\(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx\) [190]

Optimal result

Integrand size = 24, antiderivative size = 147 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{315 e (b d-a e)^3 (d+e x)^{5/2}} \] Output:

-2/9*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/63*(4*A*b*e-9*B 
*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^2/(e*x+d)^(7/2)+4/315*b*(4*A*b*e- 
9*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)^3/(e*x+d)^(5/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 (a+b x)^{9/2} \left (-35 B d e+35 A e^2+\frac {45 b B d (d+e x)}{a+b x}-\frac {90 A b e (d+e x)}{a+b x}+\frac {45 a B e (d+e x)}{a+b x}+\frac {63 A b^2 (d+e x)^2}{(a+b x)^2}-\frac {63 a b B (d+e x)^2}{(a+b x)^2}\right )}{315 (b d-a e)^3 (d+e x)^{9/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]
 

Output:

(2*(a + b*x)^(9/2)*(-35*B*d*e + 35*A*e^2 + (45*b*B*d*(d + e*x))/(a + b*x) 
- (90*A*b*e*(d + e*x))/(a + b*x) + (45*a*B*e*(d + e*x))/(a + b*x) + (63*A* 
b^2*(d + e*x)^2)/(a + b*x)^2 - (63*a*b*B*(d + e*x)^2)/(a + b*x)^2))/(315*( 
b*d - a*e)^3*(d + e*x)^(9/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 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.

\(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-9 a B e+4 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}}dx}{9 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-9 a B e+4 A b e+5 b B d) \left (\frac {2 b \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {4 b (a+b x)^{5/2}}{35 (d+e x)^{5/2} (b d-a e)^2}+\frac {2 (a+b x)^{5/2}}{7 (d+e x)^{7/2} (b d-a e)}\right ) (-9 a B e+4 A b e+5 b B d)}{9 e (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

Input:

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(11/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + ((5*b 
*B*d + 4*A*b*e - 9*a*B*e)*((2*(a + b*x)^(5/2))/(7*(b*d - a*e)*(d + e*x)^(7 
/2)) + (4*b*(a + b*x)^(5/2))/(35*(b*d - a*e)^2*(d + e*x)^(5/2))))/(9*e*(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 [A] (verified)

Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.20

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} e^{2} x^{2}-18 B a b \,e^{2} x^{2}+10 B \,b^{2} d e \,x^{2}-20 A a b \,e^{2} x +36 A \,b^{2} d e x +45 B \,a^{2} e^{2} x -106 B a b d e x +45 b^{2} B \,d^{2} x +35 a^{2} A \,e^{2}-90 A a b d e +63 A \,b^{2} d^{2}+10 B \,a^{2} d e -18 B a b \,d^{2}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{3} e^{3}-3 a^{2} b \,e^{2} d +3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\)	\(177\)
orering	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} e^{2} x^{2}-18 B a b \,e^{2} x^{2}+10 B \,b^{2} d e \,x^{2}-20 A a b \,e^{2} x +36 A \,b^{2} d e x +45 B \,a^{2} e^{2} x -106 B a b d e x +45 b^{2} B \,d^{2} x +35 a^{2} A \,e^{2}-90 A a b d e +63 A \,b^{2} d^{2}+10 B \,a^{2} d e -18 B a b \,d^{2}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{3} e^{3}-3 a^{2} b \,e^{2} d +3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\)	\(177\)
default	\(-\frac {2 \left (8 A \,b^{3} e^{2} x^{3}-18 B a \,b^{2} e^{2} x^{3}+10 B \,b^{3} d e \,x^{3}-12 A a \,b^{2} e^{2} x^{2}+36 A \,b^{3} d e \,x^{2}+27 B \,a^{2} b \,e^{2} x^{2}-96 B a \,b^{2} d e \,x^{2}+45 B \,b^{3} d^{2} x^{2}+15 A \,a^{2} b \,e^{2} x -54 A a \,b^{2} d e x +63 A \,b^{3} d^{2} x +45 B \,a^{3} e^{2} x -96 B \,a^{2} b d e x +27 B a \,b^{2} d^{2} x +35 a^{3} A \,e^{2}-90 A \,a^{2} b d e +63 A a \,b^{2} d^{2}+10 B \,a^{3} d e -18 B \,a^{2} b \,d^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a e -d b \right )^{3}}\)	\(233\)

Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

-2/315*(b*x+a)^(5/2)*(8*A*b^2*e^2*x^2-18*B*a*b*e^2*x^2+10*B*b^2*d*e*x^2-20 
*A*a*b*e^2*x+36*A*b^2*d*e*x+45*B*a^2*e^2*x-106*B*a*b*d*e*x+45*B*b^2*d^2*x+ 
35*A*a^2*e^2-90*A*a*b*d*e+63*A*b^2*d^2+10*B*a^2*d*e-18*B*a*b*d^2)/(e*x+d)^ 
(9/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (129) = 258\).

Time = 28.00 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.86 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (35 \, A a^{4} e^{2} + 2 \, {\left (5 \, B b^{4} d e - {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} e^{2}\right )} x^{4} + {\left (45 \, B b^{4} d^{2} - 2 \, {\left (43 \, B a b^{3} - 18 \, A b^{4}\right )} d e + {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{3} - 9 \, {\left (2 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d^{2} + 10 \, {\left (B a^{4} - 9 \, A a^{3} b\right )} d e + 3 \, {\left (3 \, {\left (8 \, B a b^{3} + 7 \, A b^{4}\right )} d^{2} - 2 \, {\left (32 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2}\right )} x^{2} + {\left (9 \, {\left (B a^{2} b^{2} + 14 \, A a b^{3}\right )} d^{2} - 2 \, {\left (43 \, B a^{3} b + 72 \, A a^{2} b^{2}\right )} d e + 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{3} d^{8} - 3 \, a b^{2} d^{7} e + 3 \, a^{2} b d^{6} e^{2} - a^{3} d^{5} e^{3} + {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} x^{5} + 5 \, {\left (b^{3} d^{4} e^{4} - 3 \, a b^{2} d^{3} e^{5} + 3 \, a^{2} b d^{2} e^{6} - a^{3} d e^{7}\right )} x^{4} + 10 \, {\left (b^{3} d^{5} e^{3} - 3 \, a b^{2} d^{4} e^{4} + 3 \, a^{2} b d^{3} e^{5} - a^{3} d^{2} e^{6}\right )} x^{3} + 10 \, {\left (b^{3} d^{6} e^{2} - 3 \, a b^{2} d^{5} e^{3} + 3 \, a^{2} b d^{4} e^{4} - a^{3} d^{3} e^{5}\right )} x^{2} + 5 \, {\left (b^{3} d^{7} e - 3 \, a b^{2} d^{6} e^{2} + 3 \, a^{2} b d^{5} e^{3} - a^{3} d^{4} e^{4}\right )} x\right )}} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="fricas")
 

Output:

2/315*(35*A*a^4*e^2 + 2*(5*B*b^4*d*e - (9*B*a*b^3 - 4*A*b^4)*e^2)*x^4 + (4 
5*B*b^4*d^2 - 2*(43*B*a*b^3 - 18*A*b^4)*d*e + (9*B*a^2*b^2 - 4*A*a*b^3)*e^ 
2)*x^3 - 9*(2*B*a^3*b - 7*A*a^2*b^2)*d^2 + 10*(B*a^4 - 9*A*a^3*b)*d*e + 3* 
(3*(8*B*a*b^3 + 7*A*b^4)*d^2 - 2*(32*B*a^2*b^2 + 3*A*a*b^3)*d*e + (24*B*a^ 
3*b + A*a^2*b^2)*e^2)*x^2 + (9*(B*a^2*b^2 + 14*A*a*b^3)*d^2 - 2*(43*B*a^3* 
b + 72*A*a^2*b^2)*d*e + 5*(9*B*a^4 + 10*A*a^3*b)*e^2)*x)*sqrt(b*x + a)*sqr 
t(e*x + d)/(b^3*d^8 - 3*a*b^2*d^7*e + 3*a^2*b*d^6*e^2 - a^3*d^5*e^3 + (b^3 
*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*x^5 + 5*(b^3*d^4*e^4 
 - 3*a*b^2*d^3*e^5 + 3*a^2*b*d^2*e^6 - a^3*d*e^7)*x^4 + 10*(b^3*d^5*e^3 - 
3*a*b^2*d^4*e^4 + 3*a^2*b*d^3*e^5 - a^3*d^2*e^6)*x^3 + 10*(b^3*d^6*e^2 - 3 
*a*b^2*d^5*e^3 + 3*a^2*b*d^4*e^4 - a^3*d^3*e^5)*x^2 + 5*(b^3*d^7*e - 3*a*b 
^2*d^6*e^2 + 3*a^2*b*d^5*e^3 - a^3*d^4*e^4)*x)
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(11/2),x)
 

Output:

Integral((A + B*x)*(a + b*x)**(3/2)/(d + e*x)**(11/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),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. 516 vs. \(2 (129) = 258\).

Time = 0.45 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.51 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{11} d^{2} e^{5} {\left | b \right |} - 14 \, B a b^{10} d e^{6} {\left | b \right |} + 4 \, A b^{11} d e^{6} {\left | b \right |} + 9 \, B a^{2} b^{9} e^{7} {\left | b \right |} - 4 \, A a b^{10} e^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} + \frac {9 \, {\left (5 \, B b^{12} d^{3} e^{4} {\left | b \right |} - 19 \, B a b^{11} d^{2} e^{5} {\left | b \right |} + 4 \, A b^{12} d^{2} e^{5} {\left | b \right |} + 23 \, B a^{2} b^{10} d e^{6} {\left | b \right |} - 8 \, A a b^{11} d e^{6} {\left | b \right |} - 9 \, B a^{3} b^{9} e^{7} {\left | b \right |} + 4 \, A a^{2} b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} - \frac {63 \, {\left (B a b^{12} d^{3} e^{4} {\left | b \right |} - A b^{13} d^{3} e^{4} {\left | b \right |} - 3 \, B a^{2} b^{11} d^{2} e^{5} {\left | b \right |} + 3 \, A a b^{12} d^{2} e^{5} {\left | b \right |} + 3 \, B a^{3} b^{10} d e^{6} {\left | b \right |} - 3 \, A a^{2} b^{11} d e^{6} {\left | b \right |} - B a^{4} b^{9} e^{7} {\left | b \right |} + A a^{3} b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \] Input:

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="giac")
 

Output:

2/315*((b*x + a)*(2*(5*B*b^11*d^2*e^5*abs(b) - 14*B*a*b^10*d*e^6*abs(b) + 
4*A*b^11*d*e^6*abs(b) + 9*B*a^2*b^9*e^7*abs(b) - 4*A*a*b^10*e^7*abs(b))*(b 
*x + a)/(b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e 
^7 + a^4*b^2*e^8) + 9*(5*B*b^12*d^3*e^4*abs(b) - 19*B*a*b^11*d^2*e^5*abs(b 
) + 4*A*b^12*d^2*e^5*abs(b) + 23*B*a^2*b^10*d*e^6*abs(b) - 8*A*a*b^11*d*e^ 
6*abs(b) - 9*B*a^3*b^9*e^7*abs(b) + 4*A*a^2*b^10*e^7*abs(b))/(b^6*d^4*e^4 
- 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8)) - 
63*(B*a*b^12*d^3*e^4*abs(b) - A*b^13*d^3*e^4*abs(b) - 3*B*a^2*b^11*d^2*e^5 
*abs(b) + 3*A*a*b^12*d^2*e^5*abs(b) + 3*B*a^3*b^10*d*e^6*abs(b) - 3*A*a^2* 
b^11*d*e^6*abs(b) - B*a^4*b^9*e^7*abs(b) + A*a^3*b^10*e^7*abs(b))/(b^6*d^4 
*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8 
))*(b*x + a)^(5/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)
 

Mupad [B] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.73 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e+70\,A\,a^4\,e^2-36\,B\,a^3\,b\,d^2-180\,A\,a^3\,b\,d\,e+126\,A\,a^2\,b^2\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (144\,B\,a^3\,b\,e^2-384\,B\,a^2\,b^2\,d\,e+6\,A\,a^2\,b^2\,e^2+144\,B\,a\,b^3\,d^2-36\,A\,a\,b^3\,d\,e+126\,A\,b^4\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (90\,B\,a^4\,e^2-172\,B\,a^3\,b\,d\,e+100\,A\,a^3\,b\,e^2+18\,B\,a^2\,b^2\,d^2-288\,A\,a^2\,b^2\,d\,e+252\,A\,a\,b^3\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \] Input:

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(11/2),x)
 

Output:

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(70*A*a^4*e^2 + 20*B*a^4*d*e - 36*B*a^ 
3*b*d^2 + 126*A*a^2*b^2*d^2 - 180*A*a^3*b*d*e))/(315*e^5*(a*e - b*d)^3) + 
(x^2*(a + b*x)^(1/2)*(126*A*b^4*d^2 + 144*B*a*b^3*d^2 + 144*B*a^3*b*e^2 + 
6*A*a^2*b^2*e^2 - 36*A*a*b^3*d*e - 384*B*a^2*b^2*d*e))/(315*e^5*(a*e - b*d 
)^3) + (x*(a + b*x)^(1/2)*(90*B*a^4*e^2 + 252*A*a*b^3*d^2 + 100*A*a^3*b*e^ 
2 + 18*B*a^2*b^2*d^2 - 172*B*a^3*b*d*e - 288*A*a^2*b^2*d*e))/(315*e^5*(a*e 
 - b*d)^3) + (4*b^3*x^4*(a + b*x)^(1/2)*(4*A*b*e - 9*B*a*e + 5*B*b*d))/(31 
5*e^4*(a*e - b*d)^3) - (2*b^2*x^3*(a*e - 9*b*d)*(a + b*x)^(1/2)*(4*A*b*e - 
 9*B*a*e + 5*B*b*d))/(315*e^5*(a*e - b*d)^3)))/(x^5 + d^5/e^5 + (5*d*x^4)/ 
e + (5*d^4*x)/e^4 + (10*d^2*x^3)/e^2 + (10*d^3*x^2)/e^3)
 

Reduce [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.58 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{4} e^{5}}{9}+\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} b d \,e^{4}}{7}-\frac {38 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} b \,e^{5} x}{63}+\frac {6 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b^{2} d \,e^{4} x}{7}-\frac {10 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b^{2} e^{5} x^{2}}{21}+\frac {6 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} d \,e^{4} x^{2}}{7}-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} e^{5} x^{3}}{63}+\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} d \,e^{4} x^{3}}{7}+\frac {4 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} e^{5} x^{4}}{63}-\frac {4 \sqrt {e}\, \sqrt {b}\, b^{4} d^{5}}{63}-\frac {20 \sqrt {e}\, \sqrt {b}\, b^{4} d^{4} e x}{63}-\frac {40 \sqrt {e}\, \sqrt {b}\, b^{4} d^{3} e^{2} x^{2}}{63}-\frac {40 \sqrt {e}\, \sqrt {b}\, b^{4} d^{2} e^{3} x^{3}}{63}-\frac {20 \sqrt {e}\, \sqrt {b}\, b^{4} d \,e^{4} x^{4}}{63}-\frac {4 \sqrt {e}\, \sqrt {b}\, b^{4} e^{5} x^{5}}{63}}{e^{4} \left (a^{2} e^{7} x^{5}-2 a b d \,e^{6} x^{5}+b^{2} d^{2} e^{5} x^{5}+5 a^{2} d \,e^{6} x^{4}-10 a b \,d^{2} e^{5} x^{4}+5 b^{2} d^{3} e^{4} x^{4}+10 a^{2} d^{2} e^{5} x^{3}-20 a b \,d^{3} e^{4} x^{3}+10 b^{2} d^{4} e^{3} x^{3}+10 a^{2} d^{3} e^{4} x^{2}-20 a b \,d^{4} e^{3} x^{2}+10 b^{2} d^{5} e^{2} x^{2}+5 a^{2} d^{4} e^{3} x -10 a b \,d^{5} e^{2} x +5 b^{2} d^{6} e x +a^{2} d^{5} e^{2}-2 a b \,d^{6} e +b^{2} d^{7}\right )} \] Input:

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(11/2),x)
 

Output:

(2*( - 7*sqrt(d + e*x)*sqrt(a + b*x)*a**4*e**5 + 9*sqrt(d + e*x)*sqrt(a + 
b*x)*a**3*b*d*e**4 - 19*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b*e**5*x + 27*sqr 
t(d + e*x)*sqrt(a + b*x)*a**2*b**2*d*e**4*x - 15*sqrt(d + e*x)*sqrt(a + b* 
x)*a**2*b**2*e**5*x**2 + 27*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*d*e**4*x**2 
 - sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*e**5*x**3 + 9*sqrt(d + e*x)*sqrt(a + 
 b*x)*b**4*d*e**4*x**3 + 2*sqrt(d + e*x)*sqrt(a + b*x)*b**4*e**5*x**4 - 2* 
sqrt(e)*sqrt(b)*b**4*d**5 - 10*sqrt(e)*sqrt(b)*b**4*d**4*e*x - 20*sqrt(e)* 
sqrt(b)*b**4*d**3*e**2*x**2 - 20*sqrt(e)*sqrt(b)*b**4*d**2*e**3*x**3 - 10* 
sqrt(e)*sqrt(b)*b**4*d*e**4*x**4 - 2*sqrt(e)*sqrt(b)*b**4*e**5*x**5))/(63* 
e**4*(a**2*d**5*e**2 + 5*a**2*d**4*e**3*x + 10*a**2*d**3*e**4*x**2 + 10*a* 
*2*d**2*e**5*x**3 + 5*a**2*d*e**6*x**4 + a**2*e**7*x**5 - 2*a*b*d**6*e - 1 
0*a*b*d**5*e**2*x - 20*a*b*d**4*e**3*x**2 - 20*a*b*d**3*e**4*x**3 - 10*a*b 
*d**2*e**5*x**4 - 2*a*b*d*e**6*x**5 + b**2*d**7 + 5*b**2*d**6*e*x + 10*b** 
2*d**5*e**2*x**2 + 10*b**2*d**4*e**3*x**3 + 5*b**2*d**3*e**4*x**4 + b**2*d 
**2*e**5*x**5))