\(\int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx\) [229]

Optimal result

Integrand size = 24, antiderivative size = 246 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=-\frac {2 (A b-a B)}{3 b (b d-a e) (a+b x)^{3/2} (d+e x)^{5/2}}-\frac {2 (3 b B d-8 A b e+5 a B e)}{3 b (b d-a e)^2 \sqrt {a+b x} (d+e x)^{5/2}}-\frac {4 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{5 b (b d-a e)^3 (d+e x)^{5/2}}-\frac {16 e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^4 (d+e x)^{3/2}}-\frac {32 b e (3 b B d-8 A b e+5 a B e) \sqrt {a+b x}}{15 (b d-a e)^5 \sqrt {d+e x}} \] Output:

1/3*(-2*A*b+2*B*a)/b/(-a*e+b*d)/(b*x+a)^(3/2)/(e*x+d)^(5/2)-2/3*(-8*A*b*e+ 
5*B*a*e+3*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^(1/2)/(e*x+d)^(5/2)-4/5*e*(-8*A*b* 
e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^3/(e*x+d)^(5/2)-16/15*e*(-8* 
A*b*e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(3/2)-32/15*b*e* 
(-8*A*b*e+5*B*a*e+3*B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^5/(e*x+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=-\frac {2 \left (3 B d e^3 (a+b x)^4-3 A e^4 (a+b x)^4-15 b B d e^2 (a+b x)^3 (d+e x)+20 A b e^3 (a+b x)^3 (d+e x)-5 a B e^3 (a+b x)^3 (d+e x)+45 b^2 B d e (a+b x)^2 (d+e x)^2-90 A b^2 e^2 (a+b x)^2 (d+e x)^2+45 a b B e^2 (a+b x)^2 (d+e x)^2+15 b^3 B d (a+b x) (d+e x)^3-60 A b^3 e (a+b x) (d+e x)^3+45 a b^2 B e (a+b x) (d+e x)^3+5 A b^4 (d+e x)^4-5 a b^3 B (d+e x)^4\right )}{15 (b d-a e)^5 (a+b x)^{3/2} (d+e x)^{5/2}} \] Input:

Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]
 

Output:

(-2*(3*B*d*e^3*(a + b*x)^4 - 3*A*e^4*(a + b*x)^4 - 15*b*B*d*e^2*(a + b*x)^ 
3*(d + e*x) + 20*A*b*e^3*(a + b*x)^3*(d + e*x) - 5*a*B*e^3*(a + b*x)^3*(d 
+ e*x) + 45*b^2*B*d*e*(a + b*x)^2*(d + e*x)^2 - 90*A*b^2*e^2*(a + b*x)^2*( 
d + e*x)^2 + 45*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 15*b^3*B*d*(a + b*x)*( 
d + e*x)^3 - 60*A*b^3*e*(a + b*x)*(d + e*x)^3 + 45*a*b^2*B*e*(a + b*x)*(d 
+ e*x)^3 + 5*A*b^4*(d + e*x)^4 - 5*a*b^3*B*(d + e*x)^4))/(15*(b*d - a*e)^5 
*(a + b*x)^(3/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 55, 55, 55, 48}

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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(5 a B e-8 A b e+3 b B d) \int \frac {1}{(a+b x)^{3/2} (d+e x)^{7/2}}dx}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(5 a B e-8 A b e+3 b B d) \left (-\frac {6 e \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}}dx}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(5 a B e-8 A b e+3 b B d) \left (-\frac {6 e \left (\frac {4 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(5 a B e-8 A b e+3 b B d) \left (-\frac {6 e \left (\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}\right )}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (-\frac {6 e \left (\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{b d-a e}-\frac {2}{\sqrt {a+b x} (d+e x)^{5/2} (b d-a e)}\right ) (5 a B e-8 A b e+3 b B d)}{3 b (b d-a e)}-\frac {2 (A b-a B)}{3 b (a+b x)^{3/2} (d+e x)^{5/2} (b d-a e)}\)

Input:

Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x]
 

Output:

(-2*(A*b - a*B))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)*(d + e*x)^(5/2)) + ((3*b 
*B*d - 8*A*b*e + 5*a*B*e)*(-2/((b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2)) 
- (6*e*((2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (4*b*((2*Sqrt[ 
a + b*x])/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (4*b*Sqrt[a + b*x])/(3*(b*d - 
a*e)^2*Sqrt[d + e*x])))/(5*(b*d - a*e))))/(b*d - a*e)))/(3*b*(b*d - a*e))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

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]))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(217)=434\).

Time = 0.35 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a e -d b \right )^{5}}\)	\(449\)
gosper	\(-\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)
orering	\(-\frac {2 \left (128 A \,b^{4} e^{4} x^{4}-80 B a \,b^{3} e^{4} x^{4}-48 B \,b^{4} d \,e^{3} x^{4}+192 A a \,b^{3} e^{4} x^{3}+320 A \,b^{4} d \,e^{3} x^{3}-120 B \,a^{2} b^{2} e^{4} x^{3}-272 B a \,b^{3} d \,e^{3} x^{3}-120 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}+480 A a \,b^{3} d \,e^{3} x^{2}+240 A \,b^{4} d^{2} e^{2} x^{2}-30 B \,a^{3} b \,e^{4} x^{2}-318 B \,a^{2} b^{2} d \,e^{3} x^{2}-330 B a \,b^{3} d^{2} e^{2} x^{2}-90 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +120 A \,a^{2} b^{2} d \,e^{3} x +360 A a \,b^{3} d^{2} e^{2} x +40 A \,b^{4} d^{3} e x +5 B \,a^{4} e^{4} x -72 B \,a^{3} b d \,e^{3} x -270 B \,a^{2} b^{2} d^{2} e^{2} x -160 B a \,b^{3} d^{3} e x -15 B \,b^{4} d^{4} x +3 A \,a^{4} e^{4}-20 A \,a^{3} b d \,e^{3}+90 A \,a^{2} b^{2} d^{2} e^{2}+60 A a \,b^{3} d^{3} e -5 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-30 B \,a^{3} b \,d^{2} e^{2}-90 B \,a^{2} b^{2} d^{3} e -10 B a \,b^{3} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)

Input:

int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(128*A*b^4*e^4*x^4-80*B*a*b^3*e^4*x^4-48*B*b^4*d*e^3*x^4+192*A*a*b^3 
*e^4*x^3+320*A*b^4*d*e^3*x^3-120*B*a^2*b^2*e^4*x^3-272*B*a*b^3*d*e^3*x^3-1 
20*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2+480*A*a*b^3*d*e^3*x^2+240*A*b^4* 
d^2*e^2*x^2-30*B*a^3*b*e^4*x^2-318*B*a^2*b^2*d*e^3*x^2-330*B*a*b^3*d^2*e^2 
*x^2-90*B*b^4*d^3*e*x^2-8*A*a^3*b*e^4*x+120*A*a^2*b^2*d*e^3*x+360*A*a*b^3* 
d^2*e^2*x+40*A*b^4*d^3*e*x+5*B*a^4*e^4*x-72*B*a^3*b*d*e^3*x-270*B*a^2*b^2* 
d^2*e^2*x-160*B*a*b^3*d^3*e*x-15*B*b^4*d^4*x+3*A*a^4*e^4-20*A*a^3*b*d*e^3+ 
90*A*a^2*b^2*d^2*e^2+60*A*a*b^3*d^3*e-5*A*b^4*d^4+2*B*a^4*d*e^3-30*B*a^3*b 
*d^2*e^2-90*B*a^2*b^2*d^3*e-10*B*a*b^3*d^4)/(e*x+d)^(5/2)/(b*x+a)^(3/2)/(a 
*e-b*d)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (216) = 432\).

Time = 24.16 (sec) , antiderivative size = 916, normalized size of antiderivative = 3.72 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

2/15*(3*A*a^4*e^4 - 5*(2*B*a*b^3 + A*b^4)*d^4 - 30*(3*B*a^2*b^2 - 2*A*a*b^ 
3)*d^3*e - 30*(B*a^3*b - 3*A*a^2*b^2)*d^2*e^2 + 2*(B*a^4 - 10*A*a^3*b)*d*e 
^3 - 16*(3*B*b^4*d*e^3 + (5*B*a*b^3 - 8*A*b^4)*e^4)*x^4 - 8*(15*B*b^4*d^2* 
e^2 + 2*(17*B*a*b^3 - 20*A*b^4)*d*e^3 + 3*(5*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x 
^3 - 6*(15*B*b^4*d^3*e + 5*(11*B*a*b^3 - 8*A*b^4)*d^2*e^2 + (53*B*a^2*b^2 
- 80*A*a*b^3)*d*e^3 + (5*B*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 - (15*B*b^4*d^4 + 
 40*(4*B*a*b^3 - A*b^4)*d^3*e + 90*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2 + 24* 
(3*B*a^3*b - 5*A*a^2*b^2)*d*e^3 - (5*B*a^4 - 8*A*a^3*b)*e^4)*x)*sqrt(b*x + 
 a)*sqrt(e*x + d)/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10*a^4*b^3*d^6*e^2 - 10 
*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5*e^3 - 5*a*b^6* 
d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b^3*d*e^7 - a^5* 
b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10 
*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4 
+ (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 
35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^ 
7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b^3 
*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6 
*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b 
^2*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x)
 

Sympy [F]

\[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=\int \frac {A + B x}{\left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((A + B*x)/((a + b*x)**(5/2)*(d + e*x)**(7/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),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. 2081 vs. \(2 (216) = 432\).

Time = 1.00 (sec) , antiderivative size = 2081, normalized size of antiderivative = 8.46 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

-2/15*((b*x + a)*((33*B*b^15*d^8*e^5 - 191*B*a*b^14*d^7*e^6 - 73*A*b^15*d^ 
7*e^6 + 413*B*a^2*b^13*d^6*e^7 + 511*A*a*b^14*d^6*e^7 - 315*B*a^3*b^12*d^5 
*e^8 - 1533*A*a^2*b^13*d^5*e^8 - 245*B*a^4*b^11*d^4*e^9 + 2555*A*a^3*b^12* 
d^4*e^9 + 707*B*a^5*b^10*d^3*e^10 - 2555*A*a^4*b^11*d^3*e^10 - 609*B*a^6*b 
^9*d^2*e^11 + 1533*A*a^5*b^10*d^2*e^11 + 247*B*a^7*b^8*d*e^12 - 511*A*a^6* 
b^9*d*e^12 - 40*B*a^8*b^7*e^13 + 73*A*a^7*b^8*e^13)*(b*x + a)/(b^14*d^12*e 
^2*abs(b) - 12*a*b^13*d^11*e^3*abs(b) + 66*a^2*b^12*d^10*e^4*abs(b) - 220* 
a^3*b^11*d^9*e^5*abs(b) + 495*a^4*b^10*d^8*e^6*abs(b) - 792*a^5*b^9*d^7*e^ 
7*abs(b) + 924*a^6*b^8*d^6*e^8*abs(b) - 792*a^7*b^7*d^5*e^9*abs(b) + 495*a 
^8*b^6*d^4*e^10*abs(b) - 220*a^9*b^5*d^3*e^11*abs(b) + 66*a^10*b^4*d^2*e^1 
2*abs(b) - 12*a^11*b^3*d*e^13*abs(b) + a^12*b^2*e^14*abs(b)) + 5*(15*B*b^1 
6*d^9*e^4 - 103*B*a*b^15*d^8*e^5 - 32*A*b^16*d^8*e^5 + 284*B*a^2*b^14*d^7* 
e^6 + 256*A*a*b^15*d^7*e^6 - 364*B*a^3*b^13*d^6*e^7 - 896*A*a^2*b^14*d^6*e 
^7 + 98*B*a^4*b^12*d^5*e^8 + 1792*A*a^3*b^13*d^5*e^8 + 350*B*a^5*b^11*d^4* 
e^9 - 2240*A*a^4*b^12*d^4*e^9 - 532*B*a^6*b^10*d^3*e^10 + 1792*A*a^5*b^11* 
d^3*e^10 + 356*B*a^7*b^9*d^2*e^11 - 896*A*a^6*b^10*d^2*e^11 - 121*B*a^8*b^ 
8*d*e^12 + 256*A*a^7*b^9*d*e^12 + 17*B*a^9*b^7*e^13 - 32*A*a^8*b^8*e^13)/( 
b^14*d^12*e^2*abs(b) - 12*a*b^13*d^11*e^3*abs(b) + 66*a^2*b^12*d^10*e^4*ab 
s(b) - 220*a^3*b^11*d^9*e^5*abs(b) + 495*a^4*b^10*d^8*e^6*abs(b) - 792*a^5 
*b^9*d^7*e^7*abs(b) + 924*a^6*b^8*d^6*e^8*abs(b) - 792*a^7*b^7*d^5*e^9*...
 

Mupad [B] (verification not implemented)

Time = 2.04 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {32\,b^2\,x^4\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,{\left (a\,e-b\,d\right )}^5}+\frac {-4\,B\,a^4\,d\,e^3-6\,A\,a^4\,e^4+60\,B\,a^3\,b\,d^2\,e^2+40\,A\,a^3\,b\,d\,e^3+180\,B\,a^2\,b^2\,d^3\,e-180\,A\,a^2\,b^2\,d^2\,e^2+20\,B\,a\,b^3\,d^4-120\,A\,a\,b^3\,d^3\,e+10\,A\,b^4\,d^4}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,x^2\,\left (a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{5\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )\,\left (-a^3\,e^3+15\,a^2\,b\,d\,e^2+45\,a\,b^2\,d^2\,e+5\,b^3\,d^3\right )}{15\,b\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b\,x^3\,\left (3\,a\,e+5\,b\,d\right )\,\left (5\,B\,a\,e-8\,A\,b\,e+3\,B\,b\,d\right )}{15\,e\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a\,d^3\,\sqrt {a+b\,x}}{b\,e^3}+\frac {x^3\,\left (a\,e+3\,b\,d\right )\,\sqrt {a+b\,x}}{b\,e}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e^3}} \] Input:

int((A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(7/2)),x)
 

Output:

((d + e*x)^(1/2)*((32*b^2*x^4*(5*B*a*e - 8*A*b*e + 3*B*b*d))/(15*(a*e - b* 
d)^5) + (10*A*b^4*d^4 - 6*A*a^4*e^4 + 20*B*a*b^3*d^4 - 4*B*a^4*d*e^3 + 180 
*B*a^2*b^2*d^3*e + 60*B*a^3*b*d^2*e^2 - 180*A*a^2*b^2*d^2*e^2 - 120*A*a*b^ 
3*d^3*e + 40*A*a^3*b*d*e^3)/(15*b*e^3*(a*e - b*d)^5) + (4*x^2*(a^2*e^2 + 5 
*b^2*d^2 + 10*a*b*d*e)*(5*B*a*e - 8*A*b*e + 3*B*b*d))/(5*e^2*(a*e - b*d)^5 
) + (2*x*(5*B*a*e - 8*A*b*e + 3*B*b*d)*(5*b^3*d^3 - a^3*e^3 + 45*a*b^2*d^2 
*e + 15*a^2*b*d*e^2))/(15*b*e^3*(a*e - b*d)^5) + (16*b*x^3*(3*a*e + 5*b*d) 
*(5*B*a*e - 8*A*b*e + 3*B*b*d))/(15*e*(a*e - b*d)^5)))/(x^4*(a + b*x)^(1/2 
) + (a*d^3*(a + b*x)^(1/2))/(b*e^3) + (x^3*(a*e + 3*b*d)*(a + b*x)^(1/2))/ 
(b*e) + (3*d*x^2*(a*e + b*d)*(a + b*x)^(1/2))/(b*e^2) + (d^2*x*(3*a*e + b* 
d)*(a + b*x)^(1/2))/(b*e^3))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.09 \[ \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{7/2}} \, dx=\frac {\frac {32 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} d^{3}}{5}+\frac {96 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} d^{2} e x}{5}+\frac {96 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} d \,e^{2} x^{2}}{5}+\frac {32 \sqrt {e}\, \sqrt {b}\, \sqrt {b x +a}\, b^{2} e^{3} x^{3}}{5}-\frac {2 \sqrt {e x +d}\, a^{3} e^{3}}{5}+2 \sqrt {e x +d}\, a^{2} b d \,e^{2}+\frac {4 \sqrt {e x +d}\, a^{2} b \,e^{3} x}{5}-6 \sqrt {e x +d}\, a \,b^{2} d^{2} e -8 \sqrt {e x +d}\, a \,b^{2} d \,e^{2} x -\frac {16 \sqrt {e x +d}\, a \,b^{2} e^{3} x^{2}}{5}-2 \sqrt {e x +d}\, b^{3} d^{3}-12 \sqrt {e x +d}\, b^{3} d^{2} e x -16 \sqrt {e x +d}\, b^{3} d \,e^{2} x^{2}-\frac {32 \sqrt {e x +d}\, b^{3} e^{3} x^{3}}{5}}{\sqrt {b x +a}\, \left (a^{4} e^{7} x^{3}-4 a^{3} b d \,e^{6} x^{3}+6 a^{2} b^{2} d^{2} e^{5} x^{3}-4 a \,b^{3} d^{3} e^{4} x^{3}+b^{4} d^{4} e^{3} x^{3}+3 a^{4} d \,e^{6} x^{2}-12 a^{3} b \,d^{2} e^{5} x^{2}+18 a^{2} b^{2} d^{3} e^{4} x^{2}-12 a \,b^{3} d^{4} e^{3} x^{2}+3 b^{4} d^{5} e^{2} x^{2}+3 a^{4} d^{2} e^{5} x -12 a^{3} b \,d^{3} e^{4} x +18 a^{2} b^{2} d^{4} e^{3} x -12 a \,b^{3} d^{5} e^{2} x +3 b^{4} d^{6} e x +a^{4} d^{3} e^{4}-4 a^{3} b \,d^{4} e^{3}+6 a^{2} b^{2} d^{5} e^{2}-4 a \,b^{3} d^{6} e +b^{4} d^{7}\right )} \] Input:

int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(7/2),x)
 

Output:

(2*(16*sqrt(e)*sqrt(b)*sqrt(a + b*x)*b**2*d**3 + 48*sqrt(e)*sqrt(b)*sqrt(a 
 + b*x)*b**2*d**2*e*x + 48*sqrt(e)*sqrt(b)*sqrt(a + b*x)*b**2*d*e**2*x**2 
+ 16*sqrt(e)*sqrt(b)*sqrt(a + b*x)*b**2*e**3*x**3 - sqrt(d + e*x)*a**3*e** 
3 + 5*sqrt(d + e*x)*a**2*b*d*e**2 + 2*sqrt(d + e*x)*a**2*b*e**3*x - 15*sqr 
t(d + e*x)*a*b**2*d**2*e - 20*sqrt(d + e*x)*a*b**2*d*e**2*x - 8*sqrt(d + e 
*x)*a*b**2*e**3*x**2 - 5*sqrt(d + e*x)*b**3*d**3 - 30*sqrt(d + e*x)*b**3*d 
**2*e*x - 40*sqrt(d + e*x)*b**3*d*e**2*x**2 - 16*sqrt(d + e*x)*b**3*e**3*x 
**3))/(5*sqrt(a + b*x)*(a**4*d**3*e**4 + 3*a**4*d**2*e**5*x + 3*a**4*d*e** 
6*x**2 + a**4*e**7*x**3 - 4*a**3*b*d**4*e**3 - 12*a**3*b*d**3*e**4*x - 12* 
a**3*b*d**2*e**5*x**2 - 4*a**3*b*d*e**6*x**3 + 6*a**2*b**2*d**5*e**2 + 18* 
a**2*b**2*d**4*e**3*x + 18*a**2*b**2*d**3*e**4*x**2 + 6*a**2*b**2*d**2*e** 
5*x**3 - 4*a*b**3*d**6*e - 12*a*b**3*d**5*e**2*x - 12*a*b**3*d**4*e**3*x** 
2 - 4*a*b**3*d**3*e**4*x**3 + b**4*d**7 + 3*b**4*d**6*e*x + 3*b**4*d**5*e* 
*2*x**2 + b**4*d**4*e**3*x**3))