\(\int \frac {x^{9/2} (A+B x)}{(b x+c x^2)^2} \, dx\) [89]

Optimal result

Integrand size = 22, antiderivative size = 116 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {b (7 b B-5 A c) \sqrt {x}}{c^4}-\frac {(7 b B-5 A c) x^{3/2}}{3 c^3}+\frac {2 B x^{5/2}}{5 c^2}+\frac {(b B-A c) x^{5/2}}{c^2 (b+c x)}-\frac {b^{3/2} (7 b B-5 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}} \] Output:

b*(-5*A*c+7*B*b)*x^(1/2)/c^4-1/3*(-5*A*c+7*B*b)*x^(3/2)/c^3+2/5*B*x^(5/2)/ 
c^2+(-A*c+B*b)*x^(5/2)/c^2/(c*x+b)-b^(3/2)*(-5*A*c+7*B*b)*arctan(c^(1/2)*x 
^(1/2)/b^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {\sqrt {x} \left (105 b^3 B+2 c^3 x^2 (5 A+3 B x)-2 b c^2 x (25 A+7 B x)+b^2 (-75 A c+70 B c x)\right )}{15 c^4 (b+c x)}-\frac {b^{3/2} (7 b B-5 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}} \] Input:

Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^2,x]
 

Output:

(Sqrt[x]*(105*b^3*B + 2*c^3*x^2*(5*A + 3*B*x) - 2*b*c^2*x*(25*A + 7*B*x) + 
 b^2*(-75*A*c + 70*B*c*x)))/(15*c^4*(b + c*x)) - (b^(3/2)*(7*b*B - 5*A*c)* 
ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/c^(9/2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {9, 87, 60, 60, 60, 73, 218}

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.

\(\displaystyle \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {x^{5/2} (A+B x)}{(b+c x)^2}dx\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {(7 b B-5 A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{3/2}}\right )}{c}\right )}{c}\right )}{2 b c}-\frac {x^{7/2} (b B-A c)}{b c (b+c x)}\)

Input:

Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^2,x]
 

Output:

-(((b*B - A*c)*x^(7/2))/(b*c*(b + c*x))) + ((7*b*B - 5*A*c)*((2*x^(5/2))/( 
5*c) - (b*((2*x^(3/2))/(3*c) - (b*((2*Sqrt[x])/c - (2*Sqrt[b]*ArcTan[(Sqrt 
[c]*Sqrt[x])/Sqrt[b]])/c^(3/2)))/c))/c))/(2*b*c)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 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]
 

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {2 \left (-3 B \,c^{2} x^{2}-5 A \,c^{2} x +10 B b c x +30 A b c -45 B \,b^{2}\right ) \sqrt {x}}{15 c^{4}}+\frac {b^{2} \left (\frac {2 \left (-\frac {A c}{2}+\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -7 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}}\right )}{c^{4}}\)	\(99\)
derivativedivides	\(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {5}{2}}}{5}-\frac {A \,c^{2} x^{\frac {3}{2}}}{3}+\frac {2 B b c \,x^{\frac {3}{2}}}{3}+2 A b c \sqrt {x}-3 B \,b^{2} \sqrt {x}\right )}{c^{4}}+\frac {2 b^{2} \left (\frac {\left (-\frac {A c}{2}+\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -7 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{c^{4}}\)	\(107\)
default	\(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {5}{2}}}{5}-\frac {A \,c^{2} x^{\frac {3}{2}}}{3}+\frac {2 B b c \,x^{\frac {3}{2}}}{3}+2 A b c \sqrt {x}-3 B \,b^{2} \sqrt {x}\right )}{c^{4}}+\frac {2 b^{2} \left (\frac {\left (-\frac {A c}{2}+\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -7 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{c^{4}}\)	\(107\)

Input:

int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/15*(-3*B*c^2*x^2-5*A*c^2*x+10*B*b*c*x+30*A*b*c-45*B*b^2)*x^(1/2)/c^4+b^ 
2/c^4*(2*(-1/2*A*c+1/2*B*b)*x^(1/2)/(c*x+b)+(5*A*c-7*B*b)/(b*c)^(1/2)*arct 
an(c*x^(1/2)/(b*c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.50 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\left [-\frac {15 \, {\left (7 \, B b^{3} - 5 \, A b^{2} c + {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x + 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \, {\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {x}}{30 \, {\left (c^{5} x + b c^{4}\right )}}, -\frac {15 \, {\left (7 \, B b^{3} - 5 \, A b^{2} c + {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) - {\left (6 \, B c^{3} x^{3} + 105 \, B b^{3} - 75 \, A b^{2} c - 2 \, {\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x\right )} \sqrt {x}}{15 \, {\left (c^{5} x + b c^{4}\right )}}\right ] \] Input:

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^2,x, algorithm="fricas")
 

Output:

[-1/30*(15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c - 5*A*b*c^2)*x)*sqrt(-b/c)*lo 
g((c*x + 2*c*sqrt(x)*sqrt(-b/c) - b)/(c*x + b)) - 2*(6*B*c^3*x^3 + 105*B*b 
^3 - 75*A*b^2*c - 2*(7*B*b*c^2 - 5*A*c^3)*x^2 + 10*(7*B*b^2*c - 5*A*b*c^2) 
*x)*sqrt(x))/(c^5*x + b*c^4), -1/15*(15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c 
- 5*A*b*c^2)*x)*sqrt(b/c)*arctan(c*sqrt(x)*sqrt(b/c)/b) - (6*B*c^3*x^3 + 1 
05*B*b^3 - 75*A*b^2*c - 2*(7*B*b*c^2 - 5*A*c^3)*x^2 + 10*(7*B*b^2*c - 5*A* 
b*c^2)*x)*sqrt(x))/(c^5*x + b*c^4)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (112) = 224\).

Time = 118.26 (sec) , antiderivative size = 877, normalized size of antiderivative = 7.56 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**2,x)
 

Output:

Piecewise((zoo*(2*A*x**(3/2)/3 + 2*B*x**(5/2)/5), Eq(b, 0) & Eq(c, 0)), (( 
2*A*x**(7/2)/7 + 2*B*x**(9/2)/9)/b**2, Eq(c, 0)), ((2*A*x**(3/2)/3 + 2*B*x 
**(5/2)/5)/c**2, Eq(b, 0)), (75*A*b**3*c*log(sqrt(x) - sqrt(-b/c))/(30*b*c 
**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) - 75*A*b**3*c*log(sqrt(x) + sqrt(-b 
/c))/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) - 150*A*b**2*c**2*sqrt( 
x)*sqrt(-b/c)/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) + 75*A*b**2*c* 
*2*x*log(sqrt(x) - sqrt(-b/c))/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c 
)) - 75*A*b**2*c**2*x*log(sqrt(x) + sqrt(-b/c))/(30*b*c**5*sqrt(-b/c) + 30 
*c**6*x*sqrt(-b/c)) - 100*A*b*c**3*x**(3/2)*sqrt(-b/c)/(30*b*c**5*sqrt(-b/ 
c) + 30*c**6*x*sqrt(-b/c)) + 20*A*c**4*x**(5/2)*sqrt(-b/c)/(30*b*c**5*sqrt 
(-b/c) + 30*c**6*x*sqrt(-b/c)) - 105*B*b**4*log(sqrt(x) - sqrt(-b/c))/(30* 
b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) + 105*B*b**4*log(sqrt(x) + sqrt( 
-b/c))/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) + 210*B*b**3*c*sqrt(x 
)*sqrt(-b/c)/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) - 105*B*b**3*c* 
x*log(sqrt(x) - sqrt(-b/c))/(30*b*c**5*sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)) 
+ 105*B*b**3*c*x*log(sqrt(x) + sqrt(-b/c))/(30*b*c**5*sqrt(-b/c) + 30*c**6 
*x*sqrt(-b/c)) + 140*B*b**2*c**2*x**(3/2)*sqrt(-b/c)/(30*b*c**5*sqrt(-b/c) 
 + 30*c**6*x*sqrt(-b/c)) - 28*B*b*c**3*x**(5/2)*sqrt(-b/c)/(30*b*c**5*sqrt 
(-b/c) + 30*c**6*x*sqrt(-b/c)) + 12*B*c**4*x**(7/2)*sqrt(-b/c)/(30*b*c**5* 
sqrt(-b/c) + 30*c**6*x*sqrt(-b/c)), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (B b^{3} - A b^{2} c\right )} \sqrt {x}}{c^{5} x + b c^{4}} - \frac {{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {2 \, {\left (3 \, B c^{2} x^{\frac {5}{2}} - 5 \, {\left (2 \, B b c - A c^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (3 \, B b^{2} - 2 \, A b c\right )} \sqrt {x}\right )}}{15 \, c^{4}} \] Input:

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^2,x, algorithm="maxima")
 

Output:

(B*b^3 - A*b^2*c)*sqrt(x)/(c^5*x + b*c^4) - (7*B*b^3 - 5*A*b^2*c)*arctan(c 
*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c^4) + 2/15*(3*B*c^2*x^(5/2) - 5*(2*B*b*c - 
 A*c^2)*x^(3/2) + 15*(3*B*b^2 - 2*A*b*c)*sqrt(x))/c^4
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {B b^{3} \sqrt {x} - A b^{2} c \sqrt {x}}{{\left (c x + b\right )} c^{4}} + \frac {2 \, {\left (3 \, B c^{8} x^{\frac {5}{2}} - 10 \, B b c^{7} x^{\frac {3}{2}} + 5 \, A c^{8} x^{\frac {3}{2}} + 45 \, B b^{2} c^{6} \sqrt {x} - 30 \, A b c^{7} \sqrt {x}\right )}}{15 \, c^{10}} \] Input:

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^2,x, algorithm="giac")
 

Output:

-(7*B*b^3 - 5*A*b^2*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c^4) + (B*b^ 
3*sqrt(x) - A*b^2*c*sqrt(x))/((c*x + b)*c^4) + 2/15*(3*B*c^8*x^(5/2) - 10* 
B*b*c^7*x^(3/2) + 5*A*c^8*x^(3/2) + 45*B*b^2*c^6*sqrt(x) - 30*A*b*c^7*sqrt 
(x))/c^10
 

Mupad [B] (verification not implemented)

Time = 5.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.26 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=x^{3/2}\,\left (\frac {2\,A}{3\,c^2}-\frac {4\,B\,b}{3\,c^3}\right )-\sqrt {x}\,\left (\frac {2\,b\,\left (\frac {2\,A}{c^2}-\frac {4\,B\,b}{c^3}\right )}{c}+\frac {2\,B\,b^2}{c^4}\right )+\frac {2\,B\,x^{5/2}}{5\,c^2}+\frac {\sqrt {x}\,\left (B\,b^3-A\,b^2\,c\right )}{x\,c^5+b\,c^4}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,\sqrt {x}\,\left (5\,A\,c-7\,B\,b\right )}{7\,B\,b^3-5\,A\,b^2\,c}\right )\,\left (5\,A\,c-7\,B\,b\right )}{c^{9/2}} \] Input:

int((x^(9/2)*(A + B*x))/(b*x + c*x^2)^2,x)
 

Output:

x^(3/2)*((2*A)/(3*c^2) - (4*B*b)/(3*c^3)) - x^(1/2)*((2*b*((2*A)/c^2 - (4* 
B*b)/c^3))/c + (2*B*b^2)/c^4) + (2*B*x^(5/2))/(5*c^2) + (x^(1/2)*(B*b^3 - 
A*b^2*c))/(b*c^4 + c^5*x) - (b^(3/2)*atan((b^(3/2)*c^(1/2)*x^(1/2)*(5*A*c 
- 7*B*b))/(7*B*b^3 - 5*A*b^2*c))*(5*A*c - 7*B*b))/c^(9/2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.58 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {75 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c +75 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) a b \,c^{2} x -105 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) b^{4}-105 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c x -75 \sqrt {x}\, a \,b^{2} c^{2}-50 \sqrt {x}\, a b \,c^{3} x +10 \sqrt {x}\, a \,c^{4} x^{2}+105 \sqrt {x}\, b^{4} c +70 \sqrt {x}\, b^{3} c^{2} x -14 \sqrt {x}\, b^{2} c^{3} x^{2}+6 \sqrt {x}\, b \,c^{4} x^{3}}{15 c^{5} \left (c x +b \right )} \] Input:

int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^2,x)
 

Output:

(75*sqrt(c)*sqrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*a*b**2*c + 75*sqrt 
(c)*sqrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*a*b*c**2*x - 105*sqrt(c)*s 
qrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*b**4 - 105*sqrt(c)*sqrt(b)*atan 
((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*b**3*c*x - 75*sqrt(x)*a*b**2*c**2 - 50*sqr 
t(x)*a*b*c**3*x + 10*sqrt(x)*a*c**4*x**2 + 105*sqrt(x)*b**4*c + 70*sqrt(x) 
*b**3*c**2*x - 14*sqrt(x)*b**2*c**3*x**2 + 6*sqrt(x)*b*c**4*x**3)/(15*c**5 
*(b + c*x))