\(\int \frac {(e x)^{9/2}}{(c+d x)^{5/2} (a+b x^2)^2} \, dx\) [933]

Optimal result

Integrand size = 26, antiderivative size = 601 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {c \left (3 b c^2-7 a d^2\right ) e (e x)^{7/2}}{6 a \left (b c^2+a d^2\right )^2 (c+d x)^{3/2}}-\frac {c d \left (17 b c^2-43 a d^2\right ) e^2 (e x)^{5/2}}{6 \left (b c^2+a d^2\right )^3 \sqrt {c+d x}}+\frac {a d \left (19 b c^2-a d^2\right ) e^4 \sqrt {e x} \sqrt {c+d x}}{2 b \left (b c^2+a d^2\right )^3}+\frac {c \left (7 b c^2-53 a d^2\right ) e^3 (e x)^{3/2} \sqrt {c+d x}}{6 \left (b c^2+a d^2\right )^3}+\frac {(e x)^{9/2} (a d+b c x)}{2 a \left (b c^2+a d^2\right ) (c+d x)^{3/2} \left (a+b x^2\right )}+\frac {(-a)^{3/4} \left (7 b^2 c^4+19 \sqrt {-a} b^{3/2} c^3 d-15 a b c^2 d^2-\sqrt {-a} a \sqrt {b} c d^3-2 a^2 d^4\right ) e^{9/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )^3}-\frac {(-a)^{3/4} \left (7 b^2 c^4-19 \sqrt {-a} b^{3/2} c^3 d-15 a b c^2 d^2+\sqrt {-a} a \sqrt {b} c d^3-2 a^2 d^4\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )^3} \] Output:

-1/6*c*(-7*a*d^2+3*b*c^2)*e*(e*x)^(7/2)/a/(a*d^2+b*c^2)^2/(d*x+c)^(3/2)-1/ 
6*c*d*(-43*a*d^2+17*b*c^2)*e^2*(e*x)^(5/2)/(a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1 
/2*a*d*(-a*d^2+19*b*c^2)*e^4*(e*x)^(1/2)*(d*x+c)^(1/2)/b/(a*d^2+b*c^2)^3+1 
/6*c*(-53*a*d^2+7*b*c^2)*e^3*(e*x)^(3/2)*(d*x+c)^(1/2)/(a*d^2+b*c^2)^3+1/2 
*(e*x)^(9/2)*(b*c*x+a*d)/a/(a*d^2+b*c^2)/(d*x+c)^(3/2)/(b*x^2+a)+1/4*(-a)^ 
(3/4)*(7*b^2*c^4+19*(-a)^(1/2)*b^(3/2)*c^3*d-15*a*b*c^2*d^2-(-a)^(1/2)*a*b 
^(1/2)*c*d^3-2*a^2*d^4)*e^(9/2)*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))/b^(3/2)/(b^(1/2)*c-(-a)^(1/2)*d) 
^(1/2)/(a*d^2+b*c^2)^3-1/4*(-a)^(3/4)*(7*b^2*c^4-19*(-a)^(1/2)*b^(3/2)*c^3 
*d-15*a*b*c^2*d^2+(-a)^(1/2)*a*b^(1/2)*c*d^3-2*a^2*d^4)*e^(9/2)*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)) 
/b^(3/2)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)/(a*d^2+b*c^2)^3
 

Mathematica [C] (verified)

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

Time = 2.07 (sec) , antiderivative size = 1223, normalized size of antiderivative = 2.03 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(e*x)^(9/2)/((c + d*x)^(5/2)*(a + b*x^2)^2),x]
 

Output:

((e*x)^(9/2)*((2*b*Sqrt[x]*(4*b^3*c^5*x^3 - 3*a^3*d^3*(c + d*x)^2 + a*b^2* 
c^3*x*(7*c^2 + 54*c*d*x + 55*d^2*x^2) + a^2*b*c*d*(57*c^3 + 61*c^2*d*x - 9 
*c*d^2*x^2 - 9*d^3*x^3)))/((c + d*x)^(3/2)*(a + b*x^2)) + 3*a*Sqrt[d]*Root 
Sum[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*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] + 64*a^3 
*d^6*Log[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 + 14*a*b^2*c^3*d^2*L 
og[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 40*a^2*b*c*d^4*L 
og[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) & ] - 12*a*Sqrt[d]*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 & , (2*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] - 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*Log[c + 2* 
d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 4*b^3*c^5*Log[c + 2*d*x -...
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2509\) vs. \(2(601)=1202\).

Time = 9.62 (sec) , antiderivative size = 2509, normalized size of antiderivative = 4.17, 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.

\(\displaystyle \int \frac {(e x)^{9/2}}{\left (a+b x^2\right )^2 (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b (e x)^{9/2}}{2 a \left (-a b-b^2 x^2\right ) (c+d x)^{5/2}}-\frac {b (e x)^{9/2}}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2 (c+d x)^{5/2}}-\frac {b (e x)^{9/2}}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2 (c+d x)^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(-a)^{3/4} \left (\sqrt {b} c+\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 ) e^{9/2}}{2 b^{3/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} \left (b c^2+a d^2\right )}+\frac {(-a)^{3/4} \left (9 \sqrt {b} c-4 \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 ) e^{9/2}}{4 b^{3/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^{7/2}}+\frac {\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (15 b^{3/2} c^3+3 \sqrt {-a} b d c^2+4 a \sqrt {b} d^2 c+8 (-a)^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{9/2}}{16 a b^{3/2} d^{7/2} \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}+\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (15 b^{3/2} c^3-3 \sqrt {-a} b d c^2+4 a \sqrt {b} d^2 c+8 \sqrt {-a} a d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{9/2}}{16 a b^{3/2} d^{7/2} \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}+\frac {\left (5 b^2 c^4-11 \sqrt {-a} b^{3/2} d c^3-3 a b d^2 c^2+7 (-a)^{3/2} \sqrt {b} d^3 c-4 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{9/2}}{4 a b^{3/2} d^{7/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^3}+\frac {\left (5 b^2 c^4+11 \sqrt {-a} b^{3/2} d c^3-3 a b d^2 c^2+7 \sqrt {-a} a \sqrt {b} d^3 c-4 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{9/2}}{4 a b^{3/2} d^{7/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^3}-\frac {35 c^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{9/2}}{8 a d^{7/2} \left (b c^2+a d^2\right )}-\frac {(-a)^{3/4} \left (9 \sqrt {b} c+4 \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 ) e^{9/2}}{4 b^{3/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^{7/2}}+\frac {(-a)^{3/4} \left (\sqrt {b} c-\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 ) e^{9/2}}{2 b^{3/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} \left (b c^2+a d^2\right )}-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (15 b c^2-3 \sqrt {-a} \sqrt {b} d c+4 a d^2\right ) \sqrt {e x} \sqrt {c+d x} e^4}{16 a b d^3 \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}-\frac {\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (15 b c^2+3 \sqrt {-a} \sqrt {b} d c+4 a d^2\right ) \sqrt {e x} \sqrt {c+d x} e^4}{16 a b d^3 \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}-\frac {\left (5 b^{3/2} c^3+11 \sqrt {-a} b d c^2-3 a \sqrt {b} d^2 c+2 (-a)^{3/2} d^3\right ) \sqrt {e x} \sqrt {c+d x} e^4}{4 a b d^3 \left (\sqrt {b} c+\sqrt {-a} d\right )^3}-\frac {\left (5 b^{3/2} c^3-11 \sqrt {-a} b d c^2-3 a \sqrt {b} d^2 c+2 \sqrt {-a} a d^3\right ) \sqrt {e x} \sqrt {c+d x} e^4}{4 a b d^3 \left (\sqrt {b} c-\sqrt {-a} d\right )^3}+\frac {35 c^2 \sqrt {e x} \sqrt {c+d x} e^4}{8 a d^3 \left (b c^2+a d^2\right )}+\frac {\left (5 \sqrt {b} c-\sqrt {-a} d\right ) \left (\sqrt {b} c+\sqrt {-a} d\right ) (e x)^{3/2} \sqrt {c+d x} e^3}{8 a \sqrt {b} d^2 \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}+\frac {\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (5 \sqrt {b} c+\sqrt {-a} d\right ) (e x)^{3/2} \sqrt {c+d x} e^3}{8 a \sqrt {b} d^2 \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right )}-\frac {35 c (e x)^{3/2} \sqrt {c+d x} e^3}{12 a d^2 \left (b c^2+a d^2\right )}+\frac {c \left (10 b c^2-22 \sqrt {-a} \sqrt {b} d c+3 a d^2\right ) (e x)^{3/2} e^3}{12 a \sqrt {b} d^2 \left (\sqrt {b} c-\sqrt {-a} d\right )^3 \sqrt {c+d x}}+\frac {c \left (10 b c^2+22 \sqrt {-a} \sqrt {b} d c+3 a d^2\right ) (e x)^{3/2} e^3}{12 a \sqrt {b} d^2 \left (\sqrt {b} c+\sqrt {-a} d\right )^3 \sqrt {c+d x}}-\frac {c \left (\sqrt {b} c+\sqrt {-a} d\right ) (e x)^{5/2} e^2}{2 a d \left (\sqrt {b} c-\sqrt {-a} d\right ) \left (b c^2+a d^2\right ) \sqrt {c+d x}}+\frac {7 c (e x)^{5/2} e^2}{3 a d \left (b c^2+a d^2\right ) \sqrt {c+d x}}-\frac {c \left (\sqrt {b} c-\sqrt {-a} d\right ) (e x)^{5/2} e^2}{2 a d \left (\sqrt {b} c+\sqrt {-a} d\right ) \left (b c^2+a d^2\right ) \sqrt {c+d x}}+\frac {c \left (2 \sqrt {b} c+3 \sqrt {-a} d\right ) (e x)^{5/2} e^2}{12 a \sqrt {b} d \left (\sqrt {b} c-\sqrt {-a} d\right )^2 (c+d x)^{3/2}}+\frac {c \left (2 \sqrt {b} c-3 \sqrt {-a} d\right ) (e x)^{5/2} e^2}{12 a \sqrt {b} d \left (\sqrt {b} c+\sqrt {-a} d\right )^2 (c+d x)^{3/2}}+\frac {c (e x)^{7/2} e}{3 a \left (b c^2+a d^2\right ) (c+d x)^{3/2}}+\frac {(e x)^{7/2} e}{4 \sqrt {b} \left (\sqrt {-a} \sqrt {b} c-a d\right ) \left (\sqrt {-a}-\sqrt {b} x\right ) (c+d x)^{3/2}}-\frac {\sqrt {-a} (e x)^{7/2} e}{4 \sqrt {b} \left (d (-a)^{3/2}+a \sqrt {b} c\right ) \left (\sqrt {b} x+\sqrt {-a}\right ) (c+d x)^{3/2}}\)

Input:

Int[(e*x)^(9/2)/((c + d*x)^(5/2)*(a + b*x^2)^2),x]
 

Output:

(c*(2*Sqrt[b]*c - 3*Sqrt[-a]*d)*e^2*(e*x)^(5/2))/(12*a*Sqrt[b]*d*(Sqrt[b]* 
c + Sqrt[-a]*d)^2*(c + d*x)^(3/2)) + (c*(2*Sqrt[b]*c + 3*Sqrt[-a]*d)*e^2*( 
e*x)^(5/2))/(12*a*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[-a]*d)^2*(c + d*x)^(3/2)) + 
(c*e*(e*x)^(7/2))/(3*a*(b*c^2 + a*d^2)*(c + d*x)^(3/2)) + (e*(e*x)^(7/2))/ 
(4*Sqrt[b]*(Sqrt[-a]*Sqrt[b]*c - a*d)*(Sqrt[-a] - Sqrt[b]*x)*(c + d*x)^(3/ 
2)) - (Sqrt[-a]*e*(e*x)^(7/2))/(4*Sqrt[b]*(a*Sqrt[b]*c + (-a)^(3/2)*d)*(Sq 
rt[-a] + Sqrt[b]*x)*(c + d*x)^(3/2)) + (c*(10*b*c^2 - 22*Sqrt[-a]*Sqrt[b]* 
c*d + 3*a*d^2)*e^3*(e*x)^(3/2))/(12*a*Sqrt[b]*d^2*(Sqrt[b]*c - Sqrt[-a]*d) 
^3*Sqrt[c + d*x]) + (c*(10*b*c^2 + 22*Sqrt[-a]*Sqrt[b]*c*d + 3*a*d^2)*e^3* 
(e*x)^(3/2))/(12*a*Sqrt[b]*d^2*(Sqrt[b]*c + Sqrt[-a]*d)^3*Sqrt[c + d*x]) + 
 (7*c*e^2*(e*x)^(5/2))/(3*a*d*(b*c^2 + a*d^2)*Sqrt[c + d*x]) - (c*(Sqrt[b] 
*c - Sqrt[-a]*d)*e^2*(e*x)^(5/2))/(2*a*d*(Sqrt[b]*c + Sqrt[-a]*d)*(b*c^2 + 
 a*d^2)*Sqrt[c + d*x]) - (c*(Sqrt[b]*c + Sqrt[-a]*d)*e^2*(e*x)^(5/2))/(2*a 
*d*(Sqrt[b]*c - Sqrt[-a]*d)*(b*c^2 + a*d^2)*Sqrt[c + d*x]) + (35*c^2*e^4*S 
qrt[e*x]*Sqrt[c + d*x])/(8*a*d^3*(b*c^2 + a*d^2)) - ((Sqrt[b]*c + Sqrt[-a] 
*d)*(15*b*c^2 - 3*Sqrt[-a]*Sqrt[b]*c*d + 4*a*d^2)*e^4*Sqrt[e*x]*Sqrt[c + d 
*x])/(16*a*b*d^3*(Sqrt[b]*c - Sqrt[-a]*d)*(b*c^2 + a*d^2)) - ((Sqrt[b]*c - 
 Sqrt[-a]*d)*(15*b*c^2 + 3*Sqrt[-a]*Sqrt[b]*c*d + 4*a*d^2)*e^4*Sqrt[e*x]*S 
qrt[c + d*x])/(16*a*b*d^3*(Sqrt[b]*c + Sqrt[-a]*d)*(b*c^2 + a*d^2)) - ((5* 
b^(3/2)*c^3 + 11*Sqrt[-a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + 2*(-a)^(3/2)*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. \(12174\) vs. \(2(499)=998\).

Time = 0.67 (sec) , antiderivative size = 12175, normalized size of antiderivative = 20.26

method	result	size
default	\(\text {Expression too large to display}\)	\(12175\)

Input:

int((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8458 vs. \(2 (499) = 998\).

Time = 5.32 (sec) , antiderivative size = 8458, normalized size of antiderivative = 14.07 \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(9/2)/(d*x+c)**(5/2)/(b*x**2+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((e*x)^(9/2)/((b*x^2 + a)^2*(d*x + c)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((e*x)^(9/2)/((a + b*x^2)^2*(c + d*x)^(5/2)),x)
 

Output:

int((e*x)^(9/2)/((a + b*x^2)^2*(c + d*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{9/2}}{(c+d x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int \frac {\left (e x \right )^{\frac {9}{2}}}{\left (d x +c \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{2}}d x \] Input:

int((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x)
 

Output:

int((e*x)^(9/2)/(d*x+c)^(5/2)/(b*x^2+a)^2,x)