\(\int \frac {(c+d x)^{5/2}}{\sqrt {e x} (a+b x^2)^2} \, dx\) [915]

Optimal result

Integrand size = 26, antiderivative size = 341 \[ \int \frac {(c+d x)^{5/2}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=-\frac {d^2 \sqrt {e x} \sqrt {c+d x}}{2 a b e}+\frac {\sqrt {e x} (c+d x)^{5/2}}{2 a e \left (a+b x^2\right )}-\frac {\left (2 \sqrt {-a} d \left (2 b c^2+a d^2\right )-\sqrt {b} c \left (3 b c^2+a d^2\right )\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}+\frac {\left (2 \sqrt {-a} d \left (2 b c^2+a d^2\right )+\sqrt {b} c \left (3 b c^2+a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}} \] Output:

-1/2*d^2*(e*x)^(1/2)*(d*x+c)^(1/2)/a/b/e+1/2*(e*x)^(1/2)*(d*x+c)^(5/2)/a/e 
/(b*x^2+a)-1/4*(2*(-a)^(1/2)*d*(a*d^2+2*b*c^2)-b^(1/2)*c*(a*d^2+3*b*c^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)^(7/4)/b^(3/2)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e^(1/2)+1/4*(2 
*(-a)^(1/2)*d*(a*d^2+2*b*c^2)+b^(1/2)*c*(a*d^2+3*b*c^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)^( 
7/4)/b^(3/2)/(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 = 0.69 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d x)^{5/2}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {b x \sqrt {c+d x} \left (-a d^2+b c (c+2 d x)\right )+2 a d^{7/2} \sqrt {x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {49 b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-16 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+10 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]+d^{3/2} \sqrt {x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {2 b^2 c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-97 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+32 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+2 b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-20 a b c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+2 b^2 c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2-a b d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]}{2 a b^2 \sqrt {e x} \left (a+b x^2\right )} \] Input:

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

Output:

(b*x*Sqrt[c + d*x]*(-(a*d^2) + b*c*(c + 2*d*x)) + 2*a*d^(7/2)*Sqrt[x]*(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 & , (49*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] + 
 10*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)*Sqrt[x]*(a + b*x^2)*Roo 
tSum[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^2*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 
97*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 32* 
a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 2*b^2*c^3* 
Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 20*a*b*c*d^2*Lo 
g[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 2*b^2*c^2*Log[c + 
 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 - a*b*d^2*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^2*Sqrt[e*x]*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(995\) vs. \(2(341)=682\).

Time = 4.14 (sec) , antiderivative size = 995, normalized size of antiderivative = 2.92, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {e x} \sqrt {c+d x} d^2}{2 a b e}+\frac {5 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) d^{3/2}}{2 a b \sqrt {e}}-\frac {\left (5 \sqrt {b} c-4 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) d^{3/2}}{4 a b^{3/2} \sqrt {e}}-\frac {\left (5 \sqrt {b} c+4 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) d^{3/2}}{4 a b^{3/2} \sqrt {e}}+\frac {\left (\sqrt {b} c+2 \sqrt {-a} d\right ) \sqrt {e x} \sqrt {c+d x} d}{4 (-a)^{3/2} b e}-\frac {\left (\sqrt {-a} \sqrt {b} c+2 a d\right ) \sqrt {e x} \sqrt {c+d x} d}{4 a^2 b e}+\frac {\left (\sqrt {-a} c-\frac {a d}{\sqrt {b}}\right ) \sqrt {e x} (c+d x)^{3/2}}{4 a^2 e \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\left (\sqrt {-a} c+\frac {a d}{\sqrt {b}}\right ) \sqrt {e x} (c+d x)^{3/2}}{4 a^2 e \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\left (b^{3/2} c^3+2 \sqrt {-a} b d c^2+7 a \sqrt {b} d^2 c+4 (-a)^{3/2} d^3\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} b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}+\frac {\left (b c^3-3 a d^2 c-\frac {\sqrt {-a} d \left (3 b c^2-a d^2\right )}{\sqrt {b}}\right ) \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)^{7/4} b \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}+\frac {\left (b^{3/2} c^3-2 \sqrt {-a} b d c^2+7 a \sqrt {b} d^2 c+4 \sqrt {-a} a d^3\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} b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}}+\frac {\left (b c^3-3 a d^2 c+\frac {\sqrt {-a} d \left (3 b c^2-a d^2\right )}{\sqrt {b}}\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 )}{2 (-a)^{7/4} b \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}}\)

Input:

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

Output:

(d^2*Sqrt[e*x]*Sqrt[c + d*x])/(2*a*b*e) + (d*(Sqrt[b]*c + 2*Sqrt[-a]*d)*Sq 
rt[e*x]*Sqrt[c + d*x])/(4*(-a)^(3/2)*b*e) - (d*(Sqrt[-a]*Sqrt[b]*c + 2*a*d 
)*Sqrt[e*x]*Sqrt[c + d*x])/(4*a^2*b*e) + ((Sqrt[-a]*c - (a*d)/Sqrt[b])*Sqr 
t[e*x]*(c + d*x)^(3/2))/(4*a^2*e*(Sqrt[-a] - Sqrt[b]*x)) + ((Sqrt[-a]*c + 
(a*d)/Sqrt[b])*Sqrt[e*x]*(c + d*x)^(3/2))/(4*a^2*e*(Sqrt[-a] + Sqrt[b]*x)) 
 + ((b^(3/2)*c^3 + 2*Sqrt[-a]*b*c^2*d + 7*a*Sqrt[b]*c*d^2 + 4*(-a)^(3/2)*d 
^3)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sq 
rt[c + d*x])])/(4*(-a)^(7/4)*b^(3/2)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) 
 + ((b*c^3 - 3*a*c*d^2 - (Sqrt[-a]*d*(3*b*c^2 - a*d^2))/Sqrt[b])*ArcTan[(S 
qrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])] 
)/(2*(-a)^(7/4)*b*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) + (5*c*d^(3/2)*Arc 
Tanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(2*a*b*Sqrt[e]) - (d^(3 
/2)*(5*Sqrt[b]*c - 4*Sqrt[-a]*d)*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt 
[c + d*x])])/(4*a*b^(3/2)*Sqrt[e]) - (d^(3/2)*(5*Sqrt[b]*c + 4*Sqrt[-a]*d) 
*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(4*a*b^(3/2)*Sqrt[e 
]) + ((b^(3/2)*c^3 - 2*Sqrt[-a]*b*c^2*d + 7*a*Sqrt[b]*c*d^2 + 4*Sqrt[-a]*a 
*d^3)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e] 
*Sqrt[c + d*x])])/(4*(-a)^(7/4)*b^(3/2)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[ 
e]) + ((b*c^3 - 3*a*c*d^2 + (Sqrt[-a]*d*(3*b*c^2 - a*d^2))/Sqrt[b])*ArcTan 
h[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c +...
 

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. \(3040\) vs. \(2(265)=530\).

Time = 0.50 (sec) , antiderivative size = 3041, normalized size of antiderivative = 8.92

method	result	size
default	\(\text {Expression too large to display}\)	\(3041\)

Input:

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

Output:

1/8*(d*x+c)^(1/2)*x*(-4*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^2*b^2*c^3*d^2*e*(-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^3*b*c*d^4*e*(e*(-a*d+c*(-a*b)^(1/ 
2))/b)^(1/2)+4*b^3*c^4*(-a*b)^(1/2)*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^ 
(1/2))/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^4*c^5*e*x^2*(-e*(a*d+c*(-a*b)^(1/2))/b) 
^(1/2)+4*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^2*b^2*c 
^3*d^2*e*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)-4*a^2*b*d^4*(-a*b)^(1/2)*((d*x+ 
c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/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^3*b*c*d^4 
*e*(-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^4*c^5*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^3*d^5*e*(-a*b)^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1523 vs. \(2 (264) = 528\).

Time = 0.41 (sec) , antiderivative size = 1523, normalized size of antiderivative = 4.47 \[ \int \frac {(c+d x)^{5/2}}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*b^2*e*x^2 + a^2*b*e)*sqrt(-(a^3*b^3*e*sqrt(-(81*b^2*c^10 + 90*a*b* 
c^8*d^2 + 25*a^2*c^6*d^4)/(a^7*b^3*e^2)) + 15*b^2*c^4*d + 15*a*b*c^2*d^3 + 
 4*a^2*d^5)/(a^3*b^3*e))*log(((81*b^3*c^9 + 162*a*b^2*c^7*d^2 + 101*a^2*b* 
c^5*d^4 + 20*a^3*c^3*d^6)*sqrt(d*x + c)*sqrt(e*x) + ((3*a^5*b^4*c^2 + 2*a^ 
6*b^3*d^2)*e^2*x*sqrt(-(81*b^2*c^10 + 90*a*b*c^8*d^2 + 25*a^2*c^6*d^4)/(a^ 
7*b^3*e^2)) - (9*a^2*b^3*c^6*d + 5*a^3*b^2*c^4*d^3)*e*x)*sqrt(-(a^3*b^3*e* 
sqrt(-(81*b^2*c^10 + 90*a*b*c^8*d^2 + 25*a^2*c^6*d^4)/(a^7*b^3*e^2)) + 15* 
b^2*c^4*d + 15*a*b*c^2*d^3 + 4*a^2*d^5)/(a^3*b^3*e)))/x) - (a*b^2*e*x^2 + 
a^2*b*e)*sqrt(-(a^3*b^3*e*sqrt(-(81*b^2*c^10 + 90*a*b*c^8*d^2 + 25*a^2*c^6 
*d^4)/(a^7*b^3*e^2)) + 15*b^2*c^4*d + 15*a*b*c^2*d^3 + 4*a^2*d^5)/(a^3*b^3 
*e))*log(((81*b^3*c^9 + 162*a*b^2*c^7*d^2 + 101*a^2*b*c^5*d^4 + 20*a^3*c^3 
*d^6)*sqrt(d*x + c)*sqrt(e*x) - ((3*a^5*b^4*c^2 + 2*a^6*b^3*d^2)*e^2*x*sqr 
t(-(81*b^2*c^10 + 90*a*b*c^8*d^2 + 25*a^2*c^6*d^4)/(a^7*b^3*e^2)) - (9*a^2 
*b^3*c^6*d + 5*a^3*b^2*c^4*d^3)*e*x)*sqrt(-(a^3*b^3*e*sqrt(-(81*b^2*c^10 + 
 90*a*b*c^8*d^2 + 25*a^2*c^6*d^4)/(a^7*b^3*e^2)) + 15*b^2*c^4*d + 15*a*b*c 
^2*d^3 + 4*a^2*d^5)/(a^3*b^3*e)))/x) - (a*b^2*e*x^2 + a^2*b*e)*sqrt((a^3*b 
^3*e*sqrt(-(81*b^2*c^10 + 90*a*b*c^8*d^2 + 25*a^2*c^6*d^4)/(a^7*b^3*e^2)) 
- 15*b^2*c^4*d - 15*a*b*c^2*d^3 - 4*a^2*d^5)/(a^3*b^3*e))*log(((81*b^3*c^9 
 + 162*a*b^2*c^7*d^2 + 101*a^2*b*c^5*d^4 + 20*a^3*c^3*d^6)*sqrt(d*x + c)*s 
qrt(e*x) + ((3*a^5*b^4*c^2 + 2*a^6*b^3*d^2)*e^2*x*sqrt(-(81*b^2*c^10 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((d*x+c)^(5/2)/(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)*c**2 + int((sqrt(c + d*x)*x**2)/(sqrt(x)*a**2 + 2*sqrt(x)*a*b*x**2 + s 
qrt(x)*b**2*x**4),x)*d**2 + 2*int((sqrt(c + d*x)*x)/(sqrt(x)*a**2 + 2*sqrt 
(x)*a*b*x**2 + sqrt(x)*b**2*x**4),x)*c*d)/sqrt(e)