\(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx\) [203]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{9009 e (b d-a e)^4 (d+e x)^{7/2}} \] Output:

-2/13*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(13/2)+2/143*(6*A*b*e- 
13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^2/(e*x+d)^(11/2)+8/1287*b*(6* 
A*b*e-13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(9/2)+16/9009 
*b^2*(6*A*b*e-13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d)^(7/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (693 B d e^2 (a+b x)^3-693 A e^3 (a+b x)^3-1638 b B d e (a+b x)^2 (d+e x)+2457 A b e^2 (a+b x)^2 (d+e x)-819 a B e^2 (a+b x)^2 (d+e x)+1001 b^2 B d (a+b x) (d+e x)^2-3003 A b^2 e (a+b x) (d+e x)^2+2002 a b B e (a+b x) (d+e x)^2+1287 A b^3 (d+e x)^3-1287 a b^2 B (d+e x)^3\right )}{9009 (b d-a e)^4 (d+e x)^{13/2}} \] Input:

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
 

Output:

(2*(a + b*x)^(7/2)*(693*B*d*e^2*(a + b*x)^3 - 693*A*e^3*(a + b*x)^3 - 1638 
*b*B*d*e*(a + b*x)^2*(d + e*x) + 2457*A*b*e^2*(a + b*x)^2*(d + e*x) - 819* 
a*B*e^2*(a + b*x)^2*(d + e*x) + 1001*b^2*B*d*(a + b*x)*(d + e*x)^2 - 3003* 
A*b^2*e*(a + b*x)*(d + e*x)^2 + 2002*a*b*B*e*(a + b*x)*(d + e*x)^2 + 1287* 
A*b^3*(d + e*x)^3 - 1287*a*b^2*B*(d + e*x)^3))/(9009*(b*d - a*e)^4*(d + e* 
x)^(13/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-13 a B e+6 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}}dx}{13 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-13 a B e+6 A b e+7 b B d) \left (\frac {4 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}}dx}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-13 a B e+6 A b e+7 b B d) \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{7/2}}{63 (d+e x)^{7/2} (b d-a e)^2}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}\right ) (-13 a B e+6 A b e+7 b B d)}{13 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)}\)

Input:

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + ((7 
*b*B*d + 6*A*b*e - 13*a*B*e)*((2*(a + b*x)^(7/2))/(11*(b*d - a*e)*(d + e*x 
)^(11/2)) + (4*b*((2*(a + b*x)^(7/2))/(9*(b*d - a*e)*(d + e*x)^(9/2)) + (4 
*b*(a + b*x)^(7/2))/(63*(b*d - a*e)^2*(d + e*x)^(7/2))))/(11*(b*d - a*e))) 
)/(13*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 = 322, normalized size of antiderivative = 1.60

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+104 B a \,b^{2} e^{3} x^{3}-56 B \,b^{3} d \,e^{2} x^{3}+168 A a \,b^{2} e^{3} x^{2}-312 A \,b^{3} d \,e^{2} x^{2}-364 B \,a^{2} b \,e^{3} x^{2}+872 B a \,b^{2} d \,e^{2} x^{2}-364 B \,b^{3} d^{2} e \,x^{2}-378 A \,a^{2} b \,e^{3} x +1092 A a \,b^{2} d \,e^{2} x -858 A \,b^{3} d^{2} e x +819 B \,a^{3} e^{3} x -2807 B \,a^{2} b d \,e^{2} x +3133 B a \,b^{2} d^{2} e x -1001 b^{3} B \,d^{3} x +693 a^{3} A \,e^{3}-2457 A \,a^{2} b d \,e^{2}+3003 A a \,b^{2} d^{2} e -1287 A \,b^{3} d^{3}+126 B \,a^{3} d \,e^{2}-364 B \,a^{2} b \,d^{2} e +286 B a \,b^{2} d^{3}\right )}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\)	\(322\)
orering	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+104 B a \,b^{2} e^{3} x^{3}-56 B \,b^{3} d \,e^{2} x^{3}+168 A a \,b^{2} e^{3} x^{2}-312 A \,b^{3} d \,e^{2} x^{2}-364 B \,a^{2} b \,e^{3} x^{2}+872 B a \,b^{2} d \,e^{2} x^{2}-364 B \,b^{3} d^{2} e \,x^{2}-378 A \,a^{2} b \,e^{3} x +1092 A a \,b^{2} d \,e^{2} x -858 A \,b^{3} d^{2} e x +819 B \,a^{3} e^{3} x -2807 B \,a^{2} b d \,e^{2} x +3133 B a \,b^{2} d^{2} e x -1001 b^{3} B \,d^{3} x +693 a^{3} A \,e^{3}-2457 A \,a^{2} b d \,e^{2}+3003 A a \,b^{2} d^{2} e -1287 A \,b^{3} d^{3}+126 B \,a^{3} d \,e^{2}-364 B \,a^{2} b \,d^{2} e +286 B a \,b^{2} d^{3}\right )}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\)	\(322\)
default	\(-\frac {2 \left (-48 A \,b^{5} e^{3} x^{5}+104 B a \,b^{4} e^{3} x^{5}-56 B \,b^{5} d \,e^{2} x^{5}+72 A a \,b^{4} e^{3} x^{4}-312 A \,b^{5} d \,e^{2} x^{4}-156 B \,a^{2} b^{3} e^{3} x^{4}+760 B a \,b^{4} d \,e^{2} x^{4}-364 B \,b^{5} d^{2} e \,x^{4}-90 A \,a^{2} b^{3} e^{3} x^{3}+468 A a \,b^{4} d \,e^{2} x^{3}-858 A \,b^{5} d^{2} e \,x^{3}+195 B \,a^{3} b^{2} e^{3} x^{3}-1119 B \,a^{2} b^{3} d \,e^{2} x^{3}+2405 B a \,b^{4} d^{2} e \,x^{3}-1001 B \,b^{5} d^{3} x^{3}+105 A \,a^{3} b^{2} e^{3} x^{2}-585 A \,a^{2} b^{3} d \,e^{2} x^{2}+1287 A a \,b^{4} d^{2} e \,x^{2}-1287 A \,b^{5} d^{3} x^{2}+1274 B \,a^{4} b \,e^{3} x^{2}-4616 B \,a^{3} b^{2} d \,e^{2} x^{2}+5538 B \,a^{2} b^{3} d^{2} e \,x^{2}-1716 B a \,b^{4} d^{3} x^{2}+1008 A \,a^{4} b \,e^{3} x -3822 A \,a^{3} b^{2} d \,e^{2} x +5148 A \,a^{2} b^{3} d^{2} e x -2574 A a \,b^{4} d^{3} x +819 B \,a^{5} e^{3} x -2555 B \,a^{4} b d \,e^{2} x +2405 B \,a^{3} b^{2} d^{2} e x -429 B \,a^{2} b^{3} d^{3} x +693 A \,a^{5} e^{3}-2457 A \,a^{4} b d \,e^{2}+3003 A \,a^{3} b^{2} d^{2} e -1287 A \,a^{2} b^{3} d^{3}+126 B \,a^{5} d \,e^{2}-364 B \,a^{4} b \,d^{2} e +286 B \,a^{3} b^{2} d^{3}\right ) \left (b x +a \right )^{\frac {3}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (a e -d b \right )^{4}}\)	\(525\)

Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x,method=_RETURNVERBOSE)
 

Output:

-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*e^3*x^3+104*B*a*b^2*e^3*x^3-56*B*b^3*d*e^ 
2*x^3+168*A*a*b^2*e^3*x^2-312*A*b^3*d*e^2*x^2-364*B*a^2*b*e^3*x^2+872*B*a* 
b^2*d*e^2*x^2-364*B*b^3*d^2*e*x^2-378*A*a^2*b*e^3*x+1092*A*a*b^2*d*e^2*x-8 
58*A*b^3*d^2*e*x+819*B*a^3*e^3*x-2807*B*a^2*b*d*e^2*x+3133*B*a*b^2*d^2*e*x 
-1001*B*b^3*d^3*x+693*A*a^3*e^3-2457*A*a^2*b*d*e^2+3003*A*a*b^2*d^2*e-1287 
*A*b^3*d^3+126*B*a^3*d*e^2-364*B*a^2*b*d^2*e+286*B*a*b^2*d^3)/(e*x+d)^(13/ 
2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1047 vs. \(2 (177) = 354\).

Time = 138.52 (sec) , antiderivative size = 1047, normalized size of antiderivative = 5.21 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x, algorithm="fricas")
 

Output:

-2/9009*(693*A*a^6*e^3 - 8*(7*B*b^6*d*e^2 - (13*B*a*b^5 - 6*A*b^6)*e^3)*x^ 
6 - 4*(91*B*b^6*d^2*e - 2*(88*B*a*b^5 - 39*A*b^6)*d*e^2 + (13*B*a^2*b^4 - 
6*A*a*b^5)*e^3)*x^5 - (1001*B*b^6*d^3 - 13*(157*B*a*b^5 - 66*A*b^6)*d^2*e 
+ (359*B*a^2*b^4 - 156*A*a*b^5)*d*e^2 - 3*(13*B*a^3*b^3 - 6*A*a^2*b^4)*e^3 
)*x^4 + 143*(2*B*a^4*b^2 - 9*A*a^3*b^3)*d^3 - 91*(4*B*a^5*b - 33*A*a^4*b^2 
)*d^2*e + 63*(2*B*a^6 - 39*A*a^5*b)*d*e^2 - (143*(19*B*a*b^5 + 9*A*b^6)*d^ 
3 - 13*(611*B*a^2*b^4 + 33*A*a*b^5)*d^2*e + (5735*B*a^3*b^3 + 117*A*a^2*b^ 
4)*d*e^2 - (1469*B*a^4*b^2 + 15*A*a^3*b^3)*e^3)*x^3 - (429*(5*B*a^2*b^4 + 
9*A*a*b^5)*d^3 - 13*(611*B*a^3*b^3 + 495*A*a^2*b^4)*d^2*e + (7171*B*a^4*b^ 
2 + 4407*A*a^3*b^3)*d*e^2 - 7*(299*B*a^5*b + 159*A*a^4*b^2)*e^3)*x^2 - (14 
3*(B*a^3*b^3 + 27*A*a^2*b^4)*d^3 - 13*(157*B*a^4*b^2 + 627*A*a^3*b^3)*d^2* 
e + 7*(347*B*a^5*b + 897*A*a^4*b^2)*d*e^2 - 63*(13*B*a^6 + 27*A*a^5*b)*e^3 
)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^11 - 4*a*b^3*d^10*e + 6*a^2*b^2*d^ 
9*e^2 - 4*a^3*b*d^8*e^3 + a^4*d^7*e^4 + (b^4*d^4*e^7 - 4*a*b^3*d^3*e^8 + 6 
*a^2*b^2*d^2*e^9 - 4*a^3*b*d*e^10 + a^4*e^11)*x^7 + 7*(b^4*d^5*e^6 - 4*a*b 
^3*d^4*e^7 + 6*a^2*b^2*d^3*e^8 - 4*a^3*b*d^2*e^9 + a^4*d*e^10)*x^6 + 21*(b 
^4*d^6*e^5 - 4*a*b^3*d^5*e^6 + 6*a^2*b^2*d^4*e^7 - 4*a^3*b*d^3*e^8 + a^4*d 
^2*e^9)*x^5 + 35*(b^4*d^7*e^4 - 4*a*b^3*d^6*e^5 + 6*a^2*b^2*d^5*e^6 - 4*a^ 
3*b*d^4*e^7 + a^4*d^3*e^8)*x^4 + 35*(b^4*d^8*e^3 - 4*a*b^3*d^7*e^4 + 6*a^2 
*b^2*d^6*e^5 - 4*a^3*b*d^5*e^6 + a^4*d^4*e^7)*x^3 + 21*(b^4*d^9*e^2 - 4...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),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. 1003 vs. \(2 (177) = 354\).

Time = 0.80 (sec) , antiderivative size = 1003, normalized size of antiderivative = 4.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x, algorithm="giac")
 

Output:

2/9009*((4*(b*x + a)*(2*(7*B*b^16*d^3*e^8*abs(b) - 27*B*a*b^15*d^2*e^9*abs 
(b) + 6*A*b^16*d^2*e^9*abs(b) + 33*B*a^2*b^14*d*e^10*abs(b) - 12*A*a*b^15* 
d*e^10*abs(b) - 13*B*a^3*b^13*e^11*abs(b) + 6*A*a^2*b^14*e^11*abs(b))*(b*x 
 + a)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3 
*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12) + 13*(7*B*b^ 
17*d^4*e^7*abs(b) - 34*B*a*b^16*d^3*e^8*abs(b) + 6*A*b^17*d^3*e^8*abs(b) + 
 60*B*a^2*b^15*d^2*e^9*abs(b) - 18*A*a*b^16*d^2*e^9*abs(b) - 46*B*a^3*b^14 
*d*e^10*abs(b) + 18*A*a^2*b^15*d*e^10*abs(b) + 13*B*a^4*b^13*e^11*abs(b) - 
 6*A*a^3*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4 
*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b 
^2*e^12)) + 143*(7*B*b^18*d^5*e^6*abs(b) - 41*B*a*b^17*d^4*e^7*abs(b) + 6* 
A*b^18*d^4*e^7*abs(b) + 94*B*a^2*b^16*d^3*e^8*abs(b) - 24*A*a*b^17*d^3*e^8 
*abs(b) - 106*B*a^3*b^15*d^2*e^9*abs(b) + 36*A*a^2*b^16*d^2*e^9*abs(b) + 5 
9*B*a^4*b^14*d*e^10*abs(b) - 24*A*a^3*b^15*d*e^10*abs(b) - 13*B*a^5*b^13*e 
^11*abs(b) + 6*A*a^4*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15 
*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d* 
e^11 + a^6*b^2*e^12))*(b*x + a) - 1287*(B*a*b^18*d^5*e^6*abs(b) - A*b^19*d 
^5*e^6*abs(b) - 5*B*a^2*b^17*d^4*e^7*abs(b) + 5*A*a*b^18*d^4*e^7*abs(b) + 
10*B*a^3*b^16*d^3*e^8*abs(b) - 10*A*a^2*b^17*d^3*e^8*abs(b) - 10*B*a^4*b^1 
5*d^2*e^9*abs(b) + 10*A*a^3*b^16*d^2*e^9*abs(b) + 5*B*a^5*b^14*d*e^10*a...
 

Mupad [B] (verification not implemented)

Time = 2.13 (sec) , antiderivative size = 706, normalized size of antiderivative = 3.51 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (-2938\,B\,a^4\,b^2\,e^3+11470\,B\,a^3\,b^3\,d\,e^2-30\,A\,a^3\,b^3\,e^3-15886\,B\,a^2\,b^4\,d^2\,e+234\,A\,a^2\,b^4\,d\,e^2+5434\,B\,a\,b^5\,d^3-858\,A\,a\,b^5\,d^2\,e+2574\,A\,b^6\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (252\,B\,a^6\,d\,e^2+1386\,A\,a^6\,e^3-728\,B\,a^5\,b\,d^2\,e-4914\,A\,a^5\,b\,d\,e^2+572\,B\,a^4\,b^2\,d^3+6006\,A\,a^4\,b^2\,d^2\,e-2574\,A\,a^3\,b^3\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {x\,\sqrt {a+b\,x}\,\left (1638\,B\,a^6\,e^3-4858\,B\,a^5\,b\,d\,e^2+3402\,A\,a^5\,b\,e^3+4082\,B\,a^4\,b^2\,d^2\,e-12558\,A\,a^4\,b^2\,d\,e^2-286\,B\,a^3\,b^3\,d^3+16302\,A\,a^3\,b^3\,d^2\,e-7722\,A\,a^2\,b^4\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {x^2\,\sqrt {a+b\,x}\,\left (-4186\,B\,a^5\,b\,e^3+14342\,B\,a^4\,b^2\,d\,e^2-2226\,A\,a^4\,b^2\,e^3-15886\,B\,a^3\,b^3\,d^2\,e+8814\,A\,a^3\,b^3\,d\,e^2+4290\,B\,a^2\,b^4\,d^3-12870\,A\,a^2\,b^4\,d^2\,e+7722\,A\,a\,b^5\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \] Input:

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(15/2),x)
 

Output:

((d + e*x)^(1/2)*((x^3*(a + b*x)^(1/2)*(2574*A*b^6*d^3 + 5434*B*a*b^5*d^3 
- 30*A*a^3*b^3*e^3 - 2938*B*a^4*b^2*e^3 + 234*A*a^2*b^4*d*e^2 - 15886*B*a^ 
2*b^4*d^2*e + 11470*B*a^3*b^3*d*e^2 - 858*A*a*b^5*d^2*e))/(9009*e^7*(a*e - 
 b*d)^4) - ((a + b*x)^(1/2)*(1386*A*a^6*e^3 + 252*B*a^6*d*e^2 - 2574*A*a^3 
*b^3*d^3 + 572*B*a^4*b^2*d^3 + 6006*A*a^4*b^2*d^2*e - 4914*A*a^5*b*d*e^2 - 
 728*B*a^5*b*d^2*e))/(9009*e^7*(a*e - b*d)^4) - (x*(a + b*x)^(1/2)*(1638*B 
*a^6*e^3 + 3402*A*a^5*b*e^3 - 7722*A*a^2*b^4*d^3 - 286*B*a^3*b^3*d^3 + 163 
02*A*a^3*b^3*d^2*e - 12558*A*a^4*b^2*d*e^2 + 4082*B*a^4*b^2*d^2*e - 4858*B 
*a^5*b*d*e^2))/(9009*e^7*(a*e - b*d)^4) + (x^2*(a + b*x)^(1/2)*(7722*A*a*b 
^5*d^3 - 4186*B*a^5*b*e^3 - 2226*A*a^4*b^2*e^3 + 4290*B*a^2*b^4*d^3 - 1287 
0*A*a^2*b^4*d^2*e + 8814*A*a^3*b^3*d*e^2 - 15886*B*a^3*b^3*d^2*e + 14342*B 
*a^4*b^2*d*e^2))/(9009*e^7*(a*e - b*d)^4) + (16*b^5*x^6*(a + b*x)^(1/2)*(6 
*A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^5*(a*e - b*d)^4) - (8*b^4*x^5*(a*e - 
 13*b*d)*(a + b*x)^(1/2)*(6*A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^6*(a*e - 
b*d)^4) + (2*b^3*x^4*(a + b*x)^(1/2)*(3*a^2*e^2 + 143*b^2*d^2 - 26*a*b*d*e 
)*(6*A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^7*(a*e - b*d)^4)))/(x^7 + d^7/e^ 
7 + (7*d*x^6)/e + (7*d^6*x)/e^6 + (21*d^2*x^5)/e^2 + (35*d^3*x^4)/e^3 + (3 
5*d^4*x^3)/e^4 + (21*d^5*x^2)/e^5)
 

Reduce [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 1018, normalized size of antiderivative = 5.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x)
 

Output:

(2*( - 99*sqrt(d + e*x)*sqrt(a + b*x)*a**6*e**7 + 234*sqrt(d + e*x)*sqrt(a 
 + b*x)*a**5*b*d*e**6 - 360*sqrt(d + e*x)*sqrt(a + b*x)*a**5*b*e**7*x - 14 
3*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**2*d**2*e**5 + 884*sqrt(d + e*x)*sqrt 
(a + b*x)*a**4*b**2*d*e**6*x - 458*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**2*e 
**7*x**2 - 572*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**3*d**2*e**5*x + 1196*sq 
rt(d + e*x)*sqrt(a + b*x)*a**3*b**3*d*e**6*x**2 - 212*sqrt(d + e*x)*sqrt(a 
 + b*x)*a**3*b**3*e**7*x**3 - 858*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**4*d* 
*2*e**5*x**2 + 624*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**4*d*e**6*x**3 - 3*s 
qrt(d + e*x)*sqrt(a + b*x)*a**2*b**4*e**7*x**4 - 572*sqrt(d + e*x)*sqrt(a 
+ b*x)*a*b**5*d**2*e**5*x**3 + 26*sqrt(d + e*x)*sqrt(a + b*x)*a*b**5*d*e** 
6*x**4 + 4*sqrt(d + e*x)*sqrt(a + b*x)*a*b**5*e**7*x**5 - 143*sqrt(d + e*x 
)*sqrt(a + b*x)*b**6*d**2*e**5*x**4 - 52*sqrt(d + e*x)*sqrt(a + b*x)*b**6* 
d*e**6*x**5 - 8*sqrt(d + e*x)*sqrt(a + b*x)*b**6*e**7*x**6 + 8*sqrt(e)*sqr 
t(b)*b**6*d**7 + 56*sqrt(e)*sqrt(b)*b**6*d**6*e*x + 168*sqrt(e)*sqrt(b)*b* 
*6*d**5*e**2*x**2 + 280*sqrt(e)*sqrt(b)*b**6*d**4*e**3*x**3 + 280*sqrt(e)* 
sqrt(b)*b**6*d**3*e**4*x**4 + 168*sqrt(e)*sqrt(b)*b**6*d**2*e**5*x**5 + 56 
*sqrt(e)*sqrt(b)*b**6*d*e**6*x**6 + 8*sqrt(e)*sqrt(b)*b**6*e**7*x**7))/(12 
87*e**5*(a**3*d**7*e**3 + 7*a**3*d**6*e**4*x + 21*a**3*d**5*e**5*x**2 + 35 
*a**3*d**4*e**6*x**3 + 35*a**3*d**3*e**7*x**4 + 21*a**3*d**2*e**8*x**5 + 7 
*a**3*d*e**9*x**6 + a**3*e**10*x**7 - 3*a**2*b*d**8*e**2 - 21*a**2*b*d*...