\(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\) [200]

Optimal result

Integrand size = 24, antiderivative size = 167 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {2 b^{5/2} B \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}} \] Output:

-2/7*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(7/2)-2/5*B*(b*x+a)^(5/ 
2)/e^2/(e*x+d)^(5/2)-2/3*b*B*(b*x+a)^(3/2)/e^3/(e*x+d)^(3/2)-2*b^2*B*(b*x+ 
a)^(1/2)/e^4/(e*x+d)^(1/2)+2*b^(5/2)*B*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/ 
2)/(e*x+d)^(1/2))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 \sqrt {a+b x} \left (-15 B d e^3 (a+b x)^3+15 A e^4 (a+b x)^3-21 b B d e^2 (a+b x)^2 (d+e x)+21 a B e^3 (a+b x)^2 (d+e x)-35 b^2 B d e (a+b x) (d+e x)^2+35 a b B e^2 (a+b x) (d+e x)^2-105 b^3 B d (d+e x)^3+105 a b^2 B e (d+e x)^3\right )}{105 e^4 (-b d+a e) (d+e x)^{7/2}}+\frac {2 b^{5/2} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}} \] Input:

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]
 

Output:

(-2*Sqrt[a + b*x]*(-15*B*d*e^3*(a + b*x)^3 + 15*A*e^4*(a + b*x)^3 - 21*b*B 
*d*e^2*(a + b*x)^2*(d + e*x) + 21*a*B*e^3*(a + b*x)^2*(d + e*x) - 35*b^2*B 
*d*e*(a + b*x)*(d + e*x)^2 + 35*a*b*B*e^2*(a + b*x)*(d + e*x)^2 - 105*b^3* 
B*d*(d + e*x)^3 + 105*a*b^2*B*e*(d + e*x)^3))/(105*e^4*(-(b*d) + a*e)*(d + 
 e*x)^(7/2)) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[ 
a + b*x])])/e^(9/2)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 57, 57, 57, 66, 221}

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.

\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {B \left (\frac {b \left (\frac {b \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{3/2}}-\frac {2 \sqrt {a+b x}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 (a+b x)^{3/2}}{3 e (d+e x)^{3/2}}\right )}{e}-\frac {2 (a+b x)^{5/2}}{5 e (d+e x)^{5/2}}\right )}{e}-\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}\)

Input:

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (B*(( 
-2*(a + b*x)^(5/2))/(5*e*(d + e*x)^(5/2)) + (b*((-2*(a + b*x)^(3/2))/(3*e* 
(d + e*x)^(3/2)) + (b*((-2*Sqrt[a + b*x])/(e*Sqrt[d + e*x]) + (2*Sqrt[b]*A 
rcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(3/2)))/e))/e)) 
/e
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 221

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1088\) vs. \(2(133)=266\).

Time = 0.27 (sec) , antiderivative size = 1089, normalized size of antiderivative = 6.52

method	result	size
default	\(\text {Expression too large to display}\)	\(1089\)

Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-1/105*(b*x+a)^(1/2)*(12*B*a^3*d*e^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-1 
05*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^ 
(1/2))*a*b^3*e^5*x^4+105*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e) 
^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^4*d*e^4*x^4+420*B*ln(1/2*(2*b*e*x+2*((e*x+d 
)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^4*d^2*e^3*x^3+630*B*l 
n(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2)) 
*b^4*d^3*e^2*x^2+420*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/ 
2)+a*e+d*b)/(b*e)^(1/2))*b^4*d^4*e*x-105*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x 
+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a*b^3*d^4*e+568*B*a*b^2*d*e^3 
*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+92*B*a^2*b*d*e^3*x*((e*x+d)*(b*x+ 
a))^(1/2)*(b*e)^(1/2)+476*B*a*b^2*d^2*e^2*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)+30*A*a^3*e^4*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-210*B*b^3*d^4*((e*x 
+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+30*A*b^3*e^4*x^3*((e*x+d)*(b*x+a))^(1/2)*(b 
*e)^(1/2)+42*B*a^3*e^4*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+322*B*a*b^2*e 
^4*x^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-352*B*b^3*d*e^3*x^3*((e*x+d)*(b 
*x+a))^(1/2)*(b*e)^(1/2)+90*A*a*b^2*e^4*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)+154*B*a^2*b*e^4*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-812*B*b^3*d^ 
2*e^2*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+90*A*a^2*b*e^4*x*((e*x+d)*(b 
*x+a))^(1/2)*(b*e)^(1/2)-700*B*b^3*d^3*e*x*((e*x+d)*(b*x+a))^(1/2)*(b*e)^( 
1/2)+28*B*a^2*b*d^2*e^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+140*B*a*b^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (133) = 266\).

Time = 8.13 (sec) , antiderivative size = 1053, normalized size of antiderivative = 6.31 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")
 

Output:

[1/210*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 
 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^ 
3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(b/e)*log(8*b^2*e^2*x^2 
+ b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + 
 a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(105*B*b^3*d^4 
- 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + ( 
176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 
 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e 
 - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)* 
x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 - a*e^9) 
*x^4 + 4*(b*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b* 
d^4*e^5 - a*d^3*e^6)*x), -1/105*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d 
*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3 
*d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqr 
t(-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt( 
-b/e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) + 2*(105*B*b^3*d^4 - 70*B*a 
*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + (176*B*b^ 
3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a 
*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B 
*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)*x)*sq...
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(9/2),x)
 

Output:

Integral((A + B*x)*(a + b*x)**(5/2)/(d + e*x)**(9/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(a*e-b*d)>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (133) = 266\).

Time = 0.40 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")
 

Output:

-2*B*b^2*abs(b)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)* 
b*e - a*b*e)))/(sqrt(b*e)*e^4) - 2/105*(((b*x + a)*((176*B*b^10*d^3*e^6*ab 
s(b) - 513*B*a*b^9*d^2*e^7*abs(b) - 15*A*b^10*d^2*e^7*abs(b) + 498*B*a^2*b 
^8*d*e^8*abs(b) + 30*A*a*b^9*d*e^8*abs(b) - 161*B*a^3*b^7*e^9*abs(b) - 15* 
A*a^2*b^8*e^9*abs(b))*(b*x + a)/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3 
*d*e^9 - a^3*b^2*e^10) + 406*(B*b^11*d^4*e^5*abs(b) - 4*B*a*b^10*d^3*e^6*a 
bs(b) + 6*B*a^2*b^9*d^2*e^7*abs(b) - 4*B*a^3*b^8*d*e^8*abs(b) + B*a^4*b^7* 
e^9*abs(b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^1 
0)) + 350*(B*b^12*d^5*e^4*abs(b) - 5*B*a*b^11*d^4*e^5*abs(b) + 10*B*a^2*b^ 
10*d^3*e^6*abs(b) - 10*B*a^3*b^9*d^2*e^7*abs(b) + 5*B*a^4*b^8*d*e^8*abs(b) 
 - B*a^5*b^7*e^9*abs(b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 
- a^3*b^2*e^10))*(b*x + a) + 105*(B*b^13*d^6*e^3*abs(b) - 6*B*a*b^12*d^5*e 
^4*abs(b) + 15*B*a^2*b^11*d^4*e^5*abs(b) - 20*B*a^3*b^10*d^3*e^6*abs(b) + 
15*B*a^4*b^9*d^2*e^7*abs(b) - 6*B*a^5*b^8*d*e^8*abs(b) + B*a^6*b^7*e^9*abs 
(b))/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10))*sqr 
t(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2),x)
 

Output:

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 94.66 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.49 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} e^{4}}{7}-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b d \,e^{3}}{5}-\frac {44 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b \,e^{4} x}{35}-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d^{2} e^{2}}{3}-\frac {32 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d \,e^{3} x}{15}-\frac {244 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} e^{4} x^{2}}{105}-2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{3} e -\frac {20 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{2} e^{2} x}{3}-\frac {116 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d \,e^{3} x^{2}}{15}-\frac {352 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} e^{4} x^{3}}{105}+2 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{4}+8 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{3} e x +12 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d^{2} e^{2} x^{2}+8 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} d \,e^{3} x^{3}+2 \sqrt {e}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {e x +d}}{\sqrt {a e -b d}}\right ) b^{3} e^{4} x^{4}+\frac {16 \sqrt {e}\, \sqrt {b}\, b^{3} d^{4}}{15}+\frac {64 \sqrt {e}\, \sqrt {b}\, b^{3} d^{3} e x}{15}+\frac {32 \sqrt {e}\, \sqrt {b}\, b^{3} d^{2} e^{2} x^{2}}{5}+\frac {64 \sqrt {e}\, \sqrt {b}\, b^{3} d \,e^{3} x^{3}}{15}+\frac {16 \sqrt {e}\, \sqrt {b}\, b^{3} e^{4} x^{4}}{15}}{e^{5} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x)
 

Output:

(2*( - 15*sqrt(d + e*x)*sqrt(a + b*x)*a**3*e**4 - 21*sqrt(d + e*x)*sqrt(a 
+ b*x)*a**2*b*d*e**3 - 66*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b*e**4*x - 35*s 
qrt(d + e*x)*sqrt(a + b*x)*a*b**2*d**2*e**2 - 112*sqrt(d + e*x)*sqrt(a + b 
*x)*a*b**2*d*e**3*x - 122*sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*e**4*x**2 - 1 
05*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d**3*e - 350*sqrt(d + e*x)*sqrt(a + b* 
x)*b**3*d**2*e**2*x - 406*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d*e**3*x**2 - 1 
76*sqrt(d + e*x)*sqrt(a + b*x)*b**3*e**4*x**3 + 105*sqrt(e)*sqrt(b)*log((s 
qrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**3*d**4 + 
 420*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/s 
qrt(a*e - b*d))*b**3*d**3*e*x + 630*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + 
b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**3*d**2*e**2*x**2 + 420*s 
qrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a* 
e - b*d))*b**3*d*e**3*x**3 + 105*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x 
) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*b**3*e**4*x**4 + 56*sqrt(e)*sq 
rt(b)*b**3*d**4 + 224*sqrt(e)*sqrt(b)*b**3*d**3*e*x + 336*sqrt(e)*sqrt(b)* 
b**3*d**2*e**2*x**2 + 224*sqrt(e)*sqrt(b)*b**3*d*e**3*x**3 + 56*sqrt(e)*sq 
rt(b)*b**3*e**4*x**4))/(105*e**5*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4 
*d*e**3*x**3 + e**4*x**4))