\(\int \frac {x^{5/2} (A+B x^3)}{(a+b x^3)^3} \, dx\) [153]

Optimal result

Integrand size = 22, antiderivative size = 261 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {(A b-a B) \sqrt {x}}{6 b^2 \left (a+b x^3\right )^2}+\frac {(A b-13 a B) \sqrt {x}}{36 a b^2 \left (a+b x^3\right )}-\frac {(5 A b+7 a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{11/6} b^{13/6}}+\frac {(5 A b+7 a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{11/6} b^{13/6}}+\frac {(5 A b+7 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{11/6} b^{13/6}}+\frac {(5 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{72 \sqrt {3} a^{11/6} b^{13/6}} \] Output:

-1/6*(A*b-B*a)*x^(1/2)/b^2/(b*x^3+a)^2+1/36*(A*b-13*B*a)*x^(1/2)/a/b^2/(b* 
x^3+a)+1/216*(5*A*b+7*B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/a^(1 
1/6)/b^(13/6)+1/216*(5*A*b+7*B*a)*arctan(3^(1/2)+2*b^(1/6)*x^(1/2)/a^(1/6) 
)/a^(11/6)/b^(13/6)+1/108*(5*A*b+7*B*a)*arctan(b^(1/6)*x^(1/2)/a^(1/6))/a^ 
(11/6)/b^(13/6)+1/216*(5*A*b+7*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^(11/6)/b^(13/6)
 

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.74 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-\frac {6 a^{5/6} \sqrt [6]{b} \sqrt {x} \left (7 a^2 B-A b^2 x^3+a b \left (5 A+13 B x^3\right )\right )}{\left (a+b x^3\right )^2}+2 (5 A b+7 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )-(5 A b+7 a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )+\sqrt {3} (5 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{216 a^{11/6} b^{13/6}} \] Input:

Integrate[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^3,x]
 

Output:

((-6*a^(5/6)*b^(1/6)*Sqrt[x]*(7*a^2*B - A*b^2*x^3 + a*b*(5*A + 13*B*x^3))) 
/(a + b*x^3)^2 + 2*(5*A*b + 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] - (5* 
A*b + 7*a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])] + Sqr 
t[3]*(5*A*b + 7*a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + 
b^(1/3)*x)])/(216*a^(11/6)*b^(13/6))
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {957, 817, 851, 753, 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.

\(\displaystyle \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(7 a B+5 A b) \int \frac {x^{5/2}}{\left (b x^3+a\right )^2}dx}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\int \frac {1}{\sqrt {x} \left (b x^3+a\right )}dx}{6 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\int \frac {1}{b x^3+a}d\sqrt {x}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 753

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 a^{5/6}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+2 \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(7 a B+5 A b) \left (\frac {\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}+\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}}{3 b}-\frac {\sqrt {x}}{3 b \left (a+b x^3\right )}\right )}{12 a b}+\frac {x^{7/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2}\)

Input:

Int[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^3,x]
 

Output:

((A*b - a*B)*x^(7/2))/(6*a*b*(a + b*x^3)^2) + ((5*A*b + 7*a*B)*(-1/3*Sqrt[ 
x]/(b*(a + b*x^3)) + (ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)]/(3*a^(5/6)*b^(1/6) 
) + (-(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^(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^(5/6)))/(3*b)))/(12*a*b)
 

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 753

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

rule 817

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

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 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 = 0.94 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {\left (A b -13 B a \right ) x^{\frac {7}{2}}}{36 a b}-\frac {\left (5 A b +7 B a \right ) \sqrt {x}}{36 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 A b +7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right )}{36 b^{2} a}\)	\(234\)
default	\(\frac {\frac {\left (A b -13 B a \right ) x^{\frac {7}{2}}}{36 a b}-\frac {\left (5 A b +7 B a \right ) \sqrt {x}}{36 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (5 A b +7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right )}{36 b^{2} a}\)	\(234\)

Input:

int(x^(5/2)*(B*x^3+A)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*(1/72*(A*b-13*B*a)/a/b*x^(7/2)-1/72*(5*A*b+7*B*a)/b^2*x^(1/2))/(b*x^3+a) 
^2+1/36*(5*A*b+7*B*a)/b^2/a*(1/3/a*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6)) 
+1/12/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+ 
1/6/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))-1/12/a*3^(1/2)*(a/ 
b)^(1/6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1/3))+1/6/a*(a/b)^(1/6)*a 
rctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (196) = 392\).

Time = 0.10 (sec) , antiderivative size = 1618, normalized size of antiderivative = 6.20 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

1/432*(2*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*(-(117649*B^6*a^6 + 504210* 
A*B^5*a^5*b + 900375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 459375*A^4 
*B^2*a^2*b^4 + 131250*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^13))^(1/6)*log( 
a^2*b^2*(-(117649*B^6*a^6 + 504210*A*B^5*a^5*b + 900375*A^2*B^4*a^4*b^2 + 
857500*A^3*B^3*a^3*b^3 + 459375*A^4*B^2*a^2*b^4 + 131250*A^5*B*a*b^5 + 156 
25*A^6*b^6)/(a^11*b^13))^(1/6) + (7*B*a + 5*A*b)*sqrt(x)) - 2*(a*b^4*x^6 + 
 2*a^2*b^3*x^3 + a^3*b^2)*(-(117649*B^6*a^6 + 504210*A*B^5*a^5*b + 900375* 
A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 459375*A^4*B^2*a^2*b^4 + 131250 
*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^13))^(1/6)*log(-a^2*b^2*(-(117649*B^ 
6*a^6 + 504210*A*B^5*a^5*b + 900375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b 
^3 + 459375*A^4*B^2*a^2*b^4 + 131250*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^ 
13))^(1/6) + (7*B*a + 5*A*b)*sqrt(x)) + (a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b 
^2 + sqrt(-3)*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2))*(-(117649*B^6*a^6 + 5 
04210*A*B^5*a^5*b + 900375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 4593 
75*A^4*B^2*a^2*b^4 + 131250*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11*b^13))^(1/6 
)*log((7*B*a + 5*A*b)*sqrt(x) + 1/2*(sqrt(-3)*a^2*b^2 + a^2*b^2)*(-(117649 
*B^6*a^6 + 504210*A*B^5*a^5*b + 900375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^ 
3*b^3 + 459375*A^4*B^2*a^2*b^4 + 131250*A^5*B*a*b^5 + 15625*A^6*b^6)/(a^11 
*b^13))^(1/6)) - (a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2 + sqrt(-3)*(a*b^4*x^ 
6 + 2*a^2*b^3*x^3 + a^3*b^2))*(-(117649*B^6*a^6 + 504210*A*B^5*a^5*b + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(5/2)*(B*x**3+A)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.31 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {{\left (13 \, B a b - A b^{2}\right )} x^{\frac {7}{2}} + {\left (7 \, B a^{2} + 5 \, A a b\right )} \sqrt {x}}{36 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}} + \frac {\frac {\sqrt {3} {\left (7 \, B a + 5 \, A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (7 \, B a + 5 \, A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (7 \, B a b^{\frac {1}{3}} + 5 \, A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} + 5 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} + 5 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{432 \, a b^{2}} \] Input:

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

-1/36*((13*B*a*b - A*b^2)*x^(7/2) + (7*B*a^2 + 5*A*a*b)*sqrt(x))/(a*b^4*x^ 
6 + 2*a^2*b^3*x^3 + a^3*b^2) + 1/432*(sqrt(3)*(7*B*a + 5*A*b)*log(sqrt(3)* 
a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) - sqrt(3) 
*(7*B*a + 5*A*b)*log(-sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3 
))/(a^(5/6)*b^(1/6)) + 4*(7*B*a*b^(1/3) + 5*A*b^(4/3))*arctan(b^(1/3)*sqrt 
(x)/sqrt(a^(1/3)*b^(1/3)))/(a^(2/3)*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(7* 
B*a^(4/3)*b^(1/3) + 5*A*a^(1/3)*b^(4/3))*arctan((sqrt(3)*a^(1/6)*b^(1/6) + 
 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3) 
)) + 2*(7*B*a^(4/3)*b^(1/3) + 5*A*a^(1/3)*b^(4/3))*arctan(-(sqrt(3)*a^(1/6 
)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(a*b^(1/3)*sqrt(a^(1 
/3)*b^(1/3))))/(a*b^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.26 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{2} b^{3}} - \frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{2} b^{3}} + \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{2} b^{3}} + \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{2} b^{3}} + \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a + 5 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 \, a^{2} b^{3}} - \frac {13 \, B a b x^{\frac {7}{2}} - A b^{2} x^{\frac {7}{2}} + 7 \, B a^{2} \sqrt {x} + 5 \, A a b \sqrt {x}}{36 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \] Input:

integrate(x^(5/2)*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/432*sqrt(3)*(7*(a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt 
(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^2*b^3) - 1/432*sqrt(3)*(7*(a*b^5)^(1 
/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b 
)^(1/3))/(a^2*b^3) + 1/216*(7*(a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*arc 
tan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^2*b^3) + 1/216*(7*(a 
*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sq 
rt(x))/(a/b)^(1/6))/(a^2*b^3) + 1/108*(7*(a*b^5)^(1/6)*B*a + 5*(a*b^5)^(1/ 
6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^2*b^3) - 1/36*(13*B*a*b*x^(7/2) - A 
*b^2*x^(7/2) + 7*B*a^2*sqrt(x) + 5*A*a*b*sqrt(x))/((b*x^3 + a)^2*a*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 1944, normalized size of antiderivative = 7.45 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

int((x^(5/2)*(A + B*x^3))/(a + b*x^3)^3,x)
 

Output:

(atan((((((5*A*b + 7*B*a)*(125*A^3*b^3 + 343*B^3*a^3 + 735*A*B^2*a^2*b + 5 
25*A^2*B*a*b^2))/(279936*(-a)^(23/6)*b^(19/6)) - (x^(1/2)*(625*A^4*b^4 + 2 
401*B^4*a^4 + 7350*A^2*B^2*a^2*b^2 + 6860*A*B^3*a^3*b + 3500*A^3*B*a*b^3)) 
/(279936*a^4*b^3))*(5*A*b + 7*B*a)*1i)/(216*(-a)^(11/6)*b^(13/6)) - ((((5* 
A*b + 7*B*a)*(125*A^3*b^3 + 343*B^3*a^3 + 735*A*B^2*a^2*b + 525*A^2*B*a*b^ 
2))/(279936*(-a)^(23/6)*b^(19/6)) + (x^(1/2)*(625*A^4*b^4 + 2401*B^4*a^4 + 
 7350*A^2*B^2*a^2*b^2 + 6860*A*B^3*a^3*b + 3500*A^3*B*a*b^3))/(279936*a^4* 
b^3))*(5*A*b + 7*B*a)*1i)/(216*(-a)^(11/6)*b^(13/6)))/(((((5*A*b + 7*B*a)* 
(125*A^3*b^3 + 343*B^3*a^3 + 735*A*B^2*a^2*b + 525*A^2*B*a*b^2))/(279936*( 
-a)^(23/6)*b^(19/6)) - (x^(1/2)*(625*A^4*b^4 + 2401*B^4*a^4 + 7350*A^2*B^2 
*a^2*b^2 + 6860*A*B^3*a^3*b + 3500*A^3*B*a*b^3))/(279936*a^4*b^3))*(5*A*b 
+ 7*B*a))/(216*(-a)^(11/6)*b^(13/6)) + ((((5*A*b + 7*B*a)*(125*A^3*b^3 + 3 
43*B^3*a^3 + 735*A*B^2*a^2*b + 525*A^2*B*a*b^2))/(279936*(-a)^(23/6)*b^(19 
/6)) + (x^(1/2)*(625*A^4*b^4 + 2401*B^4*a^4 + 7350*A^2*B^2*a^2*b^2 + 6860* 
A*B^3*a^3*b + 3500*A^3*B*a*b^3))/(279936*a^4*b^3))*(5*A*b + 7*B*a))/(216*( 
-a)^(11/6)*b^(13/6))))*(5*A*b + 7*B*a)*1i)/(108*(-a)^(11/6)*b^(13/6)) - (( 
x^(1/2)*(5*A*b + 7*B*a))/(36*b^2) - (x^(7/2)*(A*b - 13*B*a))/(36*a*b))/(a^ 
2 + b^2*x^6 + 2*a*b*x^3) + (atan(((((3^(1/2)*1i)/2 - 1/2)*((x^(1/2)*(625*A 
^4*b^4 + 2401*B^4*a^4 + 7350*A^2*B^2*a^2*b^2 + 6860*A*B^3*a^3*b + 3500*A^3 
*B*a*b^3))/(279936*a^4*b^3) - (((3^(1/2)*1i)/2 - 1/2)*(5*A*b + 7*B*a)*(...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.27 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-2 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-2 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}+2 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 \sqrt {x}\, b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{3}+4 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )+4 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {\sqrt {x}\, b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) x^{3}-b^{\frac {5}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )-b^{\frac {11}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}+b^{\frac {5}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+b^{\frac {11}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) x^{3}-12 \sqrt {x}\, a b}{36 a \,b^{2} \left (b \,x^{3}+a \right )} \] Input:

int(x^(5/2)*(B*x^3+A)/(b*x^3+a)^3,x)
 

Output:

( - 2*b**(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*sqrt(x)*b**(1/ 
3))/(b**(1/6)*a**(1/6)))*a - 2*b**(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*s 
qrt(3) - 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 + 2*b**(5/6)*a**( 
1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)*b**(1/3))/(b**(1/6)*a**(1 
/6)))*a + 2*b**(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*sqrt(x)* 
b**(1/3))/(b**(1/6)*a**(1/6)))*b*x**3 + 4*b**(5/6)*a**(1/6)*atan((sqrt(x)* 
b**(1/3))/(b**(1/6)*a**(1/6)))*a + 4*b**(5/6)*a**(1/6)*atan((sqrt(x)*b**(1 
/3))/(b**(1/6)*a**(1/6)))*b*x**3 - b**(5/6)*a**(1/6)*sqrt(3)*log( - sqrt(x 
)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*a - b**(5/6)*a**(1/6) 
*sqrt(3)*log( - sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x) 
*b*x**3 + b**(5/6)*a**(1/6)*sqrt(3)*log(sqrt(x)*b**(1/6)*a**(1/6)*sqrt(3) 
+ a**(1/3) + b**(1/3)*x)*a + b**(5/6)*a**(1/6)*sqrt(3)*log(sqrt(x)*b**(1/6 
)*a**(1/6)*sqrt(3) + a**(1/3) + b**(1/3)*x)*b*x**3 - 12*sqrt(x)*a*b)/(36*a 
*b**2*(a + b*x**3))