\(\int \frac {\sqrt {c+d x} (a+b x^2)^2}{(e x)^{13/2}} \, dx\) [796]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}+\frac {16 a^2 d (c+d x)^{3/2}}{99 c^2 e^2 (e x)^{9/2}}-\frac {4 a \left (33 b c^2+8 a d^2\right ) (c+d x)^{3/2}}{231 c^3 e^3 (e x)^{7/2}}+\frac {16 a d \left (33 b c^2+8 a d^2\right ) (c+d x)^{3/2}}{1155 c^4 e^4 (e x)^{5/2}}-\frac {2 \left (1155 b^2 c^4+528 a b c^2 d^2+128 a^2 d^4\right ) (c+d x)^{3/2}}{3465 c^5 e^5 (e x)^{3/2}} \] Output:

-2/11*a^2*(d*x+c)^(3/2)/c/e/(e*x)^(11/2)+16/99*a^2*d*(d*x+c)^(3/2)/c^2/e^2 
/(e*x)^(9/2)-4/231*a*(8*a*d^2+33*b*c^2)*(d*x+c)^(3/2)/c^3/e^3/(e*x)^(7/2)+ 
16/1155*a*d*(8*a*d^2+33*b*c^2)*(d*x+c)^(3/2)/c^4/e^4/(e*x)^(5/2)-2/3465*(1 
28*a^2*d^4+528*a*b*c^2*d^2+1155*b^2*c^4)*(d*x+c)^(3/2)/c^5/e^5/(e*x)^(3/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 \sqrt {e x} (c+d x)^{3/2} \left (1155 b^2 c^4 x^4+66 a b c^2 x^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )+a^2 \left (315 c^4-280 c^3 d x+240 c^2 d^2 x^2-192 c d^3 x^3+128 d^4 x^4\right )\right )}{3465 c^5 e^7 x^6} \] Input:

Integrate[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(13/2),x]
 

Output:

(-2*Sqrt[e*x]*(c + d*x)^(3/2)*(1155*b^2*c^4*x^4 + 66*a*b*c^2*x^2*(15*c^2 - 
 12*c*d*x + 8*d^2*x^2) + a^2*(315*c^4 - 280*c^3*d*x + 240*c^2*d^2*x^2 - 19 
2*c*d^3*x^3 + 128*d^4*x^4)))/(3465*c^5*e^7*x^6)
 

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x}}{(e x)^{13/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {\sqrt {c+d x} \left (-11 b^2 c x^3-22 a b c x+8 a^2 d\right )}{2 (e x)^{11/2}}dx}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} \left (-11 b^2 c x^3-22 a b c x+8 a^2 d\right )}{(e x)^{11/2}}dx}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {3 \sqrt {c+d x} \left (33 b^2 c^2 x^2+2 a \left (33 b c^2+8 a d^2\right )\right )}{2 (e x)^{9/2}}dx}{9 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\sqrt {c+d x} \left (33 b^2 c^2 x^2+2 a \left (33 b c^2+8 a d^2\right )\right )}{(e x)^{9/2}}dx}{3 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {\left (8 a d \left (33 b c^2+8 a d^2\right )-231 b^2 c^3 x\right ) \sqrt {c+d x}}{2 (e x)^{7/2}}dx}{7 c e}-\frac {4 a (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (8 a d \left (33 b c^2+8 a d^2\right )-231 b^2 c^3 x\right ) \sqrt {c+d x}}{(e x)^{7/2}}dx}{7 c e}-\frac {4 a (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (128 a^2 d^4+528 a b c^2 d^2+1155 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{(e x)^{5/2}}dx}{5 c e}-\frac {16 a d (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {-\frac {\frac {2 (c+d x)^{3/2} \left (128 a^2 d^4+528 a b c^2 d^2+1155 b^2 c^4\right )}{15 c^2 e^2 (e x)^{3/2}}-\frac {16 a d (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{3/2} \left (8 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {16 a^2 d (c+d x)^{3/2}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{3/2}}{11 c e (e x)^{11/2}}\)

Input:

Int[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(13/2),x]
 

Output:

(-2*a^2*(c + d*x)^(3/2))/(11*c*e*(e*x)^(11/2)) - ((-16*a^2*d*(c + d*x)^(3/ 
2))/(9*c*e*(e*x)^(9/2)) - ((-4*a*(33*b*c^2 + 8*a*d^2)*(c + d*x)^(3/2))/(7* 
c*e*(e*x)^(7/2)) - ((-16*a*d*(33*b*c^2 + 8*a*d^2)*(c + d*x)^(3/2))/(5*c*e* 
(e*x)^(5/2)) + (2*(1155*b^2*c^4 + 528*a*b*c^2*d^2 + 128*a^2*d^4)*(c + d*x) 
^(3/2))/(15*c^2*e^2*(e*x)^(3/2)))/(7*c*e))/(3*c*e))/(11*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 = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{4}+528 x^{4} a b \,c^{2} d^{2}+1155 x^{4} b^{2} c^{4}-192 a^{2} c \,d^{3} x^{3}-792 a b \,c^{3} d \,x^{3}+240 a^{2} c^{2} d^{2} x^{2}+990 a b \,c^{4} x^{2}-280 a^{2} d \,c^{3} x +315 a^{2} c^{4}\right )}{3465 c^{5} \left (e x \right )^{\frac {13}{2}}}\)	\(120\)
orering	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{4}+528 x^{4} a b \,c^{2} d^{2}+1155 x^{4} b^{2} c^{4}-192 a^{2} c \,d^{3} x^{3}-792 a b \,c^{3} d \,x^{3}+240 a^{2} c^{2} d^{2} x^{2}+990 a b \,c^{4} x^{2}-280 a^{2} d \,c^{3} x +315 a^{2} c^{4}\right )}{3465 c^{5} \left (e x \right )^{\frac {13}{2}}}\)	\(120\)
default	\(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{4}+528 x^{4} a b \,c^{2} d^{2}+1155 x^{4} b^{2} c^{4}-192 a^{2} c \,d^{3} x^{3}-792 a b \,c^{3} d \,x^{3}+240 a^{2} c^{2} d^{2} x^{2}+990 a b \,c^{4} x^{2}-280 a^{2} d \,c^{3} x +315 a^{2} c^{4}\right )}{3465 x^{5} c^{5} e^{6} \sqrt {e x}}\)	\(125\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (128 a^{2} d^{5} x^{5}+528 a b \,c^{2} d^{3} x^{5}+1155 b^{2} d \,x^{5} c^{4}-64 a^{2} c \,d^{4} x^{4}-264 a b \,c^{3} d^{2} x^{4}+1155 b^{2} c^{5} x^{4}+48 a^{2} c^{2} d^{3} x^{3}+198 a b \,c^{4} d \,x^{3}-40 a^{2} c^{3} d^{2} x^{2}+990 a b \,c^{5} x^{2}+35 a^{2} c^{4} d x +315 a^{2} c^{5}\right )}{3465 e^{6} \sqrt {e x}\, x^{5} c^{5}}\)	\(164\)

Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3465*x*(d*x+c)^(3/2)*(128*a^2*d^4*x^4+528*a*b*c^2*d^2*x^4+1155*b^2*c^4* 
x^4-192*a^2*c*d^3*x^3-792*a*b*c^3*d*x^3+240*a^2*c^2*d^2*x^2+990*a*b*c^4*x^ 
2-280*a^2*c^3*d*x+315*a^2*c^4)/c^5/(e*x)^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 \, {\left (35 \, a^{2} c^{4} d x + 315 \, a^{2} c^{5} + {\left (1155 \, b^{2} c^{4} d + 528 \, a b c^{2} d^{3} + 128 \, a^{2} d^{5}\right )} x^{5} + {\left (1155 \, b^{2} c^{5} - 264 \, a b c^{3} d^{2} - 64 \, a^{2} c d^{4}\right )} x^{4} + 6 \, {\left (33 \, a b c^{4} d + 8 \, a^{2} c^{2} d^{3}\right )} x^{3} + 10 \, {\left (99 \, a b c^{5} - 4 \, a^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{3465 \, c^{5} e^{7} x^{6}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(13/2),x, algorithm="fricas")
 

Output:

-2/3465*(35*a^2*c^4*d*x + 315*a^2*c^5 + (1155*b^2*c^4*d + 528*a*b*c^2*d^3 
+ 128*a^2*d^5)*x^5 + (1155*b^2*c^5 - 264*a*b*c^3*d^2 - 64*a^2*c*d^4)*x^4 + 
 6*(33*a*b*c^4*d + 8*a^2*c^2*d^3)*x^3 + 10*(99*a*b*c^5 - 4*a^2*c^3*d^2)*x^ 
2)*sqrt(d*x + c)*sqrt(e*x)/(c^5*e^7*x^6)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(b*x**2+a)**2/(e*x)**(13/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(13/2),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 = 0.18 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {{\left (1155 \, b^{2} c^{4} d e^{5} + 528 \, a b c^{2} d^{3} e^{5} + 128 \, a^{2} d^{5} e^{5}\right )} {\left (d x + c\right )}}{c^{5}} - \frac {44 \, {\left (105 \, b^{2} c^{5} d e^{5} + 66 \, a b c^{3} d^{3} e^{5} + 16 \, a^{2} c d^{5} e^{5}\right )}}{c^{5}}\right )} + \frac {198 \, {\left (35 \, b^{2} c^{6} d e^{5} + 33 \, a b c^{4} d^{3} e^{5} + 8 \, a^{2} c^{2} d^{5} e^{5}\right )}}{c^{5}}\right )} - \frac {924 \, {\left (5 \, b^{2} c^{7} d e^{5} + 7 \, a b c^{5} d^{3} e^{5} + 2 \, a^{2} c^{3} d^{5} e^{5}\right )}}{c^{5}}\right )} {\left (d x + c\right )} + \frac {1155 \, {\left (b^{2} c^{8} d e^{5} + 2 \, a b c^{6} d^{3} e^{5} + a^{2} c^{4} d^{5} e^{5}\right )}}{c^{5}}\right )} {\left (d x + c\right )}^{\frac {3}{2}} d^{7}}{3465 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {11}{2}} e^{6} {\left | d \right |}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(13/2),x, algorithm="giac")
 

Output:

-2/3465*(((d*x + c)*((d*x + c)*((1155*b^2*c^4*d*e^5 + 528*a*b*c^2*d^3*e^5 
+ 128*a^2*d^5*e^5)*(d*x + c)/c^5 - 44*(105*b^2*c^5*d*e^5 + 66*a*b*c^3*d^3* 
e^5 + 16*a^2*c*d^5*e^5)/c^5) + 198*(35*b^2*c^6*d*e^5 + 33*a*b*c^4*d^3*e^5 
+ 8*a^2*c^2*d^5*e^5)/c^5) - 924*(5*b^2*c^7*d*e^5 + 7*a*b*c^5*d^3*e^5 + 2*a 
^2*c^3*d^5*e^5)/c^5)*(d*x + c) + 1155*(b^2*c^8*d*e^5 + 2*a*b*c^6*d^3*e^5 + 
 a^2*c^4*d^5*e^5)/c^5)*(d*x + c)^(3/2)*d^7/(((d*x + c)*d*e - c*d*e)^(11/2) 
*e^6*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2}{11\,e^6}-\frac {x^2\,\left (80\,a^2\,c^3\,d^2-1980\,a\,b\,c^5\right )}{3465\,c^5\,e^6}-\frac {x^4\,\left (128\,a^2\,c\,d^4+528\,a\,b\,c^3\,d^2-2310\,b^2\,c^5\right )}{3465\,c^5\,e^6}+\frac {x^5\,\left (256\,a^2\,d^5+1056\,a\,b\,c^2\,d^3+2310\,b^2\,c^4\,d\right )}{3465\,c^5\,e^6}+\frac {2\,a^2\,d\,x}{99\,c\,e^6}+\frac {4\,a\,d\,x^3\,\left (33\,b\,c^2+8\,a\,d^2\right )}{1155\,c^3\,e^6}\right )}{x^5\,\sqrt {e\,x}} \] Input:

int(((a + b*x^2)^2*(c + d*x)^(1/2))/(e*x)^(13/2),x)
 

Output:

-((c + d*x)^(1/2)*((2*a^2)/(11*e^6) - (x^2*(80*a^2*c^3*d^2 - 1980*a*b*c^5) 
)/(3465*c^5*e^6) - (x^4*(128*a^2*c*d^4 - 2310*b^2*c^5 + 528*a*b*c^3*d^2))/ 
(3465*c^5*e^6) + (x^5*(256*a^2*d^5 + 2310*b^2*c^4*d + 1056*a*b*c^2*d^3))/( 
3465*c^5*e^6) + (2*a^2*d*x)/(99*c*e^6) + (4*a*d*x^3*(8*a*d^2 + 33*b*c^2))/ 
(1155*c^3*e^6)))/(x^5*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\frac {2 \sqrt {e}\, \left (-315 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5}-35 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d x +40 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{2} x^{2}-48 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{3} x^{3}+64 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{4} x^{4}-128 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{5} x^{5}-990 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} x^{2}-198 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d \,x^{3}+264 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{2} x^{4}-528 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x^{5}-1155 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} x^{4}-1155 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d \,x^{5}+128 \sqrt {d}\, a^{2} d^{5} x^{6}+528 \sqrt {d}\, a b \,c^{2} d^{3} x^{6}+525 \sqrt {d}\, b^{2} c^{4} d \,x^{6}\right )}{3465 c^{5} e^{7} x^{6}} \] Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(13/2),x)
 

Output:

(2*sqrt(e)*( - 315*sqrt(x)*sqrt(c + d*x)*a**2*c**5 - 35*sqrt(x)*sqrt(c + d 
*x)*a**2*c**4*d*x + 40*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**2*x**2 - 48*sqrt 
(x)*sqrt(c + d*x)*a**2*c**2*d**3*x**3 + 64*sqrt(x)*sqrt(c + d*x)*a**2*c*d* 
*4*x**4 - 128*sqrt(x)*sqrt(c + d*x)*a**2*d**5*x**5 - 990*sqrt(x)*sqrt(c + 
d*x)*a*b*c**5*x**2 - 198*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d*x**3 + 264*sqrt( 
x)*sqrt(c + d*x)*a*b*c**3*d**2*x**4 - 528*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d 
**3*x**5 - 1155*sqrt(x)*sqrt(c + d*x)*b**2*c**5*x**4 - 1155*sqrt(x)*sqrt(c 
 + d*x)*b**2*c**4*d*x**5 + 128*sqrt(d)*a**2*d**5*x**6 + 528*sqrt(d)*a*b*c* 
*2*d**3*x**6 + 525*sqrt(d)*b**2*c**4*d*x**6))/(3465*c**5*e**7*x**6)