\(\int \frac {(a+b x^2)^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx\) [832]

Optimal result

Integrand size = 26, antiderivative size = 248 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}+\frac {20 a^2 d \sqrt {c+d x}}{99 c^2 e^2 (e x)^{9/2}}-\frac {4 a \left (99 b c^2+40 a d^2\right ) \sqrt {c+d x}}{693 c^3 e^3 (e x)^{7/2}}+\frac {8 a d \left (99 b c^2+40 a d^2\right ) \sqrt {c+d x}}{1155 c^4 e^4 (e x)^{5/2}}-\frac {2 \left (1155 b^2 c^4+1584 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {c+d x}}{3465 c^5 e^5 (e x)^{3/2}}+\frac {4 d \left (1155 b^2 c^4+1584 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {c+d x}}{3465 c^6 e^6 \sqrt {e x}} \] Output:

-2/11*a^2*(d*x+c)^(1/2)/c/e/(e*x)^(11/2)+20/99*a^2*d*(d*x+c)^(1/2)/c^2/e^2 
/(e*x)^(9/2)-4/693*a*(40*a*d^2+99*b*c^2)*(d*x+c)^(1/2)/c^3/e^3/(e*x)^(7/2) 
+8/1155*a*d*(40*a*d^2+99*b*c^2)*(d*x+c)^(1/2)/c^4/e^4/(e*x)^(5/2)-2/3465*( 
640*a^2*d^4+1584*a*b*c^2*d^2+1155*b^2*c^4)*(d*x+c)^(1/2)/c^5/e^5/(e*x)^(3/ 
2)+4/3465*d*(640*a^2*d^4+1584*a*b*c^2*d^2+1155*b^2*c^4)*(d*x+c)^(1/2)/c^6/ 
e^6/(e*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {e x} \sqrt {c+d x} \left (-1155 b^2 c^4 x^4 (c-2 d x)+198 a b c^2 x^2 \left (-5 c^3+6 c^2 d x-8 c d^2 x^2+16 d^3 x^3\right )-5 a^2 \left (63 c^5-70 c^4 d x+80 c^3 d^2 x^2-96 c^2 d^3 x^3+128 c d^4 x^4-256 d^5 x^5\right )\right )}{3465 c^6 e^7 x^6} \] Input:

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

Output:

(2*Sqrt[e*x]*Sqrt[c + d*x]*(-1155*b^2*c^4*x^4*(c - 2*d*x) + 198*a*b*c^2*x^ 
2*(-5*c^3 + 6*c^2*d*x - 8*c*d^2*x^2 + 16*d^3*x^3) - 5*a^2*(63*c^5 - 70*c^4 
*d*x + 80*c^3*d^2*x^2 - 96*c^2*d^3*x^3 + 128*c*d^4*x^4 - 256*d^5*x^5)))/(3 
465*c^6*e^7*x^6)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {520, 27, 2124, 27, 520, 27, 87, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {-11 b^2 c x^3-22 a b c x+10 a^2 d}{2 (e x)^{11/2} \sqrt {c+d x}}dx}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-11 b^2 c x^3-22 a b c x+10 a^2 d}{(e x)^{11/2} \sqrt {c+d x}}dx}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {99 b^2 c^2 x^2+2 a \left (99 b c^2+40 a d^2\right )}{2 (e x)^{9/2} \sqrt {c+d x}}dx}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {99 b^2 c^2 x^2+2 a \left (99 b c^2+40 a d^2\right )}{(e x)^{9/2} \sqrt {c+d x}}dx}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {3 \left (4 a d \left (99 b c^2+40 a d^2\right )-231 b^2 c^3 x\right )}{2 (e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {4 a \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {3 \int \frac {4 a d \left (99 b c^2+40 a d^2\right )-231 b^2 c^3 x}{(e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {4 a \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {3 \left (-\frac {\left (640 a^2 d^4+1584 a b c^2 d^2+1155 b^2 c^4\right ) \int \frac {1}{(e x)^{5/2} \sqrt {c+d x}}dx}{5 c e}-\frac {8 a d \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {4 a \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {-\frac {3 \left (-\frac {\left (640 a^2 d^4+1584 a b c^2 d^2+1155 b^2 c^4\right ) \left (-\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{3 c e}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {8 a d \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {4 a \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {-\frac {3 \left (-\frac {\left (640 a^2 d^4+1584 a b c^2 d^2+1155 b^2 c^4\right ) \left (\frac {4 d \sqrt {c+d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {8 a d \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {4 a \sqrt {c+d x} \left (40 a d^2+99 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {20 a^2 d \sqrt {c+d x}}{9 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 \sqrt {c+d x}}{11 c e (e x)^{11/2}}\)

Input:

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

Output:

(-2*a^2*Sqrt[c + d*x])/(11*c*e*(e*x)^(11/2)) - ((-20*a^2*d*Sqrt[c + d*x])/ 
(9*c*e*(e*x)^(9/2)) - ((-4*a*(99*b*c^2 + 40*a*d^2)*Sqrt[c + d*x])/(7*c*e*( 
e*x)^(7/2)) - (3*((-8*a*d*(99*b*c^2 + 40*a*d^2)*Sqrt[c + d*x])/(5*c*e*(e*x 
)^(5/2)) - ((1155*b^2*c^4 + 1584*a*b*c^2*d^2 + 640*a^2*d^4)*((-2*Sqrt[c + 
d*x])/(3*c*e*(e*x)^(3/2)) + (4*d*Sqrt[c + d*x])/(3*c^2*e^2*Sqrt[e*x])))/(5 
*c*e)))/(7*c*e))/(9*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 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]))))
 

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 = 159, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {2 x \sqrt {d x +c}\, \left (-1280 a^{2} d^{5} x^{5}-3168 a b \,c^{2} d^{3} x^{5}-2310 b^{2} d \,x^{5} c^{4}+640 a^{2} c \,d^{4} x^{4}+1584 a b \,c^{3} d^{2} x^{4}+1155 b^{2} c^{5} x^{4}-480 a^{2} c^{2} d^{3} x^{3}-1188 a b \,c^{4} d \,x^{3}+400 a^{2} c^{3} d^{2} x^{2}+990 a b \,c^{5} x^{2}-350 a^{2} c^{4} d x +315 a^{2} c^{5}\right )}{3465 c^{6} \left (e x \right )^{\frac {13}{2}}}\)	\(159\)
orering	\(-\frac {2 x \sqrt {d x +c}\, \left (-1280 a^{2} d^{5} x^{5}-3168 a b \,c^{2} d^{3} x^{5}-2310 b^{2} d \,x^{5} c^{4}+640 a^{2} c \,d^{4} x^{4}+1584 a b \,c^{3} d^{2} x^{4}+1155 b^{2} c^{5} x^{4}-480 a^{2} c^{2} d^{3} x^{3}-1188 a b \,c^{4} d \,x^{3}+400 a^{2} c^{3} d^{2} x^{2}+990 a b \,c^{5} x^{2}-350 a^{2} c^{4} d x +315 a^{2} c^{5}\right )}{3465 c^{6} \left (e x \right )^{\frac {13}{2}}}\)	\(159\)
default	\(-\frac {2 \sqrt {d x +c}\, \left (-1280 a^{2} d^{5} x^{5}-3168 a b \,c^{2} d^{3} x^{5}-2310 b^{2} d \,x^{5} c^{4}+640 a^{2} c \,d^{4} x^{4}+1584 a b \,c^{3} d^{2} x^{4}+1155 b^{2} c^{5} x^{4}-480 a^{2} c^{2} d^{3} x^{3}-1188 a b \,c^{4} d \,x^{3}+400 a^{2} c^{3} d^{2} x^{2}+990 a b \,c^{5} x^{2}-350 a^{2} c^{4} d x +315 a^{2} c^{5}\right )}{3465 x^{5} c^{6} e^{6} \sqrt {e x}}\)	\(164\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-1280 a^{2} d^{5} x^{5}-3168 a b \,c^{2} d^{3} x^{5}-2310 b^{2} d \,x^{5} c^{4}+640 a^{2} c \,d^{4} x^{4}+1584 a b \,c^{3} d^{2} x^{4}+1155 b^{2} c^{5} x^{4}-480 a^{2} c^{2} d^{3} x^{3}-1188 a b \,c^{4} d \,x^{3}+400 a^{2} c^{3} d^{2} x^{2}+990 a b \,c^{5} x^{2}-350 a^{2} c^{4} d x +315 a^{2} c^{5}\right )}{3465 x^{5} c^{6} e^{6} \sqrt {e x}}\)	\(164\)

Input:

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

Output:

-2/3465*x*(d*x+c)^(1/2)*(-1280*a^2*d^5*x^5-3168*a*b*c^2*d^3*x^5-2310*b^2*c 
^4*d*x^5+640*a^2*c*d^4*x^4+1584*a*b*c^3*d^2*x^4+1155*b^2*c^5*x^4-480*a^2*c 
^2*d^3*x^3-1188*a*b*c^4*d*x^3+400*a^2*c^3*d^2*x^2+990*a*b*c^5*x^2-350*a^2* 
c^4*d*x+315*a^2*c^5)/c^6/(e*x)^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (350 \, a^{2} c^{4} d x - 315 \, a^{2} c^{5} + 2 \, {\left (1155 \, b^{2} c^{4} d + 1584 \, a b c^{2} d^{3} + 640 \, a^{2} d^{5}\right )} x^{5} - {\left (1155 \, b^{2} c^{5} + 1584 \, a b c^{3} d^{2} + 640 \, a^{2} c d^{4}\right )} x^{4} + 12 \, {\left (99 \, a b c^{4} d + 40 \, a^{2} c^{2} d^{3}\right )} x^{3} - 10 \, {\left (99 \, a b c^{5} + 40 \, a^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{3465 \, c^{6} e^{7} x^{6}} \] Input:

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

Output:

2/3465*(350*a^2*c^4*d*x - 315*a^2*c^5 + 2*(1155*b^2*c^4*d + 1584*a*b*c^2*d 
^3 + 640*a^2*d^5)*x^5 - (1155*b^2*c^5 + 1584*a*b*c^3*d^2 + 640*a^2*c*d^4)* 
x^4 + 12*(99*a*b*c^4*d + 40*a^2*c^2*d^3)*x^3 - 10*(99*a*b*c^5 + 40*a^2*c^3 
*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x)/(c^6*e^7*x^6)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(1/2),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 = 0.16 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {2 \, {\left (1155 \, b^{2} c^{4} d e^{5} + 1584 \, a b c^{2} d^{3} e^{5} + 640 \, a^{2} d^{5} e^{5}\right )} {\left (d x + c\right )}}{c^{6}} - \frac {11 \, {\left (1155 \, b^{2} c^{5} d e^{5} + 1584 \, a b c^{3} d^{3} e^{5} + 640 \, a^{2} c d^{5} e^{5}\right )}}{c^{6}}\right )} + \frac {396 \, {\left (70 \, b^{2} c^{6} d e^{5} + 99 \, a b c^{4} d^{3} e^{5} + 40 \, a^{2} c^{2} d^{5} e^{5}\right )}}{c^{6}}\right )} - \frac {462 \, {\left (65 \, b^{2} c^{7} d e^{5} + 99 \, a b c^{5} d^{3} e^{5} + 40 \, a^{2} c^{3} d^{5} e^{5}\right )}}{c^{6}}\right )} {\left (d x + c\right )} + \frac {2310 \, {\left (7 \, b^{2} c^{8} d e^{5} + 12 \, a b c^{6} d^{3} e^{5} + 5 \, a^{2} c^{4} d^{5} e^{5}\right )}}{c^{6}}\right )} {\left (d x + c\right )} - \frac {3465 \, {\left (b^{2} c^{9} d e^{5} + 2 \, a b c^{7} d^{3} e^{5} + a^{2} c^{5} d^{5} e^{5}\right )}}{c^{6}}\right )} \sqrt {d x + c} d^{7}}{3465 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {11}{2}} e^{6} {\left | d \right |}} \] Input:

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

Output:

2/3465*((((d*x + c)*((d*x + c)*(2*(1155*b^2*c^4*d*e^5 + 1584*a*b*c^2*d^3*e 
^5 + 640*a^2*d^5*e^5)*(d*x + c)/c^6 - 11*(1155*b^2*c^5*d*e^5 + 1584*a*b*c^ 
3*d^3*e^5 + 640*a^2*c*d^5*e^5)/c^6) + 396*(70*b^2*c^6*d*e^5 + 99*a*b*c^4*d 
^3*e^5 + 40*a^2*c^2*d^5*e^5)/c^6) - 462*(65*b^2*c^7*d*e^5 + 99*a*b*c^5*d^3 
*e^5 + 40*a^2*c^3*d^5*e^5)/c^6)*(d*x + c) + 2310*(7*b^2*c^8*d*e^5 + 12*a*b 
*c^6*d^3*e^5 + 5*a^2*c^4*d^5*e^5)/c^6)*(d*x + c) - 3465*(b^2*c^9*d*e^5 + 2 
*a*b*c^7*d^3*e^5 + a^2*c^5*d^5*e^5)/c^6)*sqrt(d*x + c)*d^7/(((d*x + c)*d*e 
 - c*d*e)^(11/2)*e^6*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.66 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2}{11\,c\,e^6}+\frac {x^2\,\left (800\,a^2\,c^3\,d^2+1980\,b\,a\,c^5\right )}{3465\,c^6\,e^6}+\frac {x^4\,\left (1280\,a^2\,c\,d^4+3168\,a\,b\,c^3\,d^2+2310\,b^2\,c^5\right )}{3465\,c^6\,e^6}-\frac {x^5\,\left (2560\,a^2\,d^5+6336\,a\,b\,c^2\,d^3+4620\,b^2\,c^4\,d\right )}{3465\,c^6\,e^6}-\frac {20\,a^2\,d\,x}{99\,c^2\,e^6}-\frac {8\,a\,d\,x^3\,\left (99\,b\,c^2+40\,a\,d^2\right )}{1155\,c^4\,e^6}\right )}{x^5\,\sqrt {e\,x}} \] Input:

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

Output:

-((c + d*x)^(1/2)*((2*a^2)/(11*c*e^6) + (x^2*(800*a^2*c^3*d^2 + 1980*a*b*c 
^5))/(3465*c^6*e^6) + (x^4*(2310*b^2*c^5 + 1280*a^2*c*d^4 + 3168*a*b*c^3*d 
^2))/(3465*c^6*e^6) - (x^5*(2560*a^2*d^5 + 4620*b^2*c^4*d + 6336*a*b*c^2*d 
^3))/(3465*c^6*e^6) - (20*a^2*d*x)/(99*c^2*e^6) - (8*a*d*x^3*(40*a*d^2 + 9 
9*b*c^2))/(1155*c^4*e^6)))/(x^5*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {e}\, \left (-315 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5}+350 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d x -400 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{2} x^{2}+480 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{3} x^{3}-640 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{4} x^{4}+1280 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{5} x^{5}-990 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} x^{2}+1188 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d \,x^{3}-1584 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{2} x^{4}+3168 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x^{5}-1155 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} x^{4}+2310 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d \,x^{5}-1280 \sqrt {d}\, a^{2} d^{5} x^{6}-3168 \sqrt {d}\, a b \,c^{2} d^{3} x^{6}-2310 \sqrt {d}\, b^{2} c^{4} d \,x^{6}\right )}{3465 c^{6} e^{7} x^{6}} \] Input:

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

Output:

(2*sqrt(e)*( - 315*sqrt(x)*sqrt(c + d*x)*a**2*c**5 + 350*sqrt(x)*sqrt(c + 
d*x)*a**2*c**4*d*x - 400*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**2*x**2 + 480*s 
qrt(x)*sqrt(c + d*x)*a**2*c**2*d**3*x**3 - 640*sqrt(x)*sqrt(c + d*x)*a**2* 
c*d**4*x**4 + 1280*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 + 1188*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d*x**3 - 158 
4*sqrt(x)*sqrt(c + d*x)*a*b*c**3*d**2*x**4 + 3168*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 + 2310*sqrt(x 
)*sqrt(c + d*x)*b**2*c**4*d*x**5 - 1280*sqrt(d)*a**2*d**5*x**6 - 3168*sqrt 
(d)*a*b*c**2*d**3*x**6 - 2310*sqrt(d)*b**2*c**4*d*x**6))/(3465*c**6*e**7*x 
**6)