\(\int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx\) [148]

Optimal result

Integrand size = 24, antiderivative size = 237 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=\frac {16 (b c-a d)^2 (8 b c+7 a d) (a+b x)^{7/2}}{6435 a^4 e^4 (e x)^{9/2}}-\frac {16 (2 b c-9 a d) (b c-a d)^2 (8 b c+7 a d) (a+b x)^{7/2}}{45045 a^5 c e^5 (e x)^{7/2}}-\frac {4 (b c-a d) (8 b c+7 a d) (a+b x)^{7/2} (c+d x)^2}{715 a^3 c e^3 (e x)^{11/2}}+\frac {2 (8 b c+7 a d) (a+b x)^{7/2} (c+d x)^3}{195 a^2 c e^2 (e x)^{13/2}}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}} \] Output:

16/6435*(-a*d+b*c)^2*(7*a*d+8*b*c)*(b*x+a)^(7/2)/a^4/e^4/(e*x)^(9/2)-16/45 
045*(-9*a*d+2*b*c)*(-a*d+b*c)^2*(7*a*d+8*b*c)*(b*x+a)^(7/2)/a^5/c/e^5/(e*x 
)^(7/2)-4/715*(-a*d+b*c)*(7*a*d+8*b*c)*(b*x+a)^(7/2)*(d*x+c)^2/a^3/c/e^3/( 
e*x)^(11/2)+2/195*(7*a*d+8*b*c)*(b*x+a)^(7/2)*(d*x+c)^3/a^2/c/e^2/(e*x)^(1 
3/2)-2/15*(b*x+a)^(7/2)*(d*x+c)^4/a/c/e/(e*x)^(15/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=-\frac {2 \sqrt {e x} (a+b x)^{7/2} \left (128 b^4 c^3 x^4-16 a b^3 c^2 x^3 (28 c+45 d x)+24 a^2 b^2 c x^2 \left (42 c^2+105 c d x+65 d^2 x^2\right )+7 a^4 \left (429 c^3+1485 c^2 d x+1755 c d^2 x^2+715 d^3 x^3\right )-2 a^3 b x \left (924 c^3+2835 c^2 d x+2730 c d^2 x^2+715 d^3 x^3\right )\right )}{45045 a^5 e^9 x^8} \] Input:

Integrate[((a + b*x)^(5/2)*(c + d*x)^3)/(e*x)^(17/2),x]
 

Output:

(-2*Sqrt[e*x]*(a + b*x)^(7/2)*(128*b^4*c^3*x^4 - 16*a*b^3*c^2*x^3*(28*c + 
45*d*x) + 24*a^2*b^2*c*x^2*(42*c^2 + 105*c*d*x + 65*d^2*x^2) + 7*a^4*(429* 
c^3 + 1485*c^2*d*x + 1755*c*d^2*x^2 + 715*d^3*x^3) - 2*a^3*b*x*(924*c^3 + 
2835*c^2*d*x + 2730*c*d^2*x^2 + 715*d^3*x^3)))/(45045*a^5*e^9*x^8)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {107, 105, 105, 105, 105, 100, 27, 87, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx\)

\(\Big \downarrow \) 107

\(\displaystyle -\frac {(7 a d+8 b c) \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{15/2}}dx}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2} (c+d x)^3}{(e x)^{13/2}}dx}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^3}{(e x)^{11/2}}dx}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^3}{(e x)^{9/2} \sqrt {a+b x}}dx}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {6 (b c-a d) \int \frac {(c+d x)^2}{(e x)^{7/2} \sqrt {a+b x}}dx}{7 a e}-\frac {2 \sqrt {a+b x} (c+d x)^3}{7 a e (e x)^{7/2}}\right )}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 100

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {6 (b c-a d) \left (\frac {2 \int -\frac {e^2 \left (2 c (2 b c-5 a d)-5 a d^2 x\right )}{2 (e x)^{5/2} \sqrt {a+b x}}dx}{5 a e^3}-\frac {2 c^2 \sqrt {a+b x}}{5 a e (e x)^{5/2}}\right )}{7 a e}-\frac {2 \sqrt {a+b x} (c+d x)^3}{7 a e (e x)^{7/2}}\right )}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {6 (b c-a d) \left (-\frac {\int \frac {2 c (2 b c-5 a d)-5 a d^2 x}{(e x)^{5/2} \sqrt {a+b x}}dx}{5 a e}-\frac {2 c^2 \sqrt {a+b x}}{5 a e (e x)^{5/2}}\right )}{7 a e}-\frac {2 \sqrt {a+b x} (c+d x)^3}{7 a e (e x)^{7/2}}\right )}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {6 (b c-a d) \left (-\frac {-\frac {\left (15 a^2 d^2+4 b c (2 b c-5 a d)\right ) \int \frac {1}{(e x)^{3/2} \sqrt {a+b x}}dx}{3 a e}-\frac {4 c \sqrt {a+b x} (2 b c-5 a d)}{3 a e (e x)^{3/2}}}{5 a e}-\frac {2 c^2 \sqrt {a+b x}}{5 a e (e x)^{5/2}}\right )}{7 a e}-\frac {2 \sqrt {a+b x} (c+d x)^3}{7 a e (e x)^{7/2}}\right )}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {(7 a d+8 b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {6 (b c-a d) \left (-\frac {\frac {2 \sqrt {a+b x} \left (15 a^2 d^2+4 b c (2 b c-5 a d)\right )}{3 a^2 e^2 \sqrt {e x}}-\frac {4 c \sqrt {a+b x} (2 b c-5 a d)}{3 a e (e x)^{3/2}}}{5 a e}-\frac {2 c^2 \sqrt {a+b x}}{5 a e (e x)^{5/2}}\right )}{7 a e}-\frac {2 \sqrt {a+b x} (c+d x)^3}{7 a e (e x)^{7/2}}\right )}{9 c e}-\frac {2 \sqrt {a+b x} (c+d x)^4}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (a+b x)^{3/2} (c+d x)^4}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {2 (a+b x)^{5/2} (c+d x)^4}{13 c e (e x)^{13/2}}\right )}{15 a c e}-\frac {2 (a+b x)^{7/2} (c+d x)^4}{15 a c e (e x)^{15/2}}\)

Input:

Int[((a + b*x)^(5/2)*(c + d*x)^3)/(e*x)^(17/2),x]
 

Output:

(-2*(a + b*x)^(7/2)*(c + d*x)^4)/(15*a*c*e*(e*x)^(15/2)) - ((8*b*c + 7*a*d 
)*((-2*(a + b*x)^(5/2)*(c + d*x)^4)/(13*c*e*(e*x)^(13/2)) + (5*(b*c - a*d) 
*((-2*(a + b*x)^(3/2)*(c + d*x)^4)/(11*c*e*(e*x)^(11/2)) + (3*(b*c - a*d)* 
((-2*Sqrt[a + b*x]*(c + d*x)^4)/(9*c*e*(e*x)^(9/2)) + ((b*c - a*d)*((-2*Sq 
rt[a + b*x]*(c + d*x)^3)/(7*a*e*(e*x)^(7/2)) - (6*(b*c - a*d)*((-2*c^2*Sqr 
t[a + b*x])/(5*a*e*(e*x)^(5/2)) - ((-4*c*(2*b*c - 5*a*d)*Sqrt[a + b*x])/(3 
*a*e*(e*x)^(3/2)) + (2*(15*a^2*d^2 + 4*b*c*(2*b*c - 5*a*d))*Sqrt[a + b*x]) 
/(3*a^2*e^2*Sqrt[e*x]))/(5*a*e)))/(7*a*e)))/(9*c*e)))/(11*c*e)))/(13*c*e)) 
)/(15*a*c*e)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.80

method	result	size
gosper	\(-\frac {2 x \left (b x +a \right )^{\frac {7}{2}} \left (-1430 x^{4} a^{3} b \,d^{3}+1560 x^{4} a^{2} b^{2} c \,d^{2}-720 x^{4} a \,b^{3} c^{2} d +128 b^{4} c^{3} x^{4}+5005 a^{4} d^{3} x^{3}-5460 a^{3} b c \,d^{2} x^{3}+2520 a^{2} b^{2} c^{2} d \,x^{3}-448 a \,b^{3} c^{3} x^{3}+12285 a^{4} c \,d^{2} x^{2}-5670 a^{3} b \,c^{2} d \,x^{2}+1008 a^{2} b^{2} c^{3} x^{2}+10395 a^{4} c^{2} d x -1848 a^{3} b \,c^{3} x +3003 a^{4} c^{3}\right )}{45045 a^{5} \left (e x \right )^{\frac {17}{2}}}\)	\(189\)
orering	\(-\frac {2 x \left (b x +a \right )^{\frac {7}{2}} \left (-1430 x^{4} a^{3} b \,d^{3}+1560 x^{4} a^{2} b^{2} c \,d^{2}-720 x^{4} a \,b^{3} c^{2} d +128 b^{4} c^{3} x^{4}+5005 a^{4} d^{3} x^{3}-5460 a^{3} b c \,d^{2} x^{3}+2520 a^{2} b^{2} c^{2} d \,x^{3}-448 a \,b^{3} c^{3} x^{3}+12285 a^{4} c \,d^{2} x^{2}-5670 a^{3} b \,c^{2} d \,x^{2}+1008 a^{2} b^{2} c^{3} x^{2}+10395 a^{4} c^{2} d x -1848 a^{3} b \,c^{3} x +3003 a^{4} c^{3}\right )}{45045 a^{5} \left (e x \right )^{\frac {17}{2}}}\)	\(189\)
default	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1430 a^{3} b^{3} d^{3} x^{6}+1560 a^{2} b^{4} c \,d^{2} x^{6}-720 a \,b^{5} c^{2} d \,x^{6}+128 b^{6} c^{3} x^{6}+2145 a^{4} b^{2} d^{3} x^{5}-2340 a^{3} b^{3} c \,d^{2} x^{5}+1080 a^{2} b^{4} c^{2} d \,x^{5}-192 a \,b^{5} c^{3} x^{5}+8580 a^{5} b \,d^{3} x^{4}+2925 a^{4} b^{2} c \,d^{2} x^{4}-1350 a^{3} b^{3} c^{2} d \,x^{4}+240 a^{2} b^{4} c^{3} x^{4}+5005 a^{6} d^{3} x^{3}+19110 a^{5} b c \,d^{2} x^{3}+1575 a^{4} b^{2} c^{2} d \,x^{3}-280 a^{3} b^{3} c^{3} x^{3}+12285 a^{6} c \,d^{2} x^{2}+15120 a^{5} b \,c^{2} d \,x^{2}+315 a^{4} b^{2} c^{3} x^{2}+10395 a^{6} c^{2} d x +4158 a^{5} b \,c^{3} x +3003 a^{6} c^{3}\right )}{45045 x^{7} a^{5} e^{8} \sqrt {e x}}\)	\(310\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-1430 a^{3} b^{4} d^{3} x^{7}+1560 a^{2} b^{5} c \,d^{2} x^{7}-720 a \,b^{6} c^{2} d \,x^{7}+128 b^{7} c^{3} x^{7}+715 a^{4} b^{3} d^{3} x^{6}-780 a^{3} b^{4} c \,d^{2} x^{6}+360 a^{2} b^{5} c^{2} d \,x^{6}-64 a \,b^{6} c^{3} x^{6}+10725 a^{5} b^{2} d^{3} x^{5}+585 a^{4} b^{3} c \,d^{2} x^{5}-270 a^{3} b^{4} c^{2} d \,x^{5}+48 a^{2} b^{5} c^{3} x^{5}+13585 a^{6} b \,d^{3} x^{4}+22035 a^{5} b^{2} c \,d^{2} x^{4}+225 a^{4} b^{3} c^{2} d \,x^{4}-40 a^{3} b^{4} c^{3} x^{4}+5005 a^{7} d^{3} x^{3}+31395 a^{6} b c \,d^{2} x^{3}+16695 a^{5} b^{2} c^{2} d \,x^{3}+35 a^{4} b^{3} c^{3} x^{3}+12285 a^{7} c \,d^{2} x^{2}+25515 a^{6} b \,c^{2} d \,x^{2}+4473 a^{5} b^{2} c^{3} x^{2}+10395 a^{7} c^{2} d x +7161 a^{6} b \,c^{3} x +3003 a^{7} c^{3}\right )}{45045 e^{8} \sqrt {e x}\, x^{7} a^{5}}\)	\(368\)

Input:

int((b*x+a)^(5/2)*(d*x+c)^3/(e*x)^(17/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*x*(b*x+a)^(7/2)*(-1430*a^3*b*d^3*x^4+1560*a^2*b^2*c*d^2*x^4-720*a 
*b^3*c^2*d*x^4+128*b^4*c^3*x^4+5005*a^4*d^3*x^3-5460*a^3*b*c*d^2*x^3+2520* 
a^2*b^2*c^2*d*x^3-448*a*b^3*c^3*x^3+12285*a^4*c*d^2*x^2-5670*a^3*b*c^2*d*x 
^2+1008*a^2*b^2*c^3*x^2+10395*a^4*c^2*d*x-1848*a^3*b*c^3*x+3003*a^4*c^3)/a 
^5/(e*x)^(17/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left (3003 \, a^{7} c^{3} + 2 \, {\left (64 \, b^{7} c^{3} - 360 \, a b^{6} c^{2} d + 780 \, a^{2} b^{5} c d^{2} - 715 \, a^{3} b^{4} d^{3}\right )} x^{7} - {\left (64 \, a b^{6} c^{3} - 360 \, a^{2} b^{5} c^{2} d + 780 \, a^{3} b^{4} c d^{2} - 715 \, a^{4} b^{3} d^{3}\right )} x^{6} + 3 \, {\left (16 \, a^{2} b^{5} c^{3} - 90 \, a^{3} b^{4} c^{2} d + 195 \, a^{4} b^{3} c d^{2} + 3575 \, a^{5} b^{2} d^{3}\right )} x^{5} - 5 \, {\left (8 \, a^{3} b^{4} c^{3} - 45 \, a^{4} b^{3} c^{2} d - 4407 \, a^{5} b^{2} c d^{2} - 2717 \, a^{6} b d^{3}\right )} x^{4} + 35 \, {\left (a^{4} b^{3} c^{3} + 477 \, a^{5} b^{2} c^{2} d + 897 \, a^{6} b c d^{2} + 143 \, a^{7} d^{3}\right )} x^{3} + 63 \, {\left (71 \, a^{5} b^{2} c^{3} + 405 \, a^{6} b c^{2} d + 195 \, a^{7} c d^{2}\right )} x^{2} + 231 \, {\left (31 \, a^{6} b c^{3} + 45 \, a^{7} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {e x}}{45045 \, a^{5} e^{9} x^{8}} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^3/(e*x)^(17/2),x, algorithm="fricas")
 

Output:

-2/45045*(3003*a^7*c^3 + 2*(64*b^7*c^3 - 360*a*b^6*c^2*d + 780*a^2*b^5*c*d 
^2 - 715*a^3*b^4*d^3)*x^7 - (64*a*b^6*c^3 - 360*a^2*b^5*c^2*d + 780*a^3*b^ 
4*c*d^2 - 715*a^4*b^3*d^3)*x^6 + 3*(16*a^2*b^5*c^3 - 90*a^3*b^4*c^2*d + 19 
5*a^4*b^3*c*d^2 + 3575*a^5*b^2*d^3)*x^5 - 5*(8*a^3*b^4*c^3 - 45*a^4*b^3*c^ 
2*d - 4407*a^5*b^2*c*d^2 - 2717*a^6*b*d^3)*x^4 + 35*(a^4*b^3*c^3 + 477*a^5 
*b^2*c^2*d + 897*a^6*b*c*d^2 + 143*a^7*d^3)*x^3 + 63*(71*a^5*b^2*c^3 + 405 
*a^6*b*c^2*d + 195*a^7*c*d^2)*x^2 + 231*(31*a^6*b*c^3 + 45*a^7*c^2*d)*x)*s 
qrt(b*x + a)*sqrt(e*x)/(a^5*e^9*x^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(5/2)*(d*x+c)**3/(e*x)**(17/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^3/(e*x)^(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>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (64 \, a^{2} b^{7} c^{3} e^{7} - 360 \, a^{3} b^{6} c^{2} d e^{7} + 780 \, a^{4} b^{5} c d^{2} e^{7} - 715 \, a^{5} b^{4} d^{3} e^{7}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (64 \, a^{3} b^{7} c^{3} e^{7} - 360 \, a^{4} b^{6} c^{2} d e^{7} + 780 \, a^{5} b^{5} c d^{2} e^{7} - 715 \, a^{6} b^{4} d^{3} e^{7}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (16 \, a^{4} b^{7} c^{3} e^{7} - 90 \, a^{5} b^{6} c^{2} d e^{7} + 195 \, a^{6} b^{5} c d^{2} e^{7} - 121 \, a^{7} b^{4} d^{3} e^{7}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (8 \, a^{5} b^{7} c^{3} e^{7} - 45 \, a^{6} b^{6} c^{2} d e^{7} + 66 \, a^{7} b^{5} c d^{2} e^{7} - 29 \, a^{8} b^{4} d^{3} e^{7}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (a^{6} b^{7} c^{3} e^{7} - 3 \, a^{7} b^{6} c^{2} d e^{7} + 3 \, a^{8} b^{5} c d^{2} e^{7} - a^{9} b^{4} d^{3} e^{7}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b^{9}}{45045 \, {\left ({\left (b x + a\right )} b e - a b e\right )}^{\frac {15}{2}} e^{8} {\left | b \right |}} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^3/(e*x)^(17/2),x, algorithm="giac")
 

Output:

-2/45045*(((b*x + a)*((b*x + a)*(2*(64*a^2*b^7*c^3*e^7 - 360*a^3*b^6*c^2*d 
*e^7 + 780*a^4*b^5*c*d^2*e^7 - 715*a^5*b^4*d^3*e^7)*(b*x + a)/a^7 - 15*(64 
*a^3*b^7*c^3*e^7 - 360*a^4*b^6*c^2*d*e^7 + 780*a^5*b^5*c*d^2*e^7 - 715*a^6 
*b^4*d^3*e^7)/a^7) + 195*(16*a^4*b^7*c^3*e^7 - 90*a^5*b^6*c^2*d*e^7 + 195* 
a^6*b^5*c*d^2*e^7 - 121*a^7*b^4*d^3*e^7)/a^7) - 715*(8*a^5*b^7*c^3*e^7 - 4 
5*a^6*b^6*c^2*d*e^7 + 66*a^7*b^5*c*d^2*e^7 - 29*a^8*b^4*d^3*e^7)/a^7)*(b*x 
 + a) + 6435*(a^6*b^7*c^3*e^7 - 3*a^7*b^6*c^2*d*e^7 + 3*a^8*b^5*c*d^2*e^7 
- a^9*b^4*d^3*e^7)/a^7)*(b*x + a)^(7/2)*b^9/(((b*x + a)*b*e - a*b*e)^(15/2 
)*e^8*abs(b))
 

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,a^2\,c^3}{15\,e^8}-\frac {x^6\,\left (-1430\,a^4\,b^3\,d^3+1560\,a^3\,b^4\,c\,d^2-720\,a^2\,b^5\,c^2\,d+128\,a\,b^6\,c^3\right )}{45045\,a^5\,e^8}+\frac {x^3\,\left (10010\,a^7\,d^3+62790\,a^6\,b\,c\,d^2+33390\,a^5\,b^2\,c^2\,d+70\,a^4\,b^3\,c^3\right )}{45045\,a^5\,e^8}+\frac {x^7\,\left (-2860\,a^3\,b^4\,d^3+3120\,a^2\,b^5\,c\,d^2-1440\,a\,b^6\,c^2\,d+256\,b^7\,c^3\right )}{45045\,a^5\,e^8}+\frac {2\,c\,x^2\,\left (195\,a^2\,d^2+405\,a\,b\,c\,d+71\,b^2\,c^2\right )}{715\,e^8}+\frac {2\,a\,c^2\,x\,\left (45\,a\,d+31\,b\,c\right )}{195\,e^8}+\frac {2\,b\,x^4\,\left (2717\,a^3\,d^3+4407\,a^2\,b\,c\,d^2+45\,a\,b^2\,c^2\,d-8\,b^3\,c^3\right )}{9009\,a^2\,e^8}+\frac {2\,b^2\,x^5\,\left (3575\,a^3\,d^3+195\,a^2\,b\,c\,d^2-90\,a\,b^2\,c^2\,d+16\,b^3\,c^3\right )}{15015\,a^3\,e^8}\right )}{x^7\,\sqrt {e\,x}} \] Input:

int(((a + b*x)^(5/2)*(c + d*x)^3)/(e*x)^(17/2),x)
 

Output:

-((a + b*x)^(1/2)*((2*a^2*c^3)/(15*e^8) - (x^6*(128*a*b^6*c^3 - 1430*a^4*b 
^3*d^3 - 720*a^2*b^5*c^2*d + 1560*a^3*b^4*c*d^2))/(45045*a^5*e^8) + (x^3*( 
10010*a^7*d^3 + 70*a^4*b^3*c^3 + 33390*a^5*b^2*c^2*d + 62790*a^6*b*c*d^2)) 
/(45045*a^5*e^8) + (x^7*(256*b^7*c^3 - 2860*a^3*b^4*d^3 + 3120*a^2*b^5*c*d 
^2 - 1440*a*b^6*c^2*d))/(45045*a^5*e^8) + (2*c*x^2*(195*a^2*d^2 + 71*b^2*c 
^2 + 405*a*b*c*d))/(715*e^8) + (2*a*c^2*x*(45*a*d + 31*b*c))/(195*e^8) + ( 
2*b*x^4*(2717*a^3*d^3 - 8*b^3*c^3 + 45*a*b^2*c^2*d + 4407*a^2*b*c*d^2))/(9 
009*a^2*e^8) + (2*b^2*x^5*(3575*a^3*d^3 + 16*b^3*c^3 - 90*a*b^2*c^2*d + 19 
5*a^2*b*c*d^2))/(15015*a^3*e^8)))/(x^7*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b x)^{5/2} (c+d x)^3}{(e x)^{17/2}} \, dx=\frac {2 \sqrt {e}\, \left (1560 \sqrt {b}\, a^{2} b^{5} c \,d^{2} x^{8}-720 \sqrt {b}\, a \,b^{6} c^{2} d \,x^{8}-25515 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,c^{2} d \,x^{2}-31395 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b c \,d^{2} x^{3}-16695 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c^{2} d \,x^{3}-22035 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c \,d^{2} x^{4}-225 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{2} d \,x^{4}-585 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c \,d^{2} x^{5}+270 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{2} d \,x^{5}+780 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c \,d^{2} x^{6}-360 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{2} d \,x^{6}-1560 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c \,d^{2} x^{7}+720 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{2} d \,x^{7}-3003 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c^{3}+128 \sqrt {b}\, b^{7} c^{3} x^{8}-10395 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c^{2} d x -12285 \sqrt {x}\, \sqrt {b x +a}\, a^{7} c \,d^{2} x^{2}-7161 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,c^{3} x -13585 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,d^{3} x^{4}-4473 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c^{3} x^{2}-10725 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} d^{3} x^{5}-35 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{3} x^{3}-715 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} d^{3} x^{6}+40 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{3} x^{4}+1430 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} d^{3} x^{7}-48 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{3} x^{5}+64 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{3} x^{6}-5005 \sqrt {x}\, \sqrt {b x +a}\, a^{7} d^{3} x^{3}-128 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c^{3} x^{7}-1430 \sqrt {b}\, a^{3} b^{4} d^{3} x^{8}\right )}{45045 a^{5} e^{9} x^{8}} \] Input:

int((b*x+a)^(5/2)*(d*x+c)^3/(e*x)^(17/2),x)
 

Output:

(2*sqrt(e)*( - 3003*sqrt(x)*sqrt(a + b*x)*a**7*c**3 - 10395*sqrt(x)*sqrt(a 
 + b*x)*a**7*c**2*d*x - 12285*sqrt(x)*sqrt(a + b*x)*a**7*c*d**2*x**2 - 500 
5*sqrt(x)*sqrt(a + b*x)*a**7*d**3*x**3 - 7161*sqrt(x)*sqrt(a + b*x)*a**6*b 
*c**3*x - 25515*sqrt(x)*sqrt(a + b*x)*a**6*b*c**2*d*x**2 - 31395*sqrt(x)*s 
qrt(a + b*x)*a**6*b*c*d**2*x**3 - 13585*sqrt(x)*sqrt(a + b*x)*a**6*b*d**3* 
x**4 - 4473*sqrt(x)*sqrt(a + b*x)*a**5*b**2*c**3*x**2 - 16695*sqrt(x)*sqrt 
(a + b*x)*a**5*b**2*c**2*d*x**3 - 22035*sqrt(x)*sqrt(a + b*x)*a**5*b**2*c* 
d**2*x**4 - 10725*sqrt(x)*sqrt(a + b*x)*a**5*b**2*d**3*x**5 - 35*sqrt(x)*s 
qrt(a + b*x)*a**4*b**3*c**3*x**3 - 225*sqrt(x)*sqrt(a + b*x)*a**4*b**3*c** 
2*d*x**4 - 585*sqrt(x)*sqrt(a + b*x)*a**4*b**3*c*d**2*x**5 - 715*sqrt(x)*s 
qrt(a + b*x)*a**4*b**3*d**3*x**6 + 40*sqrt(x)*sqrt(a + b*x)*a**3*b**4*c**3 
*x**4 + 270*sqrt(x)*sqrt(a + b*x)*a**3*b**4*c**2*d*x**5 + 780*sqrt(x)*sqrt 
(a + b*x)*a**3*b**4*c*d**2*x**6 + 1430*sqrt(x)*sqrt(a + b*x)*a**3*b**4*d** 
3*x**7 - 48*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c**3*x**5 - 360*sqrt(x)*sqrt(a 
 + b*x)*a**2*b**5*c**2*d*x**6 - 1560*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c*d** 
2*x**7 + 64*sqrt(x)*sqrt(a + b*x)*a*b**6*c**3*x**6 + 720*sqrt(x)*sqrt(a + 
b*x)*a*b**6*c**2*d*x**7 - 128*sqrt(x)*sqrt(a + b*x)*b**7*c**3*x**7 - 1430* 
sqrt(b)*a**3*b**4*d**3*x**8 + 1560*sqrt(b)*a**2*b**5*c*d**2*x**8 - 720*sqr 
t(b)*a*b**6*c**2*d*x**8 + 128*sqrt(b)*b**7*c**3*x**8))/(45045*a**5*e**9*x* 
*8)