\(\int \frac {\sqrt {c+d x}}{(e x)^{5/2} (a+b x^2)^2} \, dx\) [902]

Optimal result

Integrand size = 26, antiderivative size = 318 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {7 \sqrt {c+d x}}{6 a^2 e (e x)^{3/2}}-\frac {2 d \sqrt {c+d x}}{3 a^2 c e^2 \sqrt {e x}}+\frac {\sqrt {c+d x}}{2 a e (e x)^{3/2} \left (a+b x^2\right )}+\frac {\sqrt {b} \left (7 \sqrt {b} c-6 \sqrt {-a} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{11/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} e^{5/2}}+\frac {\sqrt {b} \left (7 \sqrt {b} c+6 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{11/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} e^{5/2}} \] Output:

-7/6*(d*x+c)^(1/2)/a^2/e/(e*x)^(3/2)-2/3*d*(d*x+c)^(1/2)/a^2/c/e^2/(e*x)^( 
1/2)+1/2*(d*x+c)^(1/2)/a/e/(e*x)^(3/2)/(b*x^2+a)+1/4*b^(1/2)*(7*b^(1/2)*c- 
6*(-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)^(11/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e^(5/ 
2)+1/4*b^(1/2)*(7*b^(1/2)*c+6*(-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)^(11/4)/(b^(1/2 
)*c+(-a)^(1/2)*d)^(1/2)/e^(5/2)
 

Mathematica [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\sqrt {c+d x} \left (4 a (c+d x)+b x^2 (7 c+4 d x)\right )-6 c d^{3/2} x^{3/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+16 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+2 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]-3 c d^{3/2} x^{3/2} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-32 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+4 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{6 a^2 c (e x)^{5/2} \left (a+b x^2\right )} \] Input:

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

Output:

(x*(-(Sqrt[c + d*x]*(4*a*(c + d*x) + b*x^2*(7*c + 4*d*x))) - 6*c*d^(3/2)*x 
^(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*Log[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) & ] - 3*c*d^(3/2)*x^(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]*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) & ]))/(6*a^2*c*(e*x)^(5/2)*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(846\) vs. \(2(318)=636\).

Time = 3.27 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.66, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {c+d x} d}{3 a^2 c e^2 \sqrt {e x}}+\frac {\sqrt {b} \left (5 b c^2-9 \sqrt {-a} \sqrt {b} d c-4 a d^2\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{11/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} e^{5/2}}+\frac {\sqrt {b} \sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{11/4} e^{5/2}}+\frac {\sqrt {b} \left (5 b c^2+9 \sqrt {-a} \sqrt {b} d c-4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{11/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} e^{5/2}}+\frac {\sqrt {b} \sqrt {\sqrt {b} c+\sqrt {-a} d} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{11/4} e^{5/2}}+\frac {\left (15 b c^2-17 \sqrt {-a} \sqrt {b} d c-2 a d^2\right ) \sqrt {c+d x}}{12 (-a)^{5/2} c \left (\sqrt {b} c-\sqrt {-a} d\right ) e^2 \sqrt {e x}}-\frac {\left (15 b c^2+17 \sqrt {-a} \sqrt {b} d c-2 a d^2\right ) \sqrt {c+d x}}{12 (-a)^{5/2} c \left (\sqrt {b} c+\sqrt {-a} d\right ) e^2 \sqrt {e x}}+\frac {5 \sqrt {b} \sqrt {c+d x}}{12 a^2 e^2 \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {c+d x}}{6 (-a)^{3/2} e (e x)^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {5 \sqrt {b} \sqrt {c+d x}}{12 a^2 e^2 \sqrt {e x} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {c+d x}}{6 (-a)^{3/2} e (e x)^{3/2} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {c+d x}}{3 a^2 e (e x)^{3/2}}\)

Input:

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

Output:

-1/3*Sqrt[c + d*x]/(a^2*e*(e*x)^(3/2)) - (d*Sqrt[c + d*x])/(3*a^2*c*e^2*Sq 
rt[e*x]) + ((15*b*c^2 - 17*Sqrt[-a]*Sqrt[b]*c*d - 2*a*d^2)*Sqrt[c + d*x])/ 
(12*(-a)^(5/2)*c*(Sqrt[b]*c - Sqrt[-a]*d)*e^2*Sqrt[e*x]) - ((15*b*c^2 + 17 
*Sqrt[-a]*Sqrt[b]*c*d - 2*a*d^2)*Sqrt[c + d*x])/(12*(-a)^(5/2)*c*(Sqrt[b]* 
c + Sqrt[-a]*d)*e^2*Sqrt[e*x]) - Sqrt[c + d*x]/(6*(-a)^(3/2)*e*(e*x)^(3/2) 
*(Sqrt[-a] - Sqrt[b]*x)) + (5*Sqrt[b]*Sqrt[c + d*x])/(12*a^2*e^2*Sqrt[e*x] 
*(Sqrt[-a] - Sqrt[b]*x)) - Sqrt[c + d*x]/(6*(-a)^(3/2)*e*(e*x)^(3/2)*(Sqrt 
[-a] + Sqrt[b]*x)) - (5*Sqrt[b]*Sqrt[c + d*x])/(12*a^2*e^2*Sqrt[e*x]*(Sqrt 
[-a] + Sqrt[b]*x)) + (Sqrt[b]*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*ArcTan[(Sqrt[Sq 
rt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(2*( 
-a)^(11/4)*e^(5/2)) + (Sqrt[b]*(5*b*c^2 - 9*Sqrt[-a]*Sqrt[b]*c*d - 4*a*d^2 
)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt 
[c + d*x])])/(4*(-a)^(11/4)*(Sqrt[b]*c - Sqrt[-a]*d)^(3/2)*e^(5/2)) + (Sqr 
t[b]*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sq 
rt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(2*(-a)^(11/4)*e^(5/2)) + (S 
qrt[b]*(5*b*c^2 + 9*Sqrt[-a]*Sqrt[b]*c*d - 4*a*d^2)*ArcTanh[(Sqrt[Sqrt[b]* 
c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4*(-a)^(1 
1/4)*(Sqrt[b]*c + Sqrt[-a]*d)^(3/2)*e^(5/2))
 

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. \(2439\) vs. \(2(236)=472\).

Time = 0.50 (sec) , antiderivative size = 2440, normalized size of antiderivative = 7.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2440\)
risch	\(\text {Expression too large to display}\)	\(2474\)

Input:

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

Output:

1/24*(d*x+c)^(1/2)*b*(-21*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^2*d^2*e*x^4*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)+18*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*c*d^3*e*x^4*(-a*b)^(1/2 
)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)-21*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^4*e*x^4*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)+18*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^3*d*e*x^4*(-a* 
b)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)+21*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^2*d^2*e*x^4*(-e*(a*d+c*(-a*b)^(1/2))/b) 
^(1/2)+18*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*c*d^3 
*e*x^4*(-a*b)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)+21*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^4*e*x^4*(-e*(a*d+c*(-a*b)^(1/2 
))/b)^(1/2)+18*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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2047 vs. \(2 (236) = 472\).

Time = 0.30 (sec) , antiderivative size = 2047, normalized size of antiderivative = 6.44 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/24*(3*(a^2*b*c*e^3*x^4 + a^3*c*e^3*x^2)*sqrt(-((a^5*b*c^2 + a^6*d^2)*e^ 
5*sqrt(-(2401*b^5*c^6 + 4704*a*b^4*c^4*d^2 + 2304*a^2*b^3*c^2*d^4)/((a^11* 
b^2*c^4 + 2*a^12*b*c^2*d^2 + a^13*d^4)*e^10)) + 35*b^2*c^2*d + 36*a*b*d^3) 
/((a^5*b*c^2 + a^6*d^2)*e^5))*log(((2401*b^4*c^5 + 4116*a*b^3*c^3*d^2 + 17 
28*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + ((7*a^8*b^2*c^4 + 13*a^9*b*c^2 
*d^2 + 6*a^10*d^4)*e^8*x*sqrt(-(2401*b^5*c^6 + 4704*a*b^4*c^4*d^2 + 2304*a 
^2*b^3*c^2*d^4)/((a^11*b^2*c^4 + 2*a^12*b*c^2*d^2 + a^13*d^4)*e^10)) + (49 
*a^3*b^3*c^4*d + 48*a^4*b^2*c^2*d^3)*e^3*x)*sqrt(-((a^5*b*c^2 + a^6*d^2)*e 
^5*sqrt(-(2401*b^5*c^6 + 4704*a*b^4*c^4*d^2 + 2304*a^2*b^3*c^2*d^4)/((a^11 
*b^2*c^4 + 2*a^12*b*c^2*d^2 + a^13*d^4)*e^10)) + 35*b^2*c^2*d + 36*a*b*d^3 
)/((a^5*b*c^2 + a^6*d^2)*e^5)))/x) - 3*(a^2*b*c*e^3*x^4 + a^3*c*e^3*x^2)*s 
qrt(-((a^5*b*c^2 + a^6*d^2)*e^5*sqrt(-(2401*b^5*c^6 + 4704*a*b^4*c^4*d^2 + 
 2304*a^2*b^3*c^2*d^4)/((a^11*b^2*c^4 + 2*a^12*b*c^2*d^2 + a^13*d^4)*e^10) 
) + 35*b^2*c^2*d + 36*a*b*d^3)/((a^5*b*c^2 + a^6*d^2)*e^5))*log(((2401*b^4 
*c^5 + 4116*a*b^3*c^3*d^2 + 1728*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) - 
((7*a^8*b^2*c^4 + 13*a^9*b*c^2*d^2 + 6*a^10*d^4)*e^8*x*sqrt(-(2401*b^5*c^6 
 + 4704*a*b^4*c^4*d^2 + 2304*a^2*b^3*c^2*d^4)/((a^11*b^2*c^4 + 2*a^12*b*c^ 
2*d^2 + a^13*d^4)*e^10)) + (49*a^3*b^3*c^4*d + 48*a^4*b^2*c^2*d^3)*e^3*x)* 
sqrt(-((a^5*b*c^2 + a^6*d^2)*e^5*sqrt(-(2401*b^5*c^6 + 4704*a*b^4*c^4*d^2 
+ 2304*a^2*b^3*c^2*d^4)/((a^11*b^2*c^4 + 2*a^12*b*c^2*d^2 + a^13*d^4)*e...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a^{2} x^{2}+2 \sqrt {x}\, a b \,x^{4}+\sqrt {x}\, b^{2} x^{6}}d x}{\sqrt {e}\, e^{2}} \] Input:

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

Output:

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