}\right )}{9\,a^{4/3}}-\frac {A\,{\left (b\,x^3+a\right )}^{2/3}}{3\,a\,x^3} \] Input:

int((A + B*x^3)/(x^4*(a + b*x^3)^(1/3)),x)
 

Output:

(log((A*b - 3^(1/2)*A*b*1i)^2/(36*a^(5/3)) - (A^2*b^2*(a + b*x^3)^(1/3))/( 
9*a^2))*(A*b - 3^(1/2)*A*b*1i))/(18*a^(4/3)) + (log((A*b + 3^(1/2)*A*b*1i) 
^2/(36*a^(5/3)) - (A^2*b^2*(a + b*x^3)^(1/3))/(9*a^2))*(A*b + 3^(1/2)*A*b* 
1i))/(18*a^(4/3)) - (log(B^2*(a + b*x^3)^(1/3) - (a^(1/3)*(B - 3^(1/2)*B*1 
i)^2)/4)*(B - 3^(1/2)*B*1i))/(6*a^(1/3)) - (log(B^2*(a + b*x^3)^(1/3) - (a 
^(1/3)*(B + 3^(1/2)*B*1i)^2)/4)*(B + 3^(1/2)*B*1i))/(6*a^(1/3)) + (B*log(B 
^2*(a + b*x^3)^(1/3) - B^2*a^(1/3)))/(3*a^(1/3)) - (A*b*log((a + b*x^3)^(1 
/3) - a^(1/3)))/(9*a^(4/3)) - (A*(a + b*x^3)^(2/3))/(3*a*x^3)
 

Reduce [F]

\[ \int \frac {A+B x^3}{x^4 \sqrt [3]{a+b x^3}} \, dx=\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{4}}d x \right ) a +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x}d x \right ) b \] Input:

int((B*x^3+A)/x^4/(b*x^3+a)^(1/3),x)
 

Output:

int(1/((a + b*x**3)**(1/3)*x**4),x)*a + int(1/((a + b*x**3)**(1/3)*x),x)*b