\(\int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} (a+b x^2)^2} \, dx\) [939]

Optimal result

Integrand size = 26, antiderivative size = 604 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 b c^2+4 a d^2}{2 a^2 c \left (b c^2+a d^2\right ) e \sqrt {e x} (c+d x)^{3/2}}-\frac {d \left (15 b^2 c^4+21 a b c^2 d^2+16 a^2 d^4\right ) \sqrt {e x}}{6 a^2 c^2 \left (b c^2+a d^2\right )^2 e^2 (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^6+27 a b^2 c^4 d^2+104 a^2 b c^2 d^4+32 a^3 d^6\right ) \sqrt {e x}}{6 a^2 c^3 \left (b c^2+a d^2\right )^3 e^2 \sqrt {c+d x}}+\frac {b (c-d x)}{2 a \left (b c^2+a d^2\right ) e \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )}-\frac {5 b^{3/2} \left (b^2 c^4+\sqrt {-a} b^{3/2} c^3 d+3 a b c^2 d^2+5 \sqrt {-a} a \sqrt {b} c d^3-2 a^2 d^4\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )^3 e^{3/2}}+\frac {5 b^{3/2} \left (b^2 c^4-\sqrt {-a} b^{3/2} c^3 d+3 a b c^2 d^2-5 \sqrt {-a} a \sqrt {b} c d^3-2 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{9/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )^3 e^{3/2}} \] Output:

-1/2*(4*a*d^2+5*b*c^2)/a^2/c/(a*d^2+b*c^2)/e/(e*x)^(1/2)/(d*x+c)^(3/2)-1/6 
*d*(16*a^2*d^4+21*a*b*c^2*d^2+15*b^2*c^4)*(e*x)^(1/2)/a^2/c^2/(a*d^2+b*c^2 
)^2/e^2/(d*x+c)^(3/2)-1/6*d*(32*a^3*d^6+104*a^2*b*c^2*d^4+27*a*b^2*c^4*d^2 
+15*b^3*c^6)*(e*x)^(1/2)/a^2/c^3/(a*d^2+b*c^2)^3/e^2/(d*x+c)^(1/2)+1/2*b*( 
-d*x+c)/a/(a*d^2+b*c^2)/e/(e*x)^(1/2)/(d*x+c)^(3/2)/(b*x^2+a)-5/4*b^(3/2)* 
(b^2*c^4+(-a)^(1/2)*b^(3/2)*c^3*d+3*a*b*c^2*d^2+5*(-a)^(1/2)*a*b^(1/2)*c*d 
^3-2*a^2*d^4)*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)^(9/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/(a*d^2+b 
*c^2)^3/e^(3/2)+5/4*b^(3/2)*(b^2*c^4-(-a)^(1/2)*b^(3/2)*c^3*d+3*a*b*c^2*d^ 
2-5*(-a)^(1/2)*a*b^(1/2)*c*d^3-2*a^2*d^4)*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)^(9/4)/(b^(1/2)*c 
+(-a)^(1/2)*d)^(1/2)/(a*d^2+b*c^2)^3/e^(3/2)
 

Mathematica [C] (verified)

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

Time = 2.38 (sec) , antiderivative size = 1317, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[1/((e*x)^(3/2)*(c + d*x)^(5/2)*(a + b*x^2)^2),x]
 

Output:

(x*((-2*(15*b^4*c^6*x^2*(c + d*x)^2 + 4*a^4*d^6*(3*c^2 + 12*c*d*x + 8*d^2* 
x^2) + 3*a*b^3*c^4*(c + d*x)^2*(4*c^2 + 3*c*d*x + 9*d^2*x^2) + 4*a^3*b*d^4 
*(9*c^4 + 36*c^3*d*x + 29*c^2*d^2*x^2 + 12*c*d^3*x^3 + 8*d^4*x^4) + a^2*b^ 
2*c^2*d^2*(36*c^4 + 69*c^3*d*x + 66*c^2*d^2*x^2 + 141*c*d^3*x^3 + 104*d^4* 
x^4)))/(c^3*(c + d*x)^(3/2)*(a + b*x^2)) - 12*b*Sqrt[d]*Sqrt[x]*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^3*c^6*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 19*a*b^2* 
c^4*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*a^2*b*c^ 
2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 16*a^3*d^6*L 
og[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b^3*c^5*Log[c + 2 
*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 6*a*b^2*c^3*d^2*Log[c + 
2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 20*a^2*b*c*d^4*Log[c + 
2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b^3*c^4*Log[c + 2*d*x - 
 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 + 3*a*b^2*c^2*d^2*Log[c + 2*d* 
x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 - 2*a^2*b*d^4*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) & ] - 3*b*Sqrt[d]*Sqrt[x]*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^3*c^6* 
Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 61*a*b^2*c^4*d^2*L 
og[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*a^2*b*c^2*d^4*...
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1389\) vs. \(2(604)=1208\).

Time = 6.00 (sec) , antiderivative size = 1389, normalized size of antiderivative = 2.30, 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.

\(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^2 (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b}{2 a (e x)^{3/2} \left (-a b-b^2 x^2\right ) (c+d x)^{5/2}}-\frac {b}{4 a (e x)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^2 (c+d x)^{5/2}}-\frac {b}{4 a (e x)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^2 (c+d x)^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) b^{3/2}}{2 (-a)^{9/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{5/2} e^{3/2}}-\frac {\left (3 \sqrt {b} c-8 \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 ) b^{3/2}}{4 (-a)^{9/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{7/2} e^{3/2}}+\frac {\left (3 \sqrt {b} c+8 \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 ) b^{3/2}}{4 (-a)^{9/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{7/2} e^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) b^{3/2}}{2 (-a)^{9/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{5/2} e^{3/2}}-\frac {\sqrt {b}}{4 a \left (\sqrt {-a} \sqrt {b} c-a d\right ) e \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right ) (c+d x)^{3/2}}-\frac {\sqrt {b}}{4 a \left (\sqrt {-a} \sqrt {b} c+a d\right ) e \sqrt {e x} \left (\sqrt {b} x+\sqrt {-a}\right ) (c+d x)^{3/2}}-\frac {d \left (3 b c^2-14 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \sqrt {e x}}{6 a^2 c^3 \left (\sqrt {b} c-\sqrt {-a} d\right )^2 e^2 \sqrt {c+d x}}-\frac {d \left (3 b c^2+14 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \sqrt {e x}}{6 a^2 c^3 \left (\sqrt {b} c+\sqrt {-a} d\right )^2 e^2 \sqrt {c+d x}}-\frac {d \left (9 b^{3/2} c^3+18 \sqrt {-a} b d c^2-40 a \sqrt {b} d^2 c+16 (-a)^{3/2} d^3\right ) \sqrt {e x}}{12 a^2 c^3 \left (\sqrt {b} c+\sqrt {-a} d\right )^3 e^2 \sqrt {c+d x}}-\frac {d \left (9 b^{3/2} c^3-18 \sqrt {-a} b d c^2-40 a \sqrt {b} d^2 c+16 \sqrt {-a} a d^3\right ) \sqrt {e x}}{12 a^2 c^3 \left (\sqrt {b} c-\sqrt {-a} d\right )^3 e^2 \sqrt {c+d x}}-\frac {d \left (3 \sqrt {-a} \sqrt {b} c+4 a d\right ) \sqrt {e x}}{6 a^2 c^2 \left (\sqrt {-a} \sqrt {b} c+a d\right ) e^2 (c+d x)^{3/2}}-\frac {d \left (3 \sqrt {-a} \sqrt {b} c-4 a d\right ) \sqrt {e x}}{6 a^2 c^2 \left (\sqrt {-a} \sqrt {b} c-a d\right ) e^2 (c+d x)^{3/2}}-\frac {d \left (9 b c^2-12 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \sqrt {e x}}{12 a^2 c^2 \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 (c+d x)^{3/2}}-\frac {d \left (9 b c^2+12 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \sqrt {e x}}{12 a^2 c^2 \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) e^2 (c+d x)^{3/2}}-\frac {3 \sqrt {b} c+2 \sqrt {-a} d}{4 a^2 c \left (\sqrt {b} c+\sqrt {-a} d\right ) e \sqrt {e x} (c+d x)^{3/2}}-\frac {1}{a^2 c e \sqrt {e x} (c+d x)^{3/2}}-\frac {3 \sqrt {b} c-2 \sqrt {-a} d}{4 a^2 c \left (\sqrt {b} c-\sqrt {-a} d\right ) e \sqrt {e x} (c+d x)^{3/2}}\)

Input:

Int[1/((e*x)^(3/2)*(c + d*x)^(5/2)*(a + b*x^2)^2),x]
 

Output:

-(1/(a^2*c*e*Sqrt[e*x]*(c + d*x)^(3/2))) - (3*Sqrt[b]*c - 2*Sqrt[-a]*d)/(4 
*a^2*c*(Sqrt[b]*c - Sqrt[-a]*d)*e*Sqrt[e*x]*(c + d*x)^(3/2)) - (3*Sqrt[b]* 
c + 2*Sqrt[-a]*d)/(4*a^2*c*(Sqrt[b]*c + Sqrt[-a]*d)*e*Sqrt[e*x]*(c + d*x)^ 
(3/2)) - (d*(3*Sqrt[-a]*Sqrt[b]*c - 4*a*d)*Sqrt[e*x])/(6*a^2*c^2*(Sqrt[-a] 
*Sqrt[b]*c - a*d)*e^2*(c + d*x)^(3/2)) - (d*(3*Sqrt[-a]*Sqrt[b]*c + 4*a*d) 
*Sqrt[e*x])/(6*a^2*c^2*(Sqrt[-a]*Sqrt[b]*c + a*d)*e^2*(c + d*x)^(3/2)) - ( 
d*(9*b*c^2 - 12*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*Sqrt[e*x])/(12*a^2*c^2*(b* 
c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*e^2*(c + d*x)^(3/2)) - (d*(9*b*c^2 + 
 12*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*Sqrt[e*x])/(12*a^2*c^2*(b*c^2 + 2*Sqrt 
[-a]*Sqrt[b]*c*d - a*d^2)*e^2*(c + d*x)^(3/2)) - Sqrt[b]/(4*a*(Sqrt[-a]*Sq 
rt[b]*c - a*d)*e*Sqrt[e*x]*(Sqrt[-a] - Sqrt[b]*x)*(c + d*x)^(3/2)) - Sqrt[ 
b]/(4*a*(Sqrt[-a]*Sqrt[b]*c + a*d)*e*Sqrt[e*x]*(Sqrt[-a] + Sqrt[b]*x)*(c + 
 d*x)^(3/2)) - (d*(3*b*c^2 - 14*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*Sqrt[e*x]) 
/(6*a^2*c^3*(Sqrt[b]*c - Sqrt[-a]*d)^2*e^2*Sqrt[c + d*x]) - (d*(3*b*c^2 + 
14*Sqrt[-a]*Sqrt[b]*c*d - 8*a*d^2)*Sqrt[e*x])/(6*a^2*c^3*(Sqrt[b]*c + Sqrt 
[-a]*d)^2*e^2*Sqrt[c + d*x]) - (d*(9*b^(3/2)*c^3 + 18*Sqrt[-a]*b*c^2*d - 4 
0*a*Sqrt[b]*c*d^2 + 16*(-a)^(3/2)*d^3)*Sqrt[e*x])/(12*a^2*c^3*(Sqrt[b]*c + 
 Sqrt[-a]*d)^3*e^2*Sqrt[c + d*x]) - (d*(9*b^(3/2)*c^3 - 18*Sqrt[-a]*b*c^2* 
d - 40*a*Sqrt[b]*c*d^2 + 16*Sqrt[-a]*a*d^3)*Sqrt[e*x])/(12*a^2*c^3*(Sqrt[b 
]*c - Sqrt[-a]*d)^3*e^2*Sqrt[c + d*x]) - (b^(3/2)*(3*Sqrt[b]*c - 8*Sqrt...
 

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. \(2324\) vs. \(2(508)=1016\).

Time = 0.72 (sec) , antiderivative size = 2325, normalized size of antiderivative = 3.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(2325\)
default	\(\text {Expression too large to display}\)	\(13104\)

Input:

int(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/c^3/a^2*(d*x+c)^(1/2)/e/(e*x)^(1/2)+(-2/3/c^2*b^2*d^3/((-a*b)^(1/2)*d+b 
*c)^2/((-a*b)^(1/2)*d-b*c)^2/e/(x+c/d)^2*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2) 
-4/3/c^3*b^2*d^4/((-a*b)^(1/2)*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2/e/(x+c/d)*( 
d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+1/4/a^2*b^2*(-a*b)^(1/2)/((-a*b)^(1/2)*d+ 
b*c)^2/e/(a*d-c*(-a*b)^(1/2))/(x-(-a*b)^(1/2)/b)*(d*e*(x-(-a*b)^(1/2)/b)^2 
+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^( 
1/2)+1/4/a*b^2/((-a*b)^(1/2)*d+b*c)^2/(a*d-c*(-a*b)^(1/2))/(-e*(a*d-c*(-a* 
b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/2)*d+b*c 
)/b*(x-(-a*b)^(1/2)/b)+2*(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x-(-a*b)^ 
(1/2)/b)^2+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^( 
1/2))/b)^(1/2))/(x-(-a*b)^(1/2)/b))*d-1/8/a^2*c*b^2*(-a*b)^(1/2)/((-a*b)^( 
1/2)*d+b*c)^2/(a*d-c*(-a*b)^(1/2))/(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*ln((- 
2*e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)+2 
*(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x-(-a*b)^(1/2)/b)^2+e*(2*(-a*b)^( 
1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2))/(x-(-a*b 
)^(1/2)/b))-1/4/a^2*b^2*(-a*b)^(1/2)/((-a*b)^(1/2)*d-b*c)^2/e/(a*d+c*(-a*b 
)^(1/2))/(x+(-a*b)^(1/2)/b)*(d*e*(x+(-a*b)^(1/2)/b)^2+e*(-2*(-a*b)^(1/2)*d 
+b*c)/b*(x+(-a*b)^(1/2)/b)-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)+1/4/a*b^2/((-a* 
b)^(1/2)*d-b*c)^2/(a*d+c*(-a*b)^(1/2))/(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*l 
n((-2*e*(a*d+c*(-a*b)^(1/2))/b+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9192 vs. \(2 (509) = 1018\).

Time = 22.89 (sec) , antiderivative size = 9192, normalized size of antiderivative = 15.22 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(3/2)/(d*x+c)**(5/2)/(b*x**2+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^2*(d*x + c)^(5/2)*(e*x)^(3/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(1/((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(5/2)),x)
 

Output:

int(1/((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int \frac {1}{\left (e x \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{2}}d x \] Input:

int(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x)
 

Output:

int(1/(e*x)^(3/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x)