\(\int \frac {x^{5/2} (A+B x^3)}{a+b x^3} \, dx\) [135]

Optimal result

Integrand size = 22, antiderivative size = 225 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {\sqrt [6]{a} (A b-a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {2 \sqrt [6]{a} (A b-a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{\sqrt {3} b^{13/6}} \] Output:

2*(A*b-B*a)*x^(1/2)/b^2+2/7*B*x^(7/2)/b-1/3*a^(1/6)*(A*b-B*a)*arctan(-3^(1 
/2)+2*b^(1/6)*x^(1/2)/a^(1/6))/b^(13/6)-1/3*a^(1/6)*(A*b-B*a)*arctan(3^(1/ 
2)+2*b^(1/6)*x^(1/2)/a^(1/6))/b^(13/6)-2/3*a^(1/6)*(A*b-B*a)*arctan(b^(1/6 
)*x^(1/2)/a^(1/6))/b^(13/6)-1/3*a^(1/6)*(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)/b^(13/6)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.76 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {6 \sqrt [6]{b} \sqrt {x} \left (7 A b-7 a B+b B x^3\right )+14 \sqrt [6]{a} (-A b+a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )-7 \sqrt [6]{a} (-A b+a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )+7 \sqrt {3} \sqrt [6]{a} (-A b+a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{21 b^{13/6}} \] Input:

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

Output:

(6*b^(1/6)*Sqrt[x]*(7*A*b - 7*a*B + b*B*x^3) + 14*a^(1/6)*(-(A*b) + a*B)*A 
rcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] - 7*a^(1/6)*(-(A*b) + a*B)*ArcTan[(a^(1/3 
) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])] + 7*Sqrt[3]*a^(1/6)*(-(A*b) + a* 
B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/(21*b 
^(13/6))
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {959, 843, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 843

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

\(\Big \downarrow \) 851

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

\(\Big \downarrow \) 753

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(2*B*x^(7/2))/(7*b) + ((A*b - a*B)*((2*Sqrt[x])/b - (2*a*(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))))/b))/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 843

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*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 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 959

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

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 = 0.86 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.90

method	result	size
risch	\(\frac {2 \left (b B \,x^{3}+7 A b -7 B a \right ) \sqrt {x}}{7 b^{2}}-\frac {a \left (A b -B a \right ) \left (\frac {2 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x -\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{3 a}\right )}{b^{2}}\)	\(203\)
derivativedivides	\(\frac {\frac {2 b B \,x^{\frac {7}{2}}}{7}+2 A b \sqrt {x}-2 B a \sqrt {x}}{b^{2}}-\frac {2 \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x -\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right ) a \left (A b -B a \right )}{b^{2}}\)	\(206\)
default	\(\frac {\frac {2 b B \,x^{\frac {7}{2}}}{7}+2 A b \sqrt {x}-2 B a \sqrt {x}}{b^{2}}-\frac {2 \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x -\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right ) a \left (A b -B a \right )}{b^{2}}\)	\(206\)

Input:

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

Output:

2/7*(B*b*x^3+7*A*b-7*B*a)*x^(1/2)/b^2-a*(A*b-B*a)/b^2*(2/3/a*(a/b)^(1/6)*a 
rctan(x^(1/2)/(a/b)^(1/6))-1/6/a*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1 
/6)*x^(1/2)+(a/b)^(1/3))+1/3/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)-3^ 
(1/2))+1/6/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1 
/3))+1/3/a*(a/b)^(1/6)*arctan(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. 1289 vs. \(2 (163) = 326\).

Time = 0.13 (sec) , antiderivative size = 1289, normalized size of antiderivative = 5.73 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/42*(14*b^2*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3 
*a^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6)*l 
og(b^2*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3*a^4*b^ 
3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6) - (B*a - 
 A*b)*sqrt(x)) - 14*b^2*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 
20*A^3*B^3*a^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^1 
3)^(1/6)*log(-b^2*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3 
*B^3*a^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/ 
6) - (B*a - A*b)*sqrt(x)) + 7*(sqrt(-3)*b^2 + b^2)*(-(B^6*a^7 - 6*A*B^5*a^ 
6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3*a^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5 
*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6)*log(-2*(B*a - A*b)*sqrt(x) + (sqrt(-3) 
*b^2 + b^2)*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3*a 
^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6)) - 
7*(sqrt(-3)*b^2 + b^2)*(-(B^6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 2 
0*A^3*B^3*a^4*b^3 + 15*A^4*B^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13 
)^(1/6)*log(-2*(B*a - A*b)*sqrt(x) - (sqrt(-3)*b^2 + b^2)*(-(B^6*a^7 - 6*A 
*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3*a^4*b^3 + 15*A^4*B^2*a^3*b^4 
- 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6)) + 7*(sqrt(-3)*b^2 - b^2)*(-(B^ 
6*a^7 - 6*A*B^5*a^6*b + 15*A^2*B^4*a^5*b^2 - 20*A^3*B^3*a^4*b^3 + 15*A^4*B 
^2*a^3*b^4 - 6*A^5*B*a^2*b^5 + A^6*a*b^6)/b^13)^(1/6)*log(-2*(B*a - A*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (209) = 418\).

Time = 30.32 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.69 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {13}{2}}}{13}}{a} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {for}\: a = 0 \\\frac {2 A \sqrt {x}}{b} + \frac {A \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 b} - \frac {A \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 b} + \frac {A \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b} - \frac {A \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b} - \frac {\sqrt {3} A \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 b} - \frac {\sqrt {3} A \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 b} - \frac {2 B a \sqrt {x}}{b^{2}} - \frac {B a \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{2}} + \frac {B a \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{3 b^{2}} - \frac {B a \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b^{2}} + \frac {B a \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 b^{2}} + \frac {\sqrt {3} B a \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{3 b^{2}} + \frac {\sqrt {3} B a \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 b^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*(2*A*sqrt(x) + 2*B*x**(7/2)/7), Eq(a, 0) & Eq(b, 0)), ((2*A 
*x**(7/2)/7 + 2*B*x**(13/2)/13)/a, Eq(b, 0)), ((2*A*sqrt(x) + 2*B*x**(7/2) 
/7)/b, Eq(a, 0)), (2*A*sqrt(x)/b + A*(-a/b)**(1/6)*log(sqrt(x) - (-a/b)**( 
1/6))/(3*b) - A*(-a/b)**(1/6)*log(sqrt(x) + (-a/b)**(1/6))/(3*b) + A*(-a/b 
)**(1/6)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*b) - A*( 
-a/b)**(1/6)*log(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*b) - 
sqrt(3)*A*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) - sqrt(3) 
/3)/(3*b) - sqrt(3)*A*(-a/b)**(1/6)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6 
)) + sqrt(3)/3)/(3*b) - 2*B*a*sqrt(x)/b**2 - B*a*(-a/b)**(1/6)*log(sqrt(x) 
 - (-a/b)**(1/6))/(3*b**2) + B*a*(-a/b)**(1/6)*log(sqrt(x) + (-a/b)**(1/6) 
)/(3*b**2) - B*a*(-a/b)**(1/6)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/ 
b)**(1/3))/(6*b**2) + B*a*(-a/b)**(1/6)*log(4*sqrt(x)*(-a/b)**(1/6) + 4*x 
+ 4*(-a/b)**(1/3))/(6*b**2) + sqrt(3)*B*a*(-a/b)**(1/6)*atan(2*sqrt(3)*sqr 
t(x)/(3*(-a/b)**(1/6)) - sqrt(3)/3)/(3*b**2) + sqrt(3)*B*a*(-a/b)**(1/6)*a 
tan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqrt(3)/3)/(3*b**2) + 2*B*x**(7/ 
2)/(7*b), True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.31 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {{\left (\frac {\sqrt {3} {\left (B a - A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (B a - A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (B a b^{\frac {1}{3}} - A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )} a}{6 \, b^{2}} + \frac {2 \, {\left (B b x^{\frac {7}{2}} - 7 \, {\left (B a - A b\right )} \sqrt {x}\right )}}{7 \, b^{2}} \] Input:

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

Output:

1/6*(sqrt(3)*(B*a - 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)*(B*a - 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*(B*a*b^(1/3) - A 
*b^(4/3))*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1/3)*b^(1/3)))/(a^(2/3)*b^(1/3)*s 
qrt(a^(1/3)*b^(1/3))) + 2*(B*a^(4/3)*b^(1/3) - 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*(B*a^(4/3)*b^(1/3) - A*a^(1/3)*b^(4/3))*ar 
ctan(-(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 + 2/7*(B*b*x^(7/2) - 7*(B*a - A* 
b)*sqrt(x))/b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.28 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{3}} - \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{3}} + \frac {{\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {{\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {2 \, {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {2 \, {\left (B b^{6} x^{\frac {7}{2}} - 7 \, B a b^{5} \sqrt {x} + 7 \, A b^{6} \sqrt {x}\right )}}{7 \, b^{7}} \] Input:

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

Output:

1/6*sqrt(3)*((a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a 
/b)^(1/6) + x + (a/b)^(1/3))/b^3 - 1/6*sqrt(3)*((a*b^5)^(1/6)*B*a - (a*b^5 
)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/b^3 + 1/3 
*((a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*s 
qrt(x))/(a/b)^(1/6))/b^3 + 1/3*((a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*arc 
tan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/b^3 + 2/3*((a*b^5)^(1/ 
6)*B*a - (a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/b^3 + 2/7*(B*b^6*x 
^(7/2) - 7*B*a*b^5*sqrt(x) + 7*A*b^6*sqrt(x))/b^7
 

Mupad [B] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 1933, normalized size of antiderivative = 8.59 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

x^(1/2)*((2*A)/b - (2*B*a)/b^2) + (2*B*x^(7/2))/(7*b) + ((-a)^(1/6)*atan(( 
((-a)^(1/6)*(A*b - B*a)*((96*x^(1/2)*(B^4*a^8 + A^4*a^4*b^4 + 6*A^2*B^2*a^ 
6*b^2 - 4*A*B^3*a^7*b - 4*A^3*B*a^5*b^3))/b^3 - (96*(-a)^(1/6)*(A*b - B*a) 
*(B^3*a^7 - A^3*a^4*b^3 - 3*A*B^2*a^6*b + 3*A^2*B*a^5*b^2))/b^(19/6))*1i)/ 
(3*b^(13/6)) + ((-a)^(1/6)*(A*b - B*a)*((96*x^(1/2)*(B^4*a^8 + A^4*a^4*b^4 
 + 6*A^2*B^2*a^6*b^2 - 4*A*B^3*a^7*b - 4*A^3*B*a^5*b^3))/b^3 + (96*(-a)^(1 
/6)*(A*b - B*a)*(B^3*a^7 - A^3*a^4*b^3 - 3*A*B^2*a^6*b + 3*A^2*B*a^5*b^2)) 
/b^(19/6))*1i)/(3*b^(13/6)))/(((-a)^(1/6)*(A*b - B*a)*((96*x^(1/2)*(B^4*a^ 
8 + A^4*a^4*b^4 + 6*A^2*B^2*a^6*b^2 - 4*A*B^3*a^7*b - 4*A^3*B*a^5*b^3))/b^ 
3 - (96*(-a)^(1/6)*(A*b - B*a)*(B^3*a^7 - A^3*a^4*b^3 - 3*A*B^2*a^6*b + 3* 
A^2*B*a^5*b^2))/b^(19/6)))/(3*b^(13/6)) - ((-a)^(1/6)*(A*b - B*a)*((96*x^( 
1/2)*(B^4*a^8 + A^4*a^4*b^4 + 6*A^2*B^2*a^6*b^2 - 4*A*B^3*a^7*b - 4*A^3*B* 
a^5*b^3))/b^3 + (96*(-a)^(1/6)*(A*b - B*a)*(B^3*a^7 - A^3*a^4*b^3 - 3*A*B^ 
2*a^6*b + 3*A^2*B*a^5*b^2))/b^(19/6)))/(3*b^(13/6))))*(A*b - B*a)*2i)/(3*b 
^(13/6)) + ((-a)^(1/6)*atan((((-a)^(1/6)*((3^(1/2)*1i)/2 - 1/2)*(A*b - B*a 
)*((96*x^(1/2)*(B^4*a^8 + A^4*a^4*b^4 + 6*A^2*B^2*a^6*b^2 - 4*A*B^3*a^7*b 
- 4*A^3*B*a^5*b^3))/b^3 - (96*(-a)^(1/6)*((3^(1/2)*1i)/2 - 1/2)*(A*b - B*a 
)*(B^3*a^7 - A^3*a^4*b^3 - 3*A*B^2*a^6*b + 3*A^2*B*a^5*b^2))/b^(19/6))*1i) 
/(3*b^(13/6)) + ((-a)^(1/6)*((3^(1/2)*1i)/2 - 1/2)*(A*b - B*a)*((96*x^(1/2 
)*(B^4*a^8 + A^4*a^4*b^4 + 6*A^2*B^2*a^6*b^2 - 4*A*B^3*a^7*b - 4*A^3*B*...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.03 \[ \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {2 \sqrt {x}\, x^{3}}{7} \] Input:

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

Output:

(2*sqrt(x)*x**3)/7