\(\int \frac {(e x)^{3/2} \sqrt {c+d x}}{(a+b x^2)^2} \, dx\) [898]

Optimal result

Integrand size = 26, antiderivative size = 258 \[ \int \frac {(e x)^{3/2} \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=-\frac {e \sqrt {e x} \sqrt {c+d x}}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {b} c-2 \sqrt {-a} d\right ) e^{3/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{3/4} b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (\sqrt {b} c+2 \sqrt {-a} d\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{3/4} b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.49 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.17 \[ \int \frac {(e x)^{3/2} \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \left (-b \sqrt {x} \sqrt {c+d x}+2 d^{3/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-16 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+2 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]-d^{3/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-32 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+4 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{2 b^2 x^{3/2} \left (a+b x^2\right )} \] Input:

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

Output:

((e*x)^(3/2)*(-(b*Sqrt[x]*Sqrt[c + d*x]) + 2*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] - 16*a*d^2*L 
og[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 2*b*c*Log[c + 2*d*x 
 - 2*Sqrt[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) & ] - 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*Sqrt[d]*Sqrt[x]*Sq 
rt[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*b^2*x^(3/2)*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(718\) vs. \(2(258)=516\).

Time = 2.18 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.78, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(1991\) vs. \(2(188)=376\).

Time = 0.62 (sec) , antiderivative size = 1992, normalized size of antiderivative = 7.72

method	result	size
default	\(\text {Expression too large to display}\)	\(1992\)

Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1525 vs. \(2 (188) = 376\).

Time = 0.12 (sec) , antiderivative size = 1525, normalized size of antiderivative = 5.91 \[ \int \frac {(e x)^{3/2} \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

-1/8*((b^2*x^2 + a*b)*sqrt(-((3*b*c^2*d + 4*a*d^3)*e^3 + sqrt(-c^6*e^6/(a^ 
3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*(a*b^4*c^2 + a^2*b^3*d^2))/( 
a*b^4*c^2 + a^2*b^3*d^2))*log(((b*c^5 + 4*a*c^3*d^2)*sqrt(d*x + c)*sqrt(e* 
x)*e^4 + (a*b^2*c^4*d*e^3*x - (a^2*b^5*c^4 + 3*a^3*b^4*c^2*d^2 + 2*a^4*b^3 
*d^4)*sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*x)*sq 
rt(-((3*b*c^2*d + 4*a*d^3)*e^3 + sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4*b^4*c^ 
2*d^2 + a^5*b^3*d^4))*(a*b^4*c^2 + a^2*b^3*d^2))/(a*b^4*c^2 + a^2*b^3*d^2) 
))/x) - (b^2*x^2 + a*b)*sqrt(-((3*b*c^2*d + 4*a*d^3)*e^3 + sqrt(-c^6*e^6/( 
a^3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*(a*b^4*c^2 + a^2*b^3*d^2)) 
/(a*b^4*c^2 + a^2*b^3*d^2))*log(((b*c^5 + 4*a*c^3*d^2)*sqrt(d*x + c)*sqrt( 
e*x)*e^4 - (a*b^2*c^4*d*e^3*x - (a^2*b^5*c^4 + 3*a^3*b^4*c^2*d^2 + 2*a^4*b 
^3*d^4)*sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*x)* 
sqrt(-((3*b*c^2*d + 4*a*d^3)*e^3 + sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4*b^4* 
c^2*d^2 + a^5*b^3*d^4))*(a*b^4*c^2 + a^2*b^3*d^2))/(a*b^4*c^2 + a^2*b^3*d^ 
2)))/x) + (b^2*x^2 + a*b)*sqrt(-((3*b*c^2*d + 4*a*d^3)*e^3 - sqrt(-c^6*e^6 
/(a^3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*(a*b^4*c^2 + a^2*b^3*d^2 
))/(a*b^4*c^2 + a^2*b^3*d^2))*log(((b*c^5 + 4*a*c^3*d^2)*sqrt(d*x + c)*sqr 
t(e*x)*e^4 + (a*b^2*c^4*d*e^3*x + (a^2*b^5*c^4 + 3*a^3*b^4*c^2*d^2 + 2*a^4 
*b^3*d^4)*sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4))*x 
)*sqrt(-((3*b*c^2*d + 4*a*d^3)*e^3 - sqrt(-c^6*e^6/(a^3*b^5*c^4 + 2*a^4...
 

Sympy [F(-1)]

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

integrate((e*x)**(3/2)*(d*x+c)**(1/2)/(b*x**2+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{3/2} \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x + c} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x + c)*(e*x)^(3/2)/(b*x^2 + a)^2, x)
 

Giac [F(-1)]

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

integrate((e*x)^(3/2)*(d*x+c)^(1/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(((e*x)^(3/2)*(c + d*x)^(1/2))/(a + b*x^2)^2,x)
 

Output:

int(((e*x)^(3/2)*(c + d*x)^(1/2))/(a + b*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {(e x)^{3/2} \sqrt {c+d x}}{\left (a+b x^2\right )^2} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, x}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) e \] Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)/(b*x^2+a)^2,x)
 

Output:

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