\(\int \frac {A+B x^3}{x^{3/2} (a+b x^3)} \, dx\) [138]

Optimal result

Integrand size = 22, antiderivative size = 204 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=-\frac {2 A}{a \sqrt {x}}+\frac {(A b-a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{7/6} b^{5/6}}-\frac {(A b-a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{7/6} b^{5/6}}-\frac {2 (A b-a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{7/6} b^{5/6}}+\frac {(A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{\sqrt {3} a^{7/6} b^{5/6}} \] Output:

-2*A/a/x^(1/2)-1/3*(A*b-B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^ 
(7/6)/b^(5/6)-1/3*(A*b-B*a)*arctan(3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(7 
/6)/b^(5/6)-2/3*(A*b-B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^(7/6)/b^(5/6)+ 
1/3*(A*b-B*a)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*x^(1/2)/(a^(1/3)+b^(1/3)*x)) 
*3^(1/2)/a^(7/6)/b^(5/6)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=\frac {-\frac {6 \sqrt [6]{a} A}{\sqrt {x}}+\frac {2 (-A b+a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {(A b-a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{b^{5/6}}+\frac {\sqrt {3} (A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{b^{5/6}}}{3 a^{7/6}} \] Input:

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

Output:

((-6*a^(1/6)*A)/Sqrt[x] + (2*(-(A*b) + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/ 
6)])/b^(5/6) + ((A*b - a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)* 
Sqrt[x])])/b^(5/6) + (Sqrt[3]*(A*b - a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6) 
*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/b^(5/6))/(3*a^(7/6))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.28, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {955, 851, 824, 27, 218, 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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 851

\(\displaystyle -\frac {2 (A b-a B) \int \frac {x^2}{b x^3+a}d\sqrt {x}}{a}-\frac {2 A}{a \sqrt {x}}\)

\(\Big \downarrow \) 824

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

Defintions of rubi rules used

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 218

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

rule 824

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k 
- 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 
 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k 
- 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] 
; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))   Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m 
+ 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt 
Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
 

rule 851

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

rule 955

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), 
 x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1))   Int[(e 
*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* 
c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || 
(LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[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 = 0.95 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) \left (A b -B a \right )}{a}-\frac {2 A}{a \sqrt {x}}\)	\(191\)
default	\(-\frac {2 \left (\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) \left (A b -B a \right )}{a}-\frac {2 A}{a \sqrt {x}}\)	\(191\)
risch	\(-\frac {2 A}{a \sqrt {x}}-\frac {\left (A b -B a \right ) \left (\frac {2 \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{a}\)	\(191\)

Input:

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

Output:

-2*(1/3/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+1/12/a*3^(1/2)*(a/b)^(5/ 
6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan( 
2*x^(1/2)/(a/b)^(1/6)-3^(1/2))-1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/ 
b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/ 
6)+3^(1/2)))/a*(A*b-B*a)-2*A/a/x^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1636 vs. \(2 (147) = 294\).

Time = 0.13 (sec) , antiderivative size = 1636, normalized size of antiderivative = 8.02 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(2*a*x*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a 
^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^5))^(1/6)*lo 
g(a^6*b^4*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3 
*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^5))^(5/6) - (B 
^5*a^5 - 5*A*B^4*a^4*b + 10*A^2*B^3*a^3*b^2 - 10*A^3*B^2*a^2*b^3 + 5*A^4*B 
*a*b^4 - A^5*b^5)*sqrt(x)) - 2*a*x*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4 
*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b 
^6)/(a^7*b^5))^(1/6)*log(-a^6*b^4*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4* 
a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^ 
6)/(a^7*b^5))^(5/6) - (B^5*a^5 - 5*A*B^4*a^4*b + 10*A^2*B^3*a^3*b^2 - 10*A 
^3*B^2*a^2*b^3 + 5*A^4*B*a*b^4 - A^5*b^5)*sqrt(x)) - (sqrt(-3)*a*x - a*x)* 
(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15* 
A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^5))^(1/6)*log((sqrt(-3)* 
a^6*b^4 + a^6*b^4)*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^ 
3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^5))^( 
5/6) - 2*(B^5*a^5 - 5*A*B^4*a^4*b + 10*A^2*B^3*a^3*b^2 - 10*A^3*B^2*a^2*b^ 
3 + 5*A^4*B*a*b^4 - A^5*b^5)*sqrt(x)) + (sqrt(-3)*a*x - a*x)*(-(B^6*a^6 - 
6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b 
^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^5))^(1/6)*log(-(sqrt(-3)*a^6*b^4 + a^ 
6*b^4)*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (197) = 394\).

Time = 9.88 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.75 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{\sqrt {x}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + \frac {2 B x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\- \frac {A \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 a \sqrt [6]{- \frac {a}{b}}} + \frac {A \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 a \sqrt [6]{- \frac {a}{b}}} - \frac {A \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a \sqrt [6]{- \frac {a}{b}}} + \frac {A \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3} A \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 a \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3} A \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a \sqrt [6]{- \frac {a}{b}}} - \frac {2 A}{a \sqrt {x}} + \frac {B \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 b \sqrt [6]{- \frac {a}{b}}} - \frac {B \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 b \sqrt [6]{- \frac {a}{b}}} + \frac {B \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b \sqrt [6]{- \frac {a}{b}}} - \frac {B \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3} B \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 b \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3} B \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 b \sqrt [6]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/sqrt(x)), Eq(a, 0) & Eq(b, 0)), (( 
-2*A/(7*x**(7/2)) - 2*B/sqrt(x))/b, Eq(a, 0)), ((-2*A/sqrt(x) + 2*B*x**(5/ 
2)/5)/a, Eq(b, 0)), (-A*log(sqrt(x) - (-a/b)**(1/6))/(3*a*(-a/b)**(1/6)) + 
 A*log(sqrt(x) + (-a/b)**(1/6))/(3*a*(-a/b)**(1/6)) - A*log(-4*sqrt(x)*(-a 
/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a*(-a/b)**(1/6)) + A*log(4*sqrt(x)* 
(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a*(-a/b)**(1/6)) - sqrt(3)*A*ata 
n(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(3*a*(-a/b)**(1/6)) - s 
qrt(3)*A*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(3*a*(-a/b) 
**(1/6)) - 2*A/(a*sqrt(x)) + B*log(sqrt(x) - (-a/b)**(1/6))/(3*b*(-a/b)**( 
1/6)) - B*log(sqrt(x) + (-a/b)**(1/6))/(3*b*(-a/b)**(1/6)) + B*log(-4*sqrt 
(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*b*(-a/b)**(1/6)) - B*log(4*s 
qrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*b*(-a/b)**(1/6)) + sqrt(3 
)*B*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(3*b*(-a/b)**(1/ 
6)) + sqrt(3)*B*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(3*b 
*(-a/b)**(1/6)), True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=-\frac {{\left (B a - A b\right )} {\left (\frac {\sqrt {3} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{6 \, a} - \frac {2 \, A}{a \sqrt {x}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=\frac {{\left (B a - A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, \left (a b^{5}\right )^{\frac {1}{6}} a} + \frac {{\left (B a - A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, \left (a b^{5}\right )^{\frac {1}{6}} a} + \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, \left (a b^{5}\right )^{\frac {1}{6}} a} - \frac {2 \, A}{a \sqrt {x}} - \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b^{5}} + \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {5}{6}} B a - \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b^{5}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 1700, normalized size of antiderivative = 8.33 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((A*b - B*a)^2*(32*B^3*a^12*b^3 - 32*A^3*a^9*b^6 - 96*A*B^2*a^11*b^ 
4 + 96*A^2*B*a^10*b^5 + (x^(1/2)*(A*b - B*a)*(864*A^2*a^10*b^6 + 864*B^2*a 
^12*b^4 - 1728*A*B*a^11*b^5))/(27*(-a)^(7/6)*b^(5/6)))*1i)/((-a)^(7/3)*b^( 
5/3)) + ((A*b - B*a)^2*(32*A^3*a^9*b^6 - 32*B^3*a^12*b^3 + 96*A*B^2*a^11*b 
^4 - 96*A^2*B*a^10*b^5 + (x^(1/2)*(A*b - B*a)*(864*A^2*a^10*b^6 + 864*B^2* 
a^12*b^4 - 1728*A*B*a^11*b^5))/(27*(-a)^(7/6)*b^(5/6)))*1i)/((-a)^(7/3)*b^ 
(5/3)))/(((A*b - B*a)^2*(32*B^3*a^12*b^3 - 32*A^3*a^9*b^6 - 96*A*B^2*a^11* 
b^4 + 96*A^2*B*a^10*b^5 + (x^(1/2)*(A*b - B*a)*(864*A^2*a^10*b^6 + 864*B^2 
*a^12*b^4 - 1728*A*B*a^11*b^5))/(27*(-a)^(7/6)*b^(5/6))))/((-a)^(7/3)*b^(5 
/3)) - ((A*b - B*a)^2*(32*A^3*a^9*b^6 - 32*B^3*a^12*b^3 + 96*A*B^2*a^11*b^ 
4 - 96*A^2*B*a^10*b^5 + (x^(1/2)*(A*b - B*a)*(864*A^2*a^10*b^6 + 864*B^2*a 
^12*b^4 - 1728*A*B*a^11*b^5))/(27*(-a)^(7/6)*b^(5/6))))/((-a)^(7/3)*b^(5/3 
))))*(A*b - B*a)*2i)/(3*(-a)^(7/6)*b^(5/6)) - (2*A)/(a*x^(1/2)) + (atan((( 
((3^(1/2)*1i)/2 - 1/2)^2*(A*b - B*a)^2*(32*B^3*a^12*b^3 - 32*A^3*a^9*b^6 - 
 96*A*B^2*a^11*b^4 + 96*A^2*B*a^10*b^5 + (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*( 
A*b - B*a)*(864*A^2*a^10*b^6 + 864*B^2*a^12*b^4 - 1728*A*B*a^11*b^5))/(27* 
(-a)^(7/6)*b^(5/6)))*1i)/((-a)^(7/3)*b^(5/3)) + (((3^(1/2)*1i)/2 - 1/2)^2* 
(A*b - B*a)^2*(32*A^3*a^9*b^6 - 32*B^3*a^12*b^3 + 96*A*B^2*a^11*b^4 - 96*A 
^2*B*a^10*b^5 + (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*(A*b - B*a)*(864*A^2*a^10* 
b^6 + 864*B^2*a^12*b^4 - 1728*A*B*a^11*b^5))/(27*(-a)^(7/6)*b^(5/6)))*1...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.03 \[ \int \frac {A+B x^3}{x^{3/2} \left (a+b x^3\right )} \, dx=-\frac {2}{\sqrt {x}} \] Input:

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

Output:

( - 2)/sqrt(x)