\(\int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x^2)^2} \, dx\) [900]

Optimal result

Integrand size = 26, antiderivative size = 262 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} \sqrt {c+d x}}{2 a e \left (a+b x^2\right )}+\frac {\left (3 \sqrt {b} c-2 \sqrt {-a} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {b} \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}+\frac {\left (3 \sqrt {b} c+2 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {b} \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}} \] Output:

1/2*(e*x)^(1/2)*(d*x+c)^(1/2)/a/e/(b*x^2+a)+1/4*(3*b^(1/2)*c-2*(-a)^(1/2)* 
d)*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d 
*x+c)^(1/2))/(-a)^(7/4)/b^(1/2)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e^(1/2)+1/4 
*(3*b^(1/2)*c+2*(-a)^(1/2)*d)*arctanh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x) 
^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(7/4)/b^(1/2)/(b^(1/2)*c+(-a 
)^(1/2)*d)^(1/2)/e^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.38 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {x} \left (b \sqrt {x} \sqrt {c+d x}+32 a d^{7/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]+d^{3/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-32 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+4 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{2 a b \sqrt {e x} \left (a+b x^2\right )} \] Input:

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

Output:

(Sqrt[x]*(b*Sqrt[x]*Sqrt[c + d*x] + 32*a*d^(7/2)*(a + b*x^2)*RootSum[b*c^4 
 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , Log 
[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]/(b*c^3 - 3*b*c^2*#1 - 8 
*a*d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ] + d^(3/2)*(a + b*x^2)*RootSum[b*c^4 - 
 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , (b*c^ 
2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 32*a*d^2*Log[c + 
 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 4*b*c*Log[c + 2*d*x - 2*S 
qrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x 
]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^3 - 3*b*c^2*#1 - 8*a*d^2*#1 + 3*b*c*#1^2 
- b*#1^3) & ]))/(2*a*b*Sqrt[e*x]*(a + b*x^2))
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b \sqrt {c+d x}}{2 a \sqrt {e x} \left (-a b-b^2 x^2\right )}-\frac {b \sqrt {c+d x}}{4 a \sqrt {e x} \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sqrt {c+d x}}{4 a \sqrt {e x} \left (\sqrt {-a} \sqrt {b}+b x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{7/4} \sqrt {b} \sqrt {e}}+\frac {c \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}+\frac {\sqrt {\sqrt {-a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{7/4} \sqrt {b} \sqrt {e}}+\frac {c \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}+\frac {\sqrt {e x} \sqrt {c+d x}}{4 (-a)^{3/2} e \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {e x} \sqrt {c+d x}}{4 (-a)^{3/2} e \left (\sqrt {-a}+\sqrt {b} x\right )}\)

Input:

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

Output:

(Sqrt[e*x]*Sqrt[c + d*x])/(4*(-a)^(3/2)*e*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[ 
e*x]*Sqrt[c + d*x])/(4*(-a)^(3/2)*e*(Sqrt[-a] + Sqrt[b]*x)) + (c*ArcTan[(S 
qrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])] 
)/(4*(-a)^(7/4)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) + (Sqrt[Sqrt[b]*c - 
Sqrt[-a]*d]*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sq 
rt[e]*Sqrt[c + d*x])])/(2*(-a)^(7/4)*Sqrt[b]*Sqrt[e]) + (c*ArcTanh[(Sqrt[S 
qrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4* 
(-a)^(7/4)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e]) + (Sqrt[Sqrt[b]*c + Sqrt[ 
-a]*d]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e 
]*Sqrt[c + d*x])])/(2*(-a)^(7/4)*Sqrt[b]*Sqrt[e])
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1992\) vs. \(2(192)=384\).

Time = 0.52 (sec) , antiderivative size = 1993, normalized size of antiderivative = 7.61

method	result	size
default	\(\text {Expression too large to display}\)	\(1993\)

Input:

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

Output:

1/8*(d*x+c)^(1/2)*x*(3*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^( 
1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/ 
2)))*a*b^2*c*d^2*e*x^2*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)-2*ln((-2*(-a*b)^( 
1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2) 
*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*a*b*d^3*e*x^2*(-a*b)^(1/2)*(e*(-a 
*d+c*(-a*b)^(1/2))/b)^(1/2)+3*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c) 
*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a 
*b)^(1/2)))*b^3*c^3*e*x^2*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)-2*ln((-2*(-a*b 
)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1 
/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*b^2*c^2*d*e*x^2*(-a*b)^(1/2)*( 
e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)-3*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d* 
x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x 
-(-a*b)^(1/2)))*a*b^2*c*d^2*e*x^2*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)-2*ln(( 
2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2) 
)/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*a*b*d^3*e*x^2*(-a*b)^(1 
/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)-3*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2 
*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2)) 
/(b*x-(-a*b)^(1/2)))*b^3*c^3*e*x^2*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)-2*ln( 
(2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2 
))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*b^2*c^2*d*e*x^2*(-a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1934 vs. \(2 (192) = 384\).

Time = 0.21 (sec) , antiderivative size = 1934, normalized size of antiderivative = 7.38 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*b*e*x^2 + a^2*e)*sqrt(-(3*b*c^2*d + 4*a*d^3 + (a^3*b^2*c^2 + a^4*b 
*d^2)*e*sqrt(-(81*b^2*c^6 + 144*a*b*c^4*d^2 + 64*a^2*c^2*d^4)/((a^7*b^3*c^ 
4 + 2*a^8*b^2*c^2*d^2 + a^9*b*d^4)*e^2)))/((a^3*b^2*c^2 + a^4*b*d^2)*e))*l 
og(((81*b^2*c^5 + 108*a*b*c^3*d^2 + 32*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) 
+ ((3*a^5*b^3*c^4 + 5*a^6*b^2*c^2*d^2 + 2*a^7*b*d^4)*e^2*x*sqrt(-(81*b^2*c 
^6 + 144*a*b*c^4*d^2 + 64*a^2*c^2*d^4)/((a^7*b^3*c^4 + 2*a^8*b^2*c^2*d^2 + 
 a^9*b*d^4)*e^2)) + (9*a^2*b^2*c^4*d + 8*a^3*b*c^2*d^3)*e*x)*sqrt(-(3*b*c^ 
2*d + 4*a*d^3 + (a^3*b^2*c^2 + a^4*b*d^2)*e*sqrt(-(81*b^2*c^6 + 144*a*b*c^ 
4*d^2 + 64*a^2*c^2*d^4)/((a^7*b^3*c^4 + 2*a^8*b^2*c^2*d^2 + a^9*b*d^4)*e^2 
)))/((a^3*b^2*c^2 + a^4*b*d^2)*e)))/x) - (a*b*e*x^2 + a^2*e)*sqrt(-(3*b*c^ 
2*d + 4*a*d^3 + (a^3*b^2*c^2 + a^4*b*d^2)*e*sqrt(-(81*b^2*c^6 + 144*a*b*c^ 
4*d^2 + 64*a^2*c^2*d^4)/((a^7*b^3*c^4 + 2*a^8*b^2*c^2*d^2 + a^9*b*d^4)*e^2 
)))/((a^3*b^2*c^2 + a^4*b*d^2)*e))*log(((81*b^2*c^5 + 108*a*b*c^3*d^2 + 32 
*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) - ((3*a^5*b^3*c^4 + 5*a^6*b^2*c^2*d^2 
+ 2*a^7*b*d^4)*e^2*x*sqrt(-(81*b^2*c^6 + 144*a*b*c^4*d^2 + 64*a^2*c^2*d^4) 
/((a^7*b^3*c^4 + 2*a^8*b^2*c^2*d^2 + a^9*b*d^4)*e^2)) + (9*a^2*b^2*c^4*d + 
 8*a^3*b*c^2*d^3)*e*x)*sqrt(-(3*b*c^2*d + 4*a*d^3 + (a^3*b^2*c^2 + a^4*b*d 
^2)*e*sqrt(-(81*b^2*c^6 + 144*a*b*c^4*d^2 + 64*a^2*c^2*d^4)/((a^7*b^3*c^4 
+ 2*a^8*b^2*c^2*d^2 + a^9*b*d^4)*e^2)))/((a^3*b^2*c^2 + a^4*b*d^2)*e)))/x) 
 - (a*b*e*x^2 + a^2*e)*sqrt(-(3*b*c^2*d + 4*a*d^3 - (a^3*b^2*c^2 + a^4*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(sqrt(d*x + c)/((b*x^2 + a)^2*sqrt(e*x)), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 42.42 (sec) , antiderivative size = 11288, normalized size of antiderivative = 43.08 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((((-(4*a^5*b*d^3 - 9*b*c^3*(-a^7*b^3)^(1/2) + 3*a^4*b^2*c^2*d - 8*a 
*c*d^2*(-a^7*b^3)^(1/2))/(64*(a^7*b^3*c^2*e + a^8*b^2*d^2*e)))^(1/2)*((163 
84*(344064*a^7*b^5*c^12*d^17*e^15 + 728064*a^8*b^4*c^10*d^19*e^15 + 372736 
*a^9*b^3*c^8*d^21*e^15 - 4096*a^10*b^2*c^6*d^23*e^15))/a^5 + (((16384*(419 
4304*a^10*b^6*c^12*d^16*e^16 + 7340032*a^11*b^5*c^10*d^18*e^16 + 3211264*a 
^12*b^4*c^8*d^20*e^16))/a^5 - (16384*e*x*(8388608*a^10*b^6*c^12*d^17*e^15 
+ 16252928*a^11*b^5*c^10*d^19*e^15 + 7798784*a^12*b^4*c^8*d^21*e^15))/(a^5 
*((c + d*x)^(1/2) - c^(1/2))^2))*(-(4*a^5*b*d^3 - 9*b*c^3*(-a^7*b^3)^(1/2) 
 + 3*a^4*b^2*c^2*d - 8*a*c*d^2*(-a^7*b^3)^(1/2))/(64*(a^7*b^3*c^2*e + a^8* 
b^2*d^2*e)))^(1/2) - (131072*(e*x)^(1/2)*(393216*a^7*b^6*c^13*d^16*e^15 + 
663552*a^8*b^5*c^11*d^18*e^15 + 292864*a^9*b^4*c^9*d^20*e^15 + 20480*a^10* 
b^3*c^7*d^22*e^15))/(a^4*((c + d*x)^(1/2) - c^(1/2))))*(-(4*a^5*b*d^3 - 9* 
b*c^3*(-a^7*b^3)^(1/2) + 3*a^4*b^2*c^2*d - 8*a*c*d^2*(-a^7*b^3)^(1/2))/(64 
*(a^7*b^3*c^2*e + a^8*b^2*d^2*e)))^(1/2) - (16384*e*x*(770048*a^7*b^5*c^12 
*d^18*e^14 + 1762304*a^8*b^4*c^10*d^20*e^14 + 946176*a^9*b^3*c^8*d^22*e^14 
 - 4096*a^10*b^2*c^6*d^24*e^14))/(a^5*((c + d*x)^(1/2) - c^(1/2))^2)) - (1 
31072*(e*x)^(1/2)*(36864*a^4*b^5*c^13*d^17*e^14 + 73632*a^5*b^4*c^11*d^19* 
e^14 + 35712*a^6*b^3*c^9*d^21*e^14 + 2560*a^7*b^2*c^7*d^23*e^14))/(a^4*((c 
 + d*x)^(1/2) - c^(1/2))))*(-(4*a^5*b*d^3 - 9*b*c^3*(-a^7*b^3)^(1/2) + 3*a 
^4*b^2*c^2*d - 8*a*c*d^2*(-a^7*b^3)^(1/2))/(64*(a^7*b^3*c^2*e + a^8*b^2...
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a^{2}+2 \sqrt {x}\, a b \,x^{2}+\sqrt {x}\, b^{2} x^{4}}d x}{\sqrt {e}} \] Input:

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

Output:

int(sqrt(c + d*x)/(sqrt(x)*a**2 + 2*sqrt(x)*a*b*x**2 + sqrt(x)*b**2*x**4), 
x)/sqrt(e)