\(\int \frac {(c+d x)^{5/2} (a+b x^2)^2}{(e x)^{17/2}} \, dx\) [821]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}+\frac {16 a^2 d (c+d x)^{7/2}}{195 c^2 e^2 (e x)^{13/2}}-\frac {4 a \left (65 b c^2+8 a d^2\right ) (c+d x)^{7/2}}{715 c^3 e^3 (e x)^{11/2}}+\frac {16 a d \left (65 b c^2+8 a d^2\right ) (c+d x)^{7/2}}{6435 c^4 e^4 (e x)^{9/2}}-\frac {2 \left (6435 b^2 c^4+1040 a b c^2 d^2+128 a^2 d^4\right ) (c+d x)^{7/2}}{45045 c^5 e^5 (e x)^{7/2}} \] Output:

-2/15*a^2*(d*x+c)^(7/2)/c/e/(e*x)^(15/2)+16/195*a^2*d*(d*x+c)^(7/2)/c^2/e^ 
2/(e*x)^(13/2)-4/715*a*(8*a*d^2+65*b*c^2)*(d*x+c)^(7/2)/c^3/e^3/(e*x)^(11/ 
2)+16/6435*a*d*(8*a*d^2+65*b*c^2)*(d*x+c)^(7/2)/c^4/e^4/(e*x)^(9/2)-2/4504 
5*(128*a^2*d^4+1040*a*b*c^2*d^2+6435*b^2*c^4)*(d*x+c)^(7/2)/c^5/e^5/(e*x)^ 
(7/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \sqrt {e x} (c+d x)^{7/2} \left (6435 b^2 c^4 x^4+130 a b c^2 x^2 \left (63 c^2-28 c d x+8 d^2 x^2\right )+a^2 \left (3003 c^4-1848 c^3 d x+1008 c^2 d^2 x^2-448 c d^3 x^3+128 d^4 x^4\right )\right )}{45045 c^5 e^9 x^8} \] Input:

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

Output:

(-2*Sqrt[e*x]*(c + d*x)^(7/2)*(6435*b^2*c^4*x^4 + 130*a*b*c^2*x^2*(63*c^2 
- 28*c*d*x + 8*d^2*x^2) + a^2*(3003*c^4 - 1848*c^3*d*x + 1008*c^2*d^2*x^2 
- 448*c*d^3*x^3 + 128*d^4*x^4)))/(45045*c^5*e^9*x^8)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {520, 27, 2124, 27, 520, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^{5/2} \left (-15 b^2 c x^3-30 a b c x+8 a^2 d\right )}{2 (e x)^{15/2}}dx}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} \left (-15 b^2 c x^3-30 a b c x+8 a^2 d\right )}{(e x)^{15/2}}dx}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {3 (c+d x)^{5/2} \left (65 b^2 c^2 x^2+2 a \left (65 b c^2+8 a d^2\right )\right )}{2 (e x)^{13/2}}dx}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \int \frac {(c+d x)^{5/2} \left (65 b^2 c^2 x^2+2 a \left (65 b c^2+8 a d^2\right )\right )}{(e x)^{13/2}}dx}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {3 \left (-\frac {2 \int \frac {\left (8 a d \left (65 b c^2+8 a d^2\right )-715 b^2 c^3 x\right ) (c+d x)^{5/2}}{2 (e x)^{11/2}}dx}{11 c e}-\frac {4 a (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (-\frac {\int \frac {\left (8 a d \left (65 b c^2+8 a d^2\right )-715 b^2 c^3 x\right ) (c+d x)^{5/2}}{(e x)^{11/2}}dx}{11 c e}-\frac {4 a (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {3 \left (-\frac {-\frac {\left (128 a^2 d^4+1040 a b c^2 d^2+6435 b^2 c^4\right ) \int \frac {(c+d x)^{5/2}}{(e x)^{9/2}}dx}{9 c e}-\frac {16 a d (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {3 \left (-\frac {\frac {2 (c+d x)^{7/2} \left (128 a^2 d^4+1040 a b c^2 d^2+6435 b^2 c^4\right )}{63 c^2 e^2 (e x)^{7/2}}-\frac {16 a d (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{7/2} \left (8 a d^2+65 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {16 a^2 d (c+d x)^{7/2}}{13 c e (e x)^{13/2}}}{15 c e}-\frac {2 a^2 (c+d x)^{7/2}}{15 c e (e x)^{15/2}}\)

Input:

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

Output:

(-2*a^2*(c + d*x)^(7/2))/(15*c*e*(e*x)^(15/2)) - ((-16*a^2*d*(c + d*x)^(7/ 
2))/(13*c*e*(e*x)^(13/2)) - (3*((-4*a*(65*b*c^2 + 8*a*d^2)*(c + d*x)^(7/2) 
)/(11*c*e*(e*x)^(11/2)) - ((-16*a*d*(65*b*c^2 + 8*a*d^2)*(c + d*x)^(7/2))/ 
(9*c*e*(e*x)^(9/2)) + (2*(6435*b^2*c^4 + 1040*a*b*c^2*d^2 + 128*a^2*d^4)*( 
c + d*x)^(7/2))/(63*c^2*e^2*(e*x)^(7/2)))/(11*c*e)))/(13*c*e))/(15*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 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {2 x \left (d x +c \right )^{\frac {7}{2}} \left (128 a^{2} d^{4} x^{4}+1040 x^{4} a b \,c^{2} d^{2}+6435 x^{4} b^{2} c^{4}-448 a^{2} c \,d^{3} x^{3}-3640 a b \,c^{3} d \,x^{3}+1008 a^{2} c^{2} d^{2} x^{2}+8190 a b \,c^{4} x^{2}-1848 a^{2} d \,c^{3} x +3003 a^{2} c^{4}\right )}{45045 c^{5} \left (e x \right )^{\frac {17}{2}}}\)	\(120\)
orering	\(-\frac {2 x \left (d x +c \right )^{\frac {7}{2}} \left (128 a^{2} d^{4} x^{4}+1040 x^{4} a b \,c^{2} d^{2}+6435 x^{4} b^{2} c^{4}-448 a^{2} c \,d^{3} x^{3}-3640 a b \,c^{3} d \,x^{3}+1008 a^{2} c^{2} d^{2} x^{2}+8190 a b \,c^{4} x^{2}-1848 a^{2} d \,c^{3} x +3003 a^{2} c^{4}\right )}{45045 c^{5} \left (e x \right )^{\frac {17}{2}}}\)	\(120\)
default	\(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{6} x^{6}+1040 a b \,c^{2} d^{4} x^{6}+6435 b^{2} c^{4} d^{2} x^{6}-192 a^{2} c \,d^{5} x^{5}-1560 a b \,c^{3} d^{3} x^{5}+12870 b^{2} c^{5} d \,x^{5}+240 a^{2} c^{2} d^{4} x^{4}+1950 a b \,c^{4} d^{2} x^{4}+6435 b^{2} c^{6} x^{4}-280 a^{2} c^{3} d^{3} x^{3}+12740 a b \,c^{5} d \,x^{3}+315 a^{2} c^{4} d^{2} x^{2}+8190 a b \,c^{6} x^{2}+4158 a^{2} c^{5} d x +3003 a^{2} c^{6}\right )}{45045 x^{7} c^{5} e^{8} \sqrt {e x}}\)	\(205\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (128 a^{2} d^{7} x^{7}+1040 a b \,c^{2} d^{5} x^{7}+6435 b^{2} c^{4} d^{3} x^{7}-64 a^{2} c \,d^{6} x^{6}-520 a b \,c^{3} d^{4} x^{6}+19305 b^{2} c^{5} d^{2} x^{6}+48 a^{2} c^{2} d^{5} x^{5}+390 a b \,c^{4} d^{3} x^{5}+19305 b^{2} c^{6} d \,x^{5}-40 a^{2} c^{3} d^{4} x^{4}+14690 a b \,c^{5} d^{2} x^{4}+6435 b^{2} c^{7} x^{4}+35 a^{2} c^{4} d^{3} x^{3}+20930 a b \,c^{6} d \,x^{3}+4473 a^{2} c^{5} d^{2} x^{2}+8190 a b \,c^{7} x^{2}+7161 a^{2} c^{6} d x +3003 a^{2} c^{7}\right )}{45045 e^{8} \sqrt {e x}\, x^{7} c^{5}}\)	\(246\)

Input:

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

Output:

-2/45045*x*(d*x+c)^(7/2)*(128*a^2*d^4*x^4+1040*a*b*c^2*d^2*x^4+6435*b^2*c^ 
4*x^4-448*a^2*c*d^3*x^3-3640*a*b*c^3*d*x^3+1008*a^2*c^2*d^2*x^2+8190*a*b*c 
^4*x^2-1848*a^2*c^3*d*x+3003*a^2*c^4)/c^5/(e*x)^(17/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left (7161 \, a^{2} c^{6} d x + 3003 \, a^{2} c^{7} + {\left (6435 \, b^{2} c^{4} d^{3} + 1040 \, a b c^{2} d^{5} + 128 \, a^{2} d^{7}\right )} x^{7} + {\left (19305 \, b^{2} c^{5} d^{2} - 520 \, a b c^{3} d^{4} - 64 \, a^{2} c d^{6}\right )} x^{6} + 3 \, {\left (6435 \, b^{2} c^{6} d + 130 \, a b c^{4} d^{3} + 16 \, a^{2} c^{2} d^{5}\right )} x^{5} + 5 \, {\left (1287 \, b^{2} c^{7} + 2938 \, a b c^{5} d^{2} - 8 \, a^{2} c^{3} d^{4}\right )} x^{4} + 35 \, {\left (598 \, a b c^{6} d + a^{2} c^{4} d^{3}\right )} x^{3} + 63 \, {\left (130 \, a b c^{7} + 71 \, a^{2} c^{5} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{45045 \, c^{5} e^{9} x^{8}} \] Input:

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

Output:

-2/45045*(7161*a^2*c^6*d*x + 3003*a^2*c^7 + (6435*b^2*c^4*d^3 + 1040*a*b*c 
^2*d^5 + 128*a^2*d^7)*x^7 + (19305*b^2*c^5*d^2 - 520*a*b*c^3*d^4 - 64*a^2* 
c*d^6)*x^6 + 3*(6435*b^2*c^6*d + 130*a*b*c^4*d^3 + 16*a^2*c^2*d^5)*x^5 + 5 
*(1287*b^2*c^7 + 2938*a*b*c^5*d^2 - 8*a^2*c^3*d^4)*x^4 + 35*(598*a*b*c^6*d 
 + a^2*c^4*d^3)*x^3 + 63*(130*a*b*c^7 + 71*a^2*c^5*d^2)*x^2)*sqrt(d*x + c) 
*sqrt(e*x)/(c^5*e^9*x^8)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(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 = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {{\left (6435 \, b^{2} c^{6} d^{3} e^{7} + 1040 \, a b c^{4} d^{5} e^{7} + 128 \, a^{2} c^{2} d^{7} e^{7}\right )} {\left (d x + c\right )}}{c^{7}} - \frac {60 \, {\left (429 \, b^{2} c^{7} d^{3} e^{7} + 130 \, a b c^{5} d^{5} e^{7} + 16 \, a^{2} c^{3} d^{7} e^{7}\right )}}{c^{7}}\right )} + \frac {390 \, {\left (99 \, b^{2} c^{8} d^{3} e^{7} + 65 \, a b c^{6} d^{5} e^{7} + 8 \, a^{2} c^{4} d^{7} e^{7}\right )}}{c^{7}}\right )} - \frac {2860 \, {\left (9 \, b^{2} c^{9} d^{3} e^{7} + 11 \, a b c^{7} d^{5} e^{7} + 2 \, a^{2} c^{5} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )} + \frac {6435 \, {\left (b^{2} c^{10} d^{3} e^{7} + 2 \, a b c^{8} d^{5} e^{7} + a^{2} c^{6} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )}^{\frac {7}{2}} d^{9}}{45045 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {15}{2}} e^{8} {\left | d \right |}} \] Input:

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

Output:

-2/45045*(((d*x + c)*((d*x + c)*((6435*b^2*c^6*d^3*e^7 + 1040*a*b*c^4*d^5* 
e^7 + 128*a^2*c^2*d^7*e^7)*(d*x + c)/c^7 - 60*(429*b^2*c^7*d^3*e^7 + 130*a 
*b*c^5*d^5*e^7 + 16*a^2*c^3*d^7*e^7)/c^7) + 390*(99*b^2*c^8*d^3*e^7 + 65*a 
*b*c^6*d^5*e^7 + 8*a^2*c^4*d^7*e^7)/c^7) - 2860*(9*b^2*c^9*d^3*e^7 + 11*a* 
b*c^7*d^5*e^7 + 2*a^2*c^5*d^7*e^7)/c^7)*(d*x + c) + 6435*(b^2*c^10*d^3*e^7 
 + 2*a*b*c^8*d^5*e^7 + a^2*c^6*d^7*e^7)/c^7)*(d*x + c)^(7/2)*d^9/(((d*x + 
c)*d*e - c*d*e)^(15/2)*e^8*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2\,c^2}{15\,e^8}+\frac {x^4\,\left (-80\,a^2\,c^3\,d^4+29380\,a\,b\,c^5\,d^2+12870\,b^2\,c^7\right )}{45045\,c^5\,e^8}+\frac {x^7\,\left (256\,a^2\,d^7+2080\,a\,b\,c^2\,d^5+12870\,b^2\,c^4\,d^3\right )}{45045\,c^5\,e^8}+\frac {x^5\,\left (96\,a^2\,c^2\,d^5+780\,a\,b\,c^4\,d^3+38610\,b^2\,c^6\,d\right )}{45045\,c^5\,e^8}-\frac {x^6\,\left (128\,a^2\,c\,d^6+1040\,a\,b\,c^3\,d^4-38610\,b^2\,c^5\,d^2\right )}{45045\,c^5\,e^8}+\frac {2\,a\,x^2\,\left (130\,b\,c^2+71\,a\,d^2\right )}{715\,e^8}+\frac {62\,a^2\,c\,d\,x}{195\,e^8}+\frac {2\,a\,d\,x^3\,\left (598\,b\,c^2+a\,d^2\right )}{1287\,c\,e^8}\right )}{x^7\,\sqrt {e\,x}} \] Input:

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

Output:

-((c + d*x)^(1/2)*((2*a^2*c^2)/(15*e^8) + (x^4*(12870*b^2*c^7 - 80*a^2*c^3 
*d^4 + 29380*a*b*c^5*d^2))/(45045*c^5*e^8) + (x^7*(256*a^2*d^7 + 12870*b^2 
*c^4*d^3 + 2080*a*b*c^2*d^5))/(45045*c^5*e^8) + (x^5*(38610*b^2*c^6*d + 96 
*a^2*c^2*d^5 + 780*a*b*c^4*d^3))/(45045*c^5*e^8) - (x^6*(128*a^2*c*d^6 - 3 
8610*b^2*c^5*d^2 + 1040*a*b*c^3*d^4))/(45045*c^5*e^8) + (2*a*x^2*(71*a*d^2 
 + 130*b*c^2))/(715*e^8) + (62*a^2*c*d*x)/(195*e^8) + (2*a*d*x^3*(a*d^2 + 
598*b*c^2))/(1287*c*e^8)))/(x^7*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=\frac {2 \sqrt {e}\, \left (-3003 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{7}-7161 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{6} d x -4473 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5} d^{2} x^{2}-35 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{3} x^{3}+40 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{4} x^{4}-48 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{5} x^{5}+64 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{6} x^{6}-128 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{7} x^{7}-8190 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{7} x^{2}-20930 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} d \,x^{3}-14690 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d^{2} x^{4}-390 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{3} x^{5}+520 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{4} x^{6}-1040 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{5} x^{7}-6435 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{7} x^{4}-19305 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d \,x^{5}-19305 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{2} x^{6}-6435 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{3} x^{7}+128 \sqrt {d}\, a^{2} d^{7} x^{8}+1040 \sqrt {d}\, a b \,c^{2} d^{5} x^{8}+429 \sqrt {d}\, b^{2} c^{4} d^{3} x^{8}\right )}{45045 c^{5} e^{9} x^{8}} \] Input:

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

Output:

(2*sqrt(e)*( - 3003*sqrt(x)*sqrt(c + d*x)*a**2*c**7 - 7161*sqrt(x)*sqrt(c 
+ d*x)*a**2*c**6*d*x - 4473*sqrt(x)*sqrt(c + d*x)*a**2*c**5*d**2*x**2 - 35 
*sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**3*x**3 + 40*sqrt(x)*sqrt(c + d*x)*a**2 
*c**3*d**4*x**4 - 48*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**5*x**5 + 64*sqrt(x 
)*sqrt(c + d*x)*a**2*c*d**6*x**6 - 128*sqrt(x)*sqrt(c + d*x)*a**2*d**7*x** 
7 - 8190*sqrt(x)*sqrt(c + d*x)*a*b*c**7*x**2 - 20930*sqrt(x)*sqrt(c + d*x) 
*a*b*c**6*d*x**3 - 14690*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d**2*x**4 - 390*sq 
rt(x)*sqrt(c + d*x)*a*b*c**4*d**3*x**5 + 520*sqrt(x)*sqrt(c + d*x)*a*b*c** 
3*d**4*x**6 - 1040*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**5*x**7 - 6435*sqrt(x) 
*sqrt(c + d*x)*b**2*c**7*x**4 - 19305*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d*x* 
*5 - 19305*sqrt(x)*sqrt(c + d*x)*b**2*c**5*d**2*x**6 - 6435*sqrt(x)*sqrt(c 
 + d*x)*b**2*c**4*d**3*x**7 + 128*sqrt(d)*a**2*d**7*x**8 + 1040*sqrt(d)*a* 
b*c**2*d**5*x**8 + 429*sqrt(d)*b**2*c**4*d**3*x**8))/(45045*c**5*e**9*x**8 
)