\(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx\) [204]

Optimal result

Integrand size = 24, antiderivative size = 255 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {32 b^3 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{45045 e (b d-a e)^5 (d+e x)^{7/2}} \] Output:

-2/15*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(15/2)+2/195*(8*A*b*e- 
15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^2/(e*x+d)^(13/2)+4/715*b*(8*A 
*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(11/2)+16/6435 
*b^2*(8*A*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d)^(9/2) 
+32/45045*b^3*(8*A*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^5/(e*x 
+d)^(7/2)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (-3003 B d e^3 (a+b x)^4+3003 A e^4 (a+b x)^4+10395 b B d e^2 (a+b x)^3 (d+e x)-13860 A b e^3 (a+b x)^3 (d+e x)+3465 a B e^3 (a+b x)^3 (d+e x)-12285 b^2 B d e (a+b x)^2 (d+e x)^2+24570 A b^2 e^2 (a+b x)^2 (d+e x)^2-12285 a b B e^2 (a+b x)^2 (d+e x)^2+5005 b^3 B d (a+b x) (d+e x)^3-20020 A b^3 e (a+b x) (d+e x)^3+15015 a b^2 B e (a+b x) (d+e x)^3+6435 A b^4 (d+e x)^4-6435 a b^3 B (d+e x)^4\right )}{45045 (b d-a e)^5 (d+e x)^{15/2}} \] Input:

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]
 

Output:

(2*(a + b*x)^(7/2)*(-3003*B*d*e^3*(a + b*x)^4 + 3003*A*e^4*(a + b*x)^4 + 1 
0395*b*B*d*e^2*(a + b*x)^3*(d + e*x) - 13860*A*b*e^3*(a + b*x)^3*(d + e*x) 
 + 3465*a*B*e^3*(a + b*x)^3*(d + e*x) - 12285*b^2*B*d*e*(a + b*x)^2*(d + e 
*x)^2 + 24570*A*b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 12285*a*b*B*e^2*(a + b*x 
)^2*(d + e*x)^2 + 5005*b^3*B*d*(a + b*x)*(d + e*x)^3 - 20020*A*b^3*e*(a + 
b*x)*(d + e*x)^3 + 15015*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 6435*A*b^4*(d + 
 e*x)^4 - 6435*a*b^3*B*(d + e*x)^4))/(45045*(b*d - a*e)^5*(d + e*x)^(15/2) 
)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.94, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-15 a B e+8 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{15/2}}dx}{15 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-15 a B e+8 A b e+7 b B d) \left (\frac {6 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}}dx}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-15 a B e+8 A b e+7 b B d) \left (\frac {6 b \left (\frac {4 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}}dx}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-15 a B e+8 A b e+7 b B d) \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}}dx}{9 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}\right )}{13 (b d-a e)}+\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}\right )}{15 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {2 (a+b x)^{7/2}}{13 (d+e x)^{13/2} (b d-a e)}+\frac {6 b \left (\frac {2 (a+b x)^{7/2}}{11 (d+e x)^{11/2} (b d-a e)}+\frac {4 b \left (\frac {4 b (a+b x)^{7/2}}{63 (d+e x)^{7/2} (b d-a e)^2}+\frac {2 (a+b x)^{7/2}}{9 (d+e x)^{9/2} (b d-a e)}\right )}{11 (b d-a e)}\right )}{13 (b d-a e)}\right ) (-15 a B e+8 A b e+7 b B d)}{15 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)}\)

Input:

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]
 

Output:

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + ((7 
*b*B*d + 8*A*b*e - 15*a*B*e)*((2*(a + b*x)^(7/2))/(13*(b*d - a*e)*(d + e*x 
)^(13/2)) + (6*b*((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*(b*d - 
 a*e))))/(15*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(225)=450\).

Time = 0.30 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.98

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} e^{4} x^{4}-240 B a \,b^{3} e^{4} x^{4}+112 B \,b^{4} d \,e^{3} x^{4}-448 A a \,b^{3} e^{4} x^{3}+960 A \,b^{4} d \,e^{3} x^{3}+840 B \,a^{2} b^{2} e^{4} x^{3}-2192 B a \,b^{3} d \,e^{3} x^{3}+840 B \,b^{4} d^{2} e^{2} x^{3}+1008 A \,a^{2} b^{2} e^{4} x^{2}-3360 A a \,b^{3} d \,e^{3} x^{2}+3120 A \,b^{4} d^{2} e^{2} x^{2}-1890 B \,a^{3} b \,e^{4} x^{2}+7182 B \,a^{2} b^{2} d \,e^{3} x^{2}-8790 B a \,b^{3} d^{2} e^{2} x^{2}+2730 B \,b^{4} d^{3} e \,x^{2}-1848 A \,a^{3} b \,e^{4} x +7560 A \,a^{2} b^{2} d \,e^{3} x -10920 A a \,b^{3} d^{2} e^{2} x +5720 A \,b^{4} d^{3} e x +3465 B \,a^{4} e^{4} x -15792 B \,a^{3} b d \,e^{3} x +27090 B \,a^{2} b^{2} d^{2} e^{2} x -20280 B a \,b^{3} d^{3} e x +5005 B \,b^{4} d^{4} x +3003 A \,a^{4} e^{4}-13860 A \,a^{3} b d \,e^{3}+24570 A \,a^{2} b^{2} d^{2} e^{2}-20020 A a \,b^{3} d^{3} e +6435 A \,b^{4} d^{4}+462 B \,a^{4} d \,e^{3}-1890 B \,a^{3} b \,d^{2} e^{2}+2730 B \,a^{2} b^{2} d^{3} e -1430 B a \,b^{3} d^{4}\right )}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)
orering	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} e^{4} x^{4}-240 B a \,b^{3} e^{4} x^{4}+112 B \,b^{4} d \,e^{3} x^{4}-448 A a \,b^{3} e^{4} x^{3}+960 A \,b^{4} d \,e^{3} x^{3}+840 B \,a^{2} b^{2} e^{4} x^{3}-2192 B a \,b^{3} d \,e^{3} x^{3}+840 B \,b^{4} d^{2} e^{2} x^{3}+1008 A \,a^{2} b^{2} e^{4} x^{2}-3360 A a \,b^{3} d \,e^{3} x^{2}+3120 A \,b^{4} d^{2} e^{2} x^{2}-1890 B \,a^{3} b \,e^{4} x^{2}+7182 B \,a^{2} b^{2} d \,e^{3} x^{2}-8790 B a \,b^{3} d^{2} e^{2} x^{2}+2730 B \,b^{4} d^{3} e \,x^{2}-1848 A \,a^{3} b \,e^{4} x +7560 A \,a^{2} b^{2} d \,e^{3} x -10920 A a \,b^{3} d^{2} e^{2} x +5720 A \,b^{4} d^{3} e x +3465 B \,a^{4} e^{4} x -15792 B \,a^{3} b d \,e^{3} x +27090 B \,a^{2} b^{2} d^{2} e^{2} x -20280 B a \,b^{3} d^{3} e x +5005 B \,b^{4} d^{4} x +3003 A \,a^{4} e^{4}-13860 A \,a^{3} b d \,e^{3}+24570 A \,a^{2} b^{2} d^{2} e^{2}-20020 A a \,b^{3} d^{3} e +6435 A \,b^{4} d^{4}+462 B \,a^{4} d \,e^{3}-1890 B \,a^{3} b \,d^{2} e^{2}+2730 B \,a^{2} b^{2} d^{3} e -1430 B a \,b^{3} d^{4}\right )}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)
default	\(-\frac {2 \left (128 A \,b^{6} e^{4} x^{6}-240 B a \,b^{5} e^{4} x^{6}+112 B \,b^{6} d \,e^{3} x^{6}-192 A a \,b^{5} e^{4} x^{5}+960 A \,b^{6} d \,e^{3} x^{5}+360 B \,a^{2} b^{4} e^{4} x^{5}-1968 B a \,b^{5} d \,e^{3} x^{5}+840 B \,b^{6} d^{2} e^{2} x^{5}+240 A \,a^{2} b^{4} e^{4} x^{4}-1440 A a \,b^{5} d \,e^{3} x^{4}+3120 A \,b^{6} d^{2} e^{2} x^{4}-450 B \,a^{3} b^{3} e^{4} x^{4}+2910 B \,a^{2} b^{4} d \,e^{3} x^{4}-7110 B a \,b^{5} d^{2} e^{2} x^{4}+2730 B \,b^{6} d^{3} e \,x^{4}-280 A \,a^{3} b^{3} e^{4} x^{3}+1800 A \,a^{2} b^{4} d \,e^{3} x^{3}-4680 A a \,b^{5} d^{2} e^{2} x^{3}+5720 A \,b^{6} d^{3} e \,x^{3}+525 B \,a^{4} b^{2} e^{4} x^{3}-3620 B \,a^{3} b^{3} d \,e^{3} x^{3}+10350 B \,a^{2} b^{4} d^{2} e^{2} x^{3}-14820 B a \,b^{5} d^{3} e \,x^{3}+5005 B \,b^{6} d^{4} x^{3}+315 A \,a^{4} b^{2} e^{4} x^{2}-2100 A \,a^{3} b^{3} d \,e^{3} x^{2}+5850 A \,a^{2} b^{4} d^{2} e^{2} x^{2}-8580 A a \,b^{5} d^{3} e \,x^{2}+6435 A \,b^{6} d^{4} x^{2}+5040 B \,a^{5} b \,e^{4} x^{2}-23940 B \,a^{4} b^{2} d \,e^{3} x^{2}+43500 B \,a^{3} b^{3} d^{2} e^{2} x^{2}-35100 B \,a^{2} b^{4} d^{3} e \,x^{2}+8580 B a \,b^{5} d^{4} x^{2}+4158 A \,a^{5} b \,e^{4} x -20160 A \,a^{4} b^{2} d \,e^{3} x +38220 A \,a^{3} b^{3} d^{2} e^{2} x -34320 A \,a^{2} b^{4} d^{3} e x +12870 A a \,b^{5} d^{4} x +3465 B \,a^{6} e^{4} x -14868 B \,a^{5} b d \,e^{3} x +23310 B \,a^{4} b^{2} d^{2} e^{2} x -14820 B \,a^{3} b^{3} d^{3} e x +2145 B \,a^{2} b^{4} d^{4} x +3003 a^{6} A \,e^{4}-13860 A \,a^{5} b d \,e^{3}+24570 A \,a^{4} b^{2} d^{2} e^{2}-20020 A \,a^{3} b^{3} d^{3} e +6435 A \,a^{2} b^{4} d^{4}+462 B \,a^{6} d \,e^{3}-1890 B \,a^{5} b \,d^{2} e^{2}+2730 B \,a^{4} b^{2} d^{3} e -1430 B \,a^{3} b^{3} d^{4}\right ) \left (b x +a \right )^{\frac {3}{2}}}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a e -d b \right )^{5}}\)	\(765\)

Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*e^4*x^4-240*B*a*b^3*e^4*x^4+112*B*b^4*d* 
e^3*x^4-448*A*a*b^3*e^4*x^3+960*A*b^4*d*e^3*x^3+840*B*a^2*b^2*e^4*x^3-2192 
*B*a*b^3*d*e^3*x^3+840*B*b^4*d^2*e^2*x^3+1008*A*a^2*b^2*e^4*x^2-3360*A*a*b 
^3*d*e^3*x^2+3120*A*b^4*d^2*e^2*x^2-1890*B*a^3*b*e^4*x^2+7182*B*a^2*b^2*d* 
e^3*x^2-8790*B*a*b^3*d^2*e^2*x^2+2730*B*b^4*d^3*e*x^2-1848*A*a^3*b*e^4*x+7 
560*A*a^2*b^2*d*e^3*x-10920*A*a*b^3*d^2*e^2*x+5720*A*b^4*d^3*e*x+3465*B*a^ 
4*e^4*x-15792*B*a^3*b*d*e^3*x+27090*B*a^2*b^2*d^2*e^2*x-20280*B*a*b^3*d^3* 
e*x+5005*B*b^4*d^4*x+3003*A*a^4*e^4-13860*A*a^3*b*d*e^3+24570*A*a^2*b^2*d^ 
2*e^2-20020*A*a*b^3*d^3*e+6435*A*b^4*d^4+462*B*a^4*d*e^3-1890*B*a^3*b*d^2* 
e^2+2730*B*a^2*b^2*d^3*e-1430*B*a*b^3*d^4)/(e*x+d)^(15/2)/(a^5*e^5-5*a^4*b 
*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(17/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),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. 1413 vs. \(2 (225) = 450\).

Time = 1.06 (sec) , antiderivative size = 1413, normalized size of antiderivative = 5.54 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="giac")
 

Output:

2/45045*((2*(4*(b*x + a)*(2*(7*B*b^18*d^3*e^10*abs(b) - 29*B*a*b^17*d^2*e^ 
11*abs(b) + 8*A*b^18*d^2*e^11*abs(b) + 37*B*a^2*b^16*d*e^12*abs(b) - 16*A* 
a*b^17*d*e^12*abs(b) - 15*B*a^3*b^15*e^13*abs(b) + 8*A*a^2*b^16*e^13*abs(b 
))*(b*x + a)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3* 
b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^1 
3 - a^7*b^2*e^14) + 15*(7*B*b^19*d^4*e^9*abs(b) - 36*B*a*b^18*d^3*e^10*abs 
(b) + 8*A*b^19*d^3*e^10*abs(b) + 66*B*a^2*b^17*d^2*e^11*abs(b) - 24*A*a*b^ 
18*d^2*e^11*abs(b) - 52*B*a^3*b^16*d*e^12*abs(b) + 24*A*a^2*b^17*d*e^12*ab 
s(b) + 15*B*a^4*b^15*e^13*abs(b) - 8*A*a^3*b^16*e^13*abs(b))/(b^9*d^7*e^7 
- 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5* 
d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14)) + 195*( 
7*B*b^20*d^5*e^8*abs(b) - 43*B*a*b^19*d^4*e^9*abs(b) + 8*A*b^20*d^4*e^9*ab 
s(b) + 102*B*a^2*b^18*d^3*e^10*abs(b) - 32*A*a*b^19*d^3*e^10*abs(b) - 118* 
B*a^3*b^17*d^2*e^11*abs(b) + 48*A*a^2*b^18*d^2*e^11*abs(b) + 67*B*a^4*b^16 
*d*e^12*abs(b) - 32*A*a^3*b^17*d*e^12*abs(b) - 15*B*a^5*b^15*e^13*abs(b) + 
 8*A*a^4*b^16*e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5 
*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7 
*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) + 715*(7*B*b^21*d^6*e^7*abs(b) 
- 50*B*a*b^20*d^5*e^8*abs(b) + 8*A*b^21*d^5*e^8*abs(b) + 145*B*a^2*b^19*d^ 
4*e^9*abs(b) - 40*A*a*b^20*d^4*e^9*abs(b) - 220*B*a^3*b^18*d^3*e^10*abs...
 

Mupad [B] (verification not implemented)

Time = 2.55 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(17/2),x)
 

Output:

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(6006*A*a^7*e^4 + 924*B*a^7*d*e^3 + 12 
870*A*a^3*b^4*d^4 - 2860*B*a^4*b^3*d^4 - 40040*A*a^4*b^3*d^3*e + 5460*B*a^ 
5*b^2*d^3*e - 3780*B*a^6*b*d^2*e^2 + 49140*A*a^5*b^2*d^2*e^2 - 27720*A*a^6 
*b*d*e^3))/(45045*e^8*(a*e - b*d)^5) + (x^2*(a + b*x)^(1/2)*(38610*A*a*b^6 
*d^4 + 17010*B*a^6*b*e^4 + 8946*A*a^5*b^2*e^4 + 21450*B*a^2*b^5*d^4 - 8580 
0*A*a^2*b^5*d^3*e - 44520*A*a^4*b^3*d*e^3 - 99840*B*a^3*b^4*d^3*e - 77616* 
B*a^5*b^2*d*e^3 + 88140*A*a^3*b^4*d^2*e^2 + 133620*B*a^4*b^3*d^2*e^2))/(45 
045*e^8*(a*e - b*d)^5) + (x^3*(a + b*x)^(1/2)*(12870*A*b^7*d^4 + 27170*B*a 
*b^6*d^4 + 70*A*a^4*b^3*e^4 + 11130*B*a^5*b^2*e^4 - 600*A*a^3*b^4*d*e^3 - 
99840*B*a^2*b^5*d^3*e - 55120*B*a^4*b^3*d*e^3 + 2340*A*a^2*b^5*d^2*e^2 + 1 
07700*B*a^3*b^4*d^2*e^2 - 5720*A*a*b^6*d^3*e))/(45045*e^8*(a*e - b*d)^5) + 
 (x*(a + b*x)^(1/2)*(6930*B*a^7*e^4 + 14322*A*a^6*b*e^4 + 38610*A*a^2*b^5* 
d^4 + 1430*B*a^3*b^4*d^4 - 108680*A*a^3*b^4*d^3*e - 68040*A*a^5*b^2*d*e^3 
- 24180*B*a^4*b^3*d^3*e + 125580*A*a^4*b^3*d^2*e^2 + 42840*B*a^5*b^2*d^2*e 
^2 - 28812*B*a^6*b*d*e^3))/(45045*e^8*(a*e - b*d)^5) + (32*b^6*x^7*(a + b* 
x)^(1/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d))/(45045*e^5*(a*e - b*d)^5) - (2*b^ 
3*x^4*(a + b*x)^(1/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d)*(a^3*e^3 - 143*b^3*d^ 
3 + 39*a*b^2*d^2*e - 9*a^2*b*d*e^2))/(9009*e^8*(a*e - b*d)^5) - (16*b^5*x^ 
6*(a*e - 15*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d))/(45045*e^ 
6*(a*e - b*d)^5) + (4*b^4*x^5*(a + b*x)^(1/2)*(a^2*e^2 + 65*b^2*d^2 - 1...
 

Reduce [B] (verification not implemented)

Time = 2.06 (sec) , antiderivative size = 1457, normalized size of antiderivative = 5.71 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x)
 

Output:

(2*( - 429*sqrt(d + e*x)*sqrt(a + b*x)*a**7*e**8 + 1485*sqrt(d + e*x)*sqrt 
(a + b*x)*a**6*b*d*e**7 - 1518*sqrt(d + e*x)*sqrt(a + b*x)*a**6*b*e**8*x - 
 1755*sqrt(d + e*x)*sqrt(a + b*x)*a**5*b**2*d**2*e**6 + 5400*sqrt(d + e*x) 
*sqrt(a + b*x)*a**5*b**2*d*e**7*x - 1854*sqrt(d + e*x)*sqrt(a + b*x)*a**5* 
b**2*e**8*x**2 + 715*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**3*d**3*e**5 - 663 
0*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**3*d**2*e**6*x + 6870*sqrt(d + e*x)*s 
qrt(a + b*x)*a**4*b**3*d*e**7*x**2 - 800*sqrt(d + e*x)*sqrt(a + b*x)*a**4* 
b**3*e**8*x**3 + 2860*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**4*d**3*e**5*x - 
8970*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**4*d**2*e**6*x**2 + 3180*sqrt(d + 
e*x)*sqrt(a + b*x)*a**3*b**4*d*e**7*x**3 - 5*sqrt(d + e*x)*sqrt(a + b*x)*a 
**3*b**4*e**8*x**4 + 4290*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**5*d**3*e**5* 
x**2 - 4680*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**5*d**2*e**6*x**3 + 45*sqrt 
(d + e*x)*sqrt(a + b*x)*a**2*b**5*d*e**7*x**4 + 6*sqrt(d + e*x)*sqrt(a + b 
*x)*a**2*b**5*e**8*x**5 + 2860*sqrt(d + e*x)*sqrt(a + b*x)*a*b**6*d**3*e** 
5*x**3 - 195*sqrt(d + e*x)*sqrt(a + b*x)*a*b**6*d**2*e**6*x**4 - 60*sqrt(d 
 + e*x)*sqrt(a + b*x)*a*b**6*d*e**7*x**5 - 8*sqrt(d + e*x)*sqrt(a + b*x)*a 
*b**6*e**8*x**6 + 715*sqrt(d + e*x)*sqrt(a + b*x)*b**7*d**3*e**5*x**4 + 39 
0*sqrt(d + e*x)*sqrt(a + b*x)*b**7*d**2*e**6*x**5 + 120*sqrt(d + e*x)*sqrt 
(a + b*x)*b**7*d*e**7*x**6 + 16*sqrt(d + e*x)*sqrt(a + b*x)*b**7*e**8*x**7 
 - 16*sqrt(e)*sqrt(b)*b**7*d**8 - 128*sqrt(e)*sqrt(b)*b**7*d**7*e*x - 4...