\(\int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx\) [221]

Optimal result

Integrand size = 24, antiderivative size = 237 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{7/2}}+\frac {2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{7 b (b d-a e)^2 (d+e x)^{7/2}}+\frac {12 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^2 (b B d-8 A b e+7 a B e) \sqrt {a+b x}}{35 (b d-a e)^5 \sqrt {d+e x}} \] Output:

(-2*A*b+2*B*a)/b/(-a*e+b*d)/(b*x+a)^(1/2)/(e*x+d)^(7/2)+2/7*(-8*A*b*e+7*B* 
a*e+B*b*d)*(b*x+a)^(1/2)/b/(-a*e+b*d)^2/(e*x+d)^(7/2)+12/35*(-8*A*b*e+7*B* 
a*e+B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^3/(e*x+d)^(5/2)+16/35*b*(-8*A*b*e+7*B* 
a*e+B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^4/(e*x+d)^(3/2)+32/35*b^2*(-8*A*b*e+7* 
B*a*e+B*b*d)*(b*x+a)^(1/2)/(-a*e+b*d)^5/(e*x+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=-\frac {2 \left (5 B d e^3 (a+b x)^4-5 A e^4 (a+b x)^4-21 b B d e^2 (a+b x)^3 (d+e x)+28 A b e^3 (a+b x)^3 (d+e x)-7 a B e^3 (a+b x)^3 (d+e x)+35 b^2 B d e (a+b x)^2 (d+e x)^2-70 A b^2 e^2 (a+b x)^2 (d+e x)^2+35 a b B e^2 (a+b x)^2 (d+e x)^2-35 b^3 B d (a+b x) (d+e x)^3+140 A b^3 e (a+b x) (d+e x)^3-105 a b^2 B e (a+b x) (d+e x)^3+35 A b^4 (d+e x)^4-35 a b^3 B (d+e x)^4\right )}{35 (b d-a e)^5 \sqrt {a+b x} (d+e x)^{7/2}} \] Input:

Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]
 

Output:

(-2*(5*B*d*e^3*(a + b*x)^4 - 5*A*e^4*(a + b*x)^4 - 21*b*B*d*e^2*(a + b*x)^ 
3*(d + e*x) + 28*A*b*e^3*(a + b*x)^3*(d + e*x) - 7*a*B*e^3*(a + b*x)^3*(d 
+ e*x) + 35*b^2*B*d*e*(a + b*x)^2*(d + e*x)^2 - 70*A*b^2*e^2*(a + b*x)^2*( 
d + e*x)^2 + 35*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 - 35*b^3*B*d*(a + b*x)*( 
d + e*x)^3 + 140*A*b^3*e*(a + b*x)*(d + e*x)^3 - 105*a*b^2*B*e*(a + b*x)*( 
d + e*x)^3 + 35*A*b^4*(d + e*x)^4 - 35*a*b^3*B*(d + e*x)^4))/(35*(b*d - a* 
e)^5*Sqrt[a + b*x]*(d + e*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.98, 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.

\(\displaystyle \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(7 a B e-8 A b e+b B d) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}}dx}{b (b d-a e)}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\)

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

\(\displaystyle \frac {(7 a B e-8 A b e+b B d) \left (\frac {6 b \left (\frac {4 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b (b d-a e)}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(7 a B e-8 A b e+b B d) \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{b (b d-a e)}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {6 b \left (\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right ) (7 a B e-8 A b e+b B d)}{b (b d-a e)}-\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{7/2} (b d-a e)}\)

Input:

Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x]
 

Output:

(-2*(A*b - a*B))/(b*(b*d - a*e)*Sqrt[a + b*x]*(d + e*x)^(7/2)) + ((b*B*d - 
 8*A*b*e + 7*a*B*e)*((2*Sqrt[a + b*x])/(7*(b*d - a*e)*(d + e*x)^(7/2)) + ( 
6*b*((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))))/(7*(b*d - a*e))))/(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. \(2(209)=418\).

Time = 0.34 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.89

method	result	size
default	\(-\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} \sqrt {b x +a}\, \left (a e -d b \right )^{5}}\)	\(449\)
gosper	\(-\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} \sqrt {b x +a}\, \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)
orering	\(-\frac {2 \left (-128 A \,b^{4} e^{4} x^{4}+112 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}-448 A \,b^{4} d \,e^{3} x^{3}+56 B \,a^{2} b^{2} e^{4} x^{3}+400 B a \,b^{3} d \,e^{3} x^{3}+56 B \,b^{4} d^{2} e^{2} x^{3}+16 A \,a^{2} b^{2} e^{4} x^{2}-224 A a \,b^{3} d \,e^{3} x^{2}-560 A \,b^{4} d^{2} e^{2} x^{2}-14 B \,a^{3} b \,e^{4} x^{2}+194 B \,a^{2} b^{2} d \,e^{3} x^{2}+518 B a \,b^{3} d^{2} e^{2} x^{2}+70 B \,b^{4} d^{3} e \,x^{2}-8 A \,a^{3} b \,e^{4} x +56 A \,a^{2} b^{2} d \,e^{3} x -280 A a \,b^{3} d^{2} e^{2} x -280 A \,b^{4} d^{3} e x +7 B \,a^{4} e^{4} x -48 B \,a^{3} b d \,e^{3} x +238 B \,a^{2} b^{2} d^{2} e^{2} x +280 B a \,b^{3} d^{3} e x +35 B \,b^{4} d^{4} x +5 A \,a^{4} e^{4}-28 A \,a^{3} b d \,e^{3}+70 A \,a^{2} b^{2} d^{2} e^{2}-140 A a \,b^{3} d^{3} e -35 A \,b^{4} d^{4}+2 B \,a^{4} d \,e^{3}-14 B \,a^{3} b \,d^{2} e^{2}+70 B \,a^{2} b^{2} d^{3} e +70 B a \,b^{3} d^{4}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} \sqrt {b x +a}\, \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)

Input:

int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-2/35*(-128*A*b^4*e^4*x^4+112*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3*x^4-64*A*a*b^ 
3*e^4*x^3-448*A*b^4*d*e^3*x^3+56*B*a^2*b^2*e^4*x^3+400*B*a*b^3*d*e^3*x^3+5 
6*B*b^4*d^2*e^2*x^3+16*A*a^2*b^2*e^4*x^2-224*A*a*b^3*d*e^3*x^2-560*A*b^4*d 
^2*e^2*x^2-14*B*a^3*b*e^4*x^2+194*B*a^2*b^2*d*e^3*x^2+518*B*a*b^3*d^2*e^2* 
x^2+70*B*b^4*d^3*e*x^2-8*A*a^3*b*e^4*x+56*A*a^2*b^2*d*e^3*x-280*A*a*b^3*d^ 
2*e^2*x-280*A*b^4*d^3*e*x+7*B*a^4*e^4*x-48*B*a^3*b*d*e^3*x+238*B*a^2*b^2*d 
^2*e^2*x+280*B*a*b^3*d^3*e*x+35*B*b^4*d^4*x+5*A*a^4*e^4-28*A*a^3*b*d*e^3+7 
0*A*a^2*b^2*d^2*e^2-140*A*a*b^3*d^3*e-35*A*b^4*d^4+2*B*a^4*d*e^3-14*B*a^3* 
b*d^2*e^2+70*B*a^2*b^2*d^3*e+70*B*a*b^3*d^4)/(e*x+d)^(7/2)/(b*x+a)^(1/2)/( 
a*e-b*d)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (209) = 418\).

Time = 17.48 (sec) , antiderivative size = 887, normalized size of antiderivative = 3.74 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")
 

Output:

2/35*(5*A*a^4*e^4 + 35*(2*B*a*b^3 - A*b^4)*d^4 + 70*(B*a^2*b^2 - 2*A*a*b^3 
)*d^3*e - 14*(B*a^3*b - 5*A*a^2*b^2)*d^2*e^2 + 2*(B*a^4 - 14*A*a^3*b)*d*e^ 
3 + 16*(B*b^4*d*e^3 + (7*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(7*B*b^4*d^2*e^2 
+ 2*(25*B*a*b^3 - 28*A*b^4)*d*e^3 + (7*B*a^2*b^2 - 8*A*a*b^3)*e^4)*x^3 + 2 
*(35*B*b^4*d^3*e + 7*(37*B*a*b^3 - 40*A*b^4)*d^2*e^2 + (97*B*a^2*b^2 - 112 
*A*a*b^3)*d*e^3 - (7*B*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 + (35*B*b^4*d^4 + 280 
*(B*a*b^3 - A*b^4)*d^3*e + 14*(17*B*a^2*b^2 - 20*A*a*b^3)*d^2*e^2 - 8*(6*B 
*a^3*b - 7*A*a^2*b^2)*d*e^3 + (7*B*a^4 - 8*A*a^3*b)*e^4)*x)*sqrt(b*x + a)* 
sqrt(e*x + d)/(a*b^5*d^9 - 5*a^2*b^4*d^8*e + 10*a^3*b^3*d^7*e^2 - 10*a^4*b 
^2*d^6*e^3 + 5*a^5*b*d^5*e^4 - a^6*d^4*e^5 + (b^6*d^5*e^4 - 5*a*b^5*d^4*e^ 
5 + 10*a^2*b^4*d^3*e^6 - 10*a^3*b^3*d^2*e^7 + 5*a^4*b^2*d*e^8 - a^5*b*e^9) 
*x^5 + (4*b^6*d^6*e^3 - 19*a*b^5*d^5*e^4 + 35*a^2*b^4*d^4*e^5 - 30*a^3*b^3 
*d^3*e^6 + 10*a^4*b^2*d^2*e^7 + a^5*b*d*e^8 - a^6*e^9)*x^4 + 2*(3*b^6*d^7* 
e^2 - 13*a*b^5*d^6*e^3 + 20*a^2*b^4*d^5*e^4 - 10*a^3*b^3*d^4*e^5 - 5*a^4*b 
^2*d^3*e^6 + 7*a^5*b*d^2*e^7 - 2*a^6*d*e^8)*x^3 + 2*(2*b^6*d^8*e - 7*a*b^5 
*d^7*e^2 + 5*a^2*b^4*d^6*e^3 + 10*a^3*b^3*d^5*e^4 - 20*a^4*b^2*d^4*e^5 + 1 
3*a^5*b*d^3*e^6 - 3*a^6*d^2*e^7)*x^2 + (b^6*d^9 - a*b^5*d^8*e - 10*a^2*b^4 
*d^7*e^2 + 30*a^3*b^3*d^6*e^3 - 35*a^4*b^2*d^5*e^4 + 19*a^5*b*d^4*e^5 - 4* 
a^6*d^3*e^6)*x)
 

Sympy [F]

\[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=\int \frac {A + B x}{\left (a + b x\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(9/2),x)
 

Output:

Integral((A + B*x)/((a + b*x)**(3/2)*(d + e*x)**(9/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(b*x+a)^(3/2)/(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. 2645 vs. \(2 (209) = 418\).

Time = 1.00 (sec) , antiderivative size = 2645, normalized size of antiderivative = 11.16 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="giac")
 

Output:

4*(B^2*a^2*b^9*e - 2*A*B*a*b^10*e + A^2*b^11*e)/((sqrt(b*e)*B*a*b^6*d - sq 
rt(b*e)*A*b^7*d - sqrt(b*e)*B*a^2*b^5*e + sqrt(b*e)*A*a*b^6*e - sqrt(b*e)* 
(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^4 
+ sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e) 
)^2*A*b^5)*(b^4*d^4*abs(b) - 4*a*b^3*d^3*e*abs(b) + 6*a^2*b^2*d^2*e^2*abs( 
b) - 4*a^3*b*d*e^3*abs(b) + a^4*e^4*abs(b))) + 2/35*(((b*x + a)*((16*B*b^1 
9*d^10*e^6*abs(b) - 67*B*a*b^18*d^9*e^7*abs(b) - 93*A*b^19*d^9*e^7*abs(b) 
- 117*B*a^2*b^17*d^8*e^8*abs(b) + 837*A*a*b^18*d^8*e^8*abs(b) + 1428*B*a^3 
*b^16*d^7*e^9*abs(b) - 3348*A*a^2*b^17*d^7*e^9*abs(b) - 4452*B*a^4*b^15*d^ 
6*e^10*abs(b) + 7812*A*a^3*b^16*d^6*e^10*abs(b) + 7686*B*a^5*b^14*d^5*e^11 
*abs(b) - 11718*A*a^4*b^15*d^5*e^11*abs(b) - 8358*B*a^6*b^13*d^4*e^12*abs( 
b) + 11718*A*a^5*b^14*d^4*e^12*abs(b) + 5892*B*a^7*b^12*d^3*e^13*abs(b) - 
7812*A*a^6*b^13*d^3*e^13*abs(b) - 2628*B*a^8*b^11*d^2*e^14*abs(b) + 3348*A 
*a^7*b^12*d^2*e^14*abs(b) + 677*B*a^9*b^10*d*e^15*abs(b) - 837*A*a^8*b^11* 
d*e^15*abs(b) - 77*B*a^10*b^9*e^16*abs(b) + 93*A*a^9*b^10*e^16*abs(b))*(b* 
x + a)/(b^18*d^14*e^3 - 14*a*b^17*d^13*e^4 + 91*a^2*b^16*d^12*e^5 - 364*a^ 
3*b^15*d^11*e^6 + 1001*a^4*b^14*d^10*e^7 - 2002*a^5*b^13*d^9*e^8 + 3003*a^ 
6*b^12*d^8*e^9 - 3432*a^7*b^11*d^7*e^10 + 3003*a^8*b^10*d^6*e^11 - 2002*a^ 
9*b^9*d^5*e^12 + 1001*a^10*b^8*d^4*e^13 - 364*a^11*b^7*d^3*e^14 + 91*a^12* 
b^6*d^2*e^15 - 14*a^13*b^5*d*e^16 + a^14*b^4*e^17) + 28*(2*B*b^20*d^11*...
 

Mupad [B] (verification not implemented)

Time = 1.98 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^4\,d\,e^3+10\,A\,a^4\,e^4-28\,B\,a^3\,b\,d^2\,e^2-56\,A\,a^3\,b\,d\,e^3+140\,B\,a^2\,b^2\,d^3\,e+140\,A\,a^2\,b^2\,d^2\,e^2+140\,B\,a\,b^3\,d^4-280\,A\,a\,b^3\,d^3\,e-70\,A\,b^4\,d^4}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^3\,x^4\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,x\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )\,\left (a^3\,e^3-7\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+35\,b^3\,d^3\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b^2\,x^3\,\left (a\,e+7\,b\,d\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b\,x^2\,\left (-a^2\,e^2+14\,a\,b\,d\,e+35\,b^2\,d^2\right )\,\left (7\,B\,a\,e-8\,A\,b\,e+B\,b\,d\right )}{35\,e^3\,{\left (a\,e-b\,d\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {d^4\,\sqrt {a+b\,x}}{e^4}+\frac {6\,d^2\,x^2\,\sqrt {a+b\,x}}{e^2}+\frac {4\,d\,x^3\,\sqrt {a+b\,x}}{e}+\frac {4\,d^3\,x\,\sqrt {a+b\,x}}{e^3}} \] Input:

int((A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(9/2)),x)
 

Output:

-((d + e*x)^(1/2)*((10*A*a^4*e^4 - 70*A*b^4*d^4 + 140*B*a*b^3*d^4 + 4*B*a^ 
4*d*e^3 + 140*B*a^2*b^2*d^3*e - 28*B*a^3*b*d^2*e^2 + 140*A*a^2*b^2*d^2*e^2 
 - 280*A*a*b^3*d^3*e - 56*A*a^3*b*d*e^3)/(35*e^4*(a*e - b*d)^5) + (32*b^3* 
x^4*(7*B*a*e - 8*A*b*e + B*b*d))/(35*e*(a*e - b*d)^5) + (2*x*(7*B*a*e - 8* 
A*b*e + B*b*d)*(a^3*e^3 + 35*b^3*d^3 + 35*a*b^2*d^2*e - 7*a^2*b*d*e^2))/(3 
5*e^4*(a*e - b*d)^5) + (16*b^2*x^3*(a*e + 7*b*d)*(7*B*a*e - 8*A*b*e + B*b* 
d))/(35*e^2*(a*e - b*d)^5) + (4*b*x^2*(35*b^2*d^2 - a^2*e^2 + 14*a*b*d*e)* 
(7*B*a*e - 8*A*b*e + B*b*d))/(35*e^3*(a*e - b*d)^5)))/(x^4*(a + b*x)^(1/2) 
 + (d^4*(a + b*x)^(1/2))/e^4 + (6*d^2*x^2*(a + b*x)^(1/2))/e^2 + (4*d*x^3* 
(a + b*x)^(1/2))/e + (4*d^3*x*(a + b*x)^(1/2))/e^3)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.71 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{9/2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} e^{4}}{7}+\frac {6 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b d \,e^{3}}{5}+\frac {12 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b \,e^{4} x}{35}-2 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d^{2} e^{2}-\frac {8 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} d \,e^{3} x}{5}-\frac {16 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{2} e^{4} x^{2}}{35}+2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{3} e +4 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d^{2} e^{2} x +\frac {16 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} d \,e^{3} x^{2}}{5}+\frac {32 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{3} e^{4} x^{3}}{35}-\frac {32 \sqrt {e}\, \sqrt {b}\, b^{3} d^{4}}{35}-\frac {128 \sqrt {e}\, \sqrt {b}\, b^{3} d^{3} e x}{35}-\frac {192 \sqrt {e}\, \sqrt {b}\, b^{3} d^{2} e^{2} x^{2}}{35}-\frac {128 \sqrt {e}\, \sqrt {b}\, b^{3} d \,e^{3} x^{3}}{35}-\frac {32 \sqrt {e}\, \sqrt {b}\, b^{3} e^{4} x^{4}}{35}}{e \left (a^{4} e^{8} x^{4}-4 a^{3} b d \,e^{7} x^{4}+6 a^{2} b^{2} d^{2} e^{6} x^{4}-4 a \,b^{3} d^{3} e^{5} x^{4}+b^{4} d^{4} e^{4} x^{4}+4 a^{4} d \,e^{7} x^{3}-16 a^{3} b \,d^{2} e^{6} x^{3}+24 a^{2} b^{2} d^{3} e^{5} x^{3}-16 a \,b^{3} d^{4} e^{4} x^{3}+4 b^{4} d^{5} e^{3} x^{3}+6 a^{4} d^{2} e^{6} x^{2}-24 a^{3} b \,d^{3} e^{5} x^{2}+36 a^{2} b^{2} d^{4} e^{4} x^{2}-24 a \,b^{3} d^{5} e^{3} x^{2}+6 b^{4} d^{6} e^{2} x^{2}+4 a^{4} d^{3} e^{5} x -16 a^{3} b \,d^{4} e^{4} x +24 a^{2} b^{2} d^{5} e^{3} x -16 a \,b^{3} d^{6} e^{2} x +4 b^{4} d^{7} e x +a^{4} d^{4} e^{4}-4 a^{3} b \,d^{5} e^{3}+6 a^{2} b^{2} d^{6} e^{2}-4 a \,b^{3} d^{7} e +b^{4} d^{8}\right )} \] Input:

int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(9/2),x)
 

Output:

(2*( - 5*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 + 6*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b*e**4*x - 35*sqr 
t(d + e*x)*sqrt(a + b*x)*a*b**2*d**2*e**2 - 28*sqrt(d + e*x)*sqrt(a + b*x) 
*a*b**2*d*e**3*x - 8*sqrt(d + e*x)*sqrt(a + b*x)*a*b**2*e**4*x**2 + 35*sqr 
t(d + e*x)*sqrt(a + b*x)*b**3*d**3*e + 70*sqrt(d + e*x)*sqrt(a + b*x)*b**3 
*d**2*e**2*x + 56*sqrt(d + e*x)*sqrt(a + b*x)*b**3*d*e**3*x**2 + 16*sqrt(d 
 + e*x)*sqrt(a + b*x)*b**3*e**4*x**3 - 16*sqrt(e)*sqrt(b)*b**3*d**4 - 64*s 
qrt(e)*sqrt(b)*b**3*d**3*e*x - 96*sqrt(e)*sqrt(b)*b**3*d**2*e**2*x**2 - 64 
*sqrt(e)*sqrt(b)*b**3*d*e**3*x**3 - 16*sqrt(e)*sqrt(b)*b**3*e**4*x**4))/(3 
5*e*(a**4*d**4*e**4 + 4*a**4*d**3*e**5*x + 6*a**4*d**2*e**6*x**2 + 4*a**4* 
d*e**7*x**3 + a**4*e**8*x**4 - 4*a**3*b*d**5*e**3 - 16*a**3*b*d**4*e**4*x 
- 24*a**3*b*d**3*e**5*x**2 - 16*a**3*b*d**2*e**6*x**3 - 4*a**3*b*d*e**7*x* 
*4 + 6*a**2*b**2*d**6*e**2 + 24*a**2*b**2*d**5*e**3*x + 36*a**2*b**2*d**4* 
e**4*x**2 + 24*a**2*b**2*d**3*e**5*x**3 + 6*a**2*b**2*d**2*e**6*x**4 - 4*a 
*b**3*d**7*e - 16*a*b**3*d**6*e**2*x - 24*a*b**3*d**5*e**3*x**2 - 16*a*b** 
3*d**4*e**4*x**3 - 4*a*b**3*d**3*e**5*x**4 + b**4*d**8 + 4*b**4*d**7*e*x + 
 6*b**4*d**6*e**2*x**2 + 4*b**4*d**5*e**3*x**3 + b**4*d**4*e**4*x**4))