\(\int \frac {\sqrt {e x} (c+d x)^{5/2}}{(a+b x^2)^2} \, dx\) [914]

Optimal result

Integrand size = 26, antiderivative size = 418 \[ \int \frac {\sqrt {e x} (c+d x)^{5/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {c d \sqrt {e x} \sqrt {c+d x}}{a b}-\frac {d^2 (e x)^{3/2} \sqrt {c+d x}}{2 a b e}+\frac {(e x)^{3/2} (c+d x)^{5/2}}{2 a e \left (a+b x^2\right )}-\frac {\left (b^{3/2} c^3+2 \sqrt {-a} b c^2 d+7 a \sqrt {b} c d^2+4 (-a)^{3/2} d^3\right ) \sqrt {e} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{5/4} b^2 \sqrt {\sqrt {b} c-\sqrt {-a} d}}+\frac {2 d^{5/2} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b^2}+\frac {\left (b^{3/2} c^3-2 \sqrt {-a} b c^2 d+7 a \sqrt {b} c d^2+4 \sqrt {-a} a d^3\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{5/4} b^2 \sqrt {\sqrt {b} c+\sqrt {-a} d}} \] Output:

-c*d*(e*x)^(1/2)*(d*x+c)^(1/2)/a/b-1/2*d^2*(e*x)^(3/2)*(d*x+c)^(1/2)/a/b/e 
+1/2*(e*x)^(3/2)*(d*x+c)^(5/2)/a/e/(b*x^2+a)-1/4*(b^(3/2)*c^3+2*(-a)^(1/2) 
*b*c^2*d+7*a*b^(1/2)*c*d^2+4*(-a)^(3/2)*d^3)*e^(1/2)*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)^(5/4)/ 
b^2/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)+2*d^(5/2)*e^(1/2)*arctanh(d^(1/2)*(e*x) 
^(1/2)/e^(1/2)/(d*x+c)^(1/2))/b^2+1/4*(b^(3/2)*c^3-2*(-a)^(1/2)*b*c^2*d+7* 
a*b^(1/2)*c*d^2+4*(-a)^(1/2)*a*d^3)*e^(1/2)*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)^(5/4)/b^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 = 0.85 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {e x} (c+d x)^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (\frac {2 b \sqrt {x} \sqrt {c+d x} \left (b c^2 x-a d (2 c+d x)\right )}{a \left (a+b x^2\right )}-8 d^{5/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )+4 d^{5/2} \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 {19 b c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-48 a c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+6 b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-8 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+3 b c \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 ]+\frac {\sqrt {d} \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^2 c^5 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-69 a b c^3 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+192 a^2 c d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b^2 c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-30 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+16 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2-5 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 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 ]}{a}\right )}{4 b^2 \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*((2*b*Sqrt[x]*Sqrt[c + d*x]*(b*c^2*x - a*d*(2*c + d*x)))/(a*(a 
+ b*x^2)) - 8*d^(5/2)*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d*x]] + 4*d^(5/2)* 
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 & , (19*b*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] 
- 48*a*c*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 6*b*c 
^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 8*a*d^2*Log[ 
c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 3*b*c*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) & ] + (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 & , (b^2*c^5*Log[c + 2*d 
*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 69*a*b*c^3*d^2*Log[c + 2*d*x 
- 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 192*a^2*c*d^4*Log[c + 2*d*x - 2* 
Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b^2*c^4*Log[c + 2*d*x - 2*Sqrt[d]* 
Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 30*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]* 
Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 16*a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt 
[x]*Sqrt[c + d*x] - #1]*#1 + b^2*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqr 
t[c + d*x] - #1]*#1^2 - 5*a*b*c*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) & ])/a))/(4*b^2*Sqrt[x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1121\) vs. \(2(418)=836\).

Time = 4.07 (sec) , antiderivative size = 1121, normalized size of antiderivative = 2.68, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {e x} (c+d x)^{5/2}}{4 a \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {e x} (c+d x)^{5/2}}{4 a \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {d \sqrt {e x} (c+d x)^{3/2}}{2 a b}+\frac {7 c d \sqrt {e x} \sqrt {c+d x}}{8 a b}-\frac {d \left (11 \sqrt {b} c-12 \sqrt {-a} d\right ) \sqrt {e x} \sqrt {c+d x}}{16 a b^{3/2}}-\frac {d \left (11 \sqrt {b} c+12 \sqrt {-a} d\right ) \sqrt {e x} \sqrt {c+d x}}{16 a b^{3/2}}-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \sqrt {e} \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)^{5/4} b^2}+\frac {\left (b^{3/2} c^3-8 \sqrt {-a} b d c^2-13 a \sqrt {b} d^2 c+6 \sqrt {-a} a d^3\right ) \sqrt {e} \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)^{5/4} b^2 \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\sqrt {d} \left (15 b c^2-40 \sqrt {-a} \sqrt {b} d c-24 a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{16 a b^2}-\frac {\sqrt {d} \left (15 b c^2+40 \sqrt {-a} \sqrt {b} d c-24 a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{16 a b^2}+\frac {\sqrt {d} \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{2 a b^2}+\frac {\sqrt {d} \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{2 a b^2}+\frac {7 c^2 \sqrt {d} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{8 a b}+\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \sqrt {e} \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)^{5/4} b^2}-\frac {\left (b^{3/2} c^3+8 \sqrt {-a} b d c^2-13 a \sqrt {b} d^2 c+6 (-a)^{3/2} d^3\right ) \sqrt {e} \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)^{5/4} b^2 \sqrt {\sqrt {b} c+\sqrt {-a} d}}\)

Input:

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

Output:

(7*c*d*Sqrt[e*x]*Sqrt[c + d*x])/(8*a*b) - (d*(11*Sqrt[b]*c - 12*Sqrt[-a]*d 
)*Sqrt[e*x]*Sqrt[c + d*x])/(16*a*b^(3/2)) - (d*(11*Sqrt[b]*c + 12*Sqrt[-a] 
*d)*Sqrt[e*x]*Sqrt[c + d*x])/(16*a*b^(3/2)) - (d*Sqrt[e*x]*(c + d*x)^(3/2) 
)/(2*a*b) - (Sqrt[e*x]*(c + d*x)^(5/2))/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x 
)) + (Sqrt[e*x]*(c + d*x)^(5/2))/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (S 
qrt[Sqrt[b]*c - Sqrt[-a]*d]*(b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*Sqrt[ 
e]*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqr 
t[c + d*x])])/(2*(-a)^(5/4)*b^2) + ((b^(3/2)*c^3 - 8*Sqrt[-a]*b*c^2*d - 13 
*a*Sqrt[b]*c*d^2 + 6*Sqrt[-a]*a*d^3)*Sqrt[e]*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt 
[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4*(-a)^(5/4)*b^2* 
Sqrt[Sqrt[b]*c - Sqrt[-a]*d]) + (7*c^2*Sqrt[d]*Sqrt[e]*ArcTanh[(Sqrt[d]*Sq 
rt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(8*a*b) - (Sqrt[d]*(15*b*c^2 - 40*Sqrt[ 
-a]*Sqrt[b]*c*d - 24*a*d^2)*Sqrt[e]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*S 
qrt[c + d*x])])/(16*a*b^2) - (Sqrt[d]*(15*b*c^2 + 40*Sqrt[-a]*Sqrt[b]*c*d 
- 24*a*d^2)*Sqrt[e]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/ 
(16*a*b^2) + (Sqrt[d]*(b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*Arc 
Tanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(2*a*b^2) + (Sqrt[d]*(b 
*c^2 + 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*ArcTanh[(Sqrt[d]*Sqrt[e*x]) 
/(Sqrt[e]*Sqrt[c + d*x])])/(2*a*b^2) + (Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*(b*c^ 
2 + 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*ArcTanh[(Sqrt[Sqrt[b]*c + S...
 

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. \(3558\) vs. \(2(322)=644\).

Time = 0.55 (sec) , antiderivative size = 3559, normalized size of antiderivative = 8.51

method	result	size
default	\(\text {Expression too large to display}\)	\(3559\)

Input:

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

Output:

1/8*(e*x)^(1/2)*(d*x+c)^(1/2)*(-7*(d*e)^(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)))*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*a^3*(-a*b)^( 
1/2)*c*d^4*e-8*(d*e)^(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)*a^2*b*(-a*b)^(1/2)*c*d^3-4*(d*e)^ 
(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)*a^2*b*(-a*b)^(1/2)*d^4*x+2*(d*e)^(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^2*d^3*e-7*(d*e)^(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*(-a*b)^(1/2)*c*d^4*e-2*(d 
*e)^(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^2*b^2*c^4*d*e+4*(d*e)^(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*d^5*e*x^2-(d*e)^(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*(-a*b)^(1/2)*c^5*e*x^2-(d*e)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1961 vs. \(2 (322) = 644\).

Time = 2.66 (sec) , antiderivative size = 3921, normalized size of antiderivative = 9.38 \[ \int \frac {\sqrt {e x} (c+d x)^{5/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1525 vs. \(2 (322) = 644\).

Time = 1.51 (sec) , antiderivative size = 1525, normalized size of antiderivative = 3.65 \[ \int \frac {\sqrt {e x} (c+d x)^{5/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-sqrt(d*e)*d*abs(d)*log((sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c* 
d*e))^2)/b^2 + (sqrt(d*e)*b^2*c^6*d^3*e^4*abs(d) - sqrt(d*e)*a*b*c^4*d^5*e 
^4*abs(d) - 3*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c* 
d*e))^2*b^2*c^5*d^2*e^3*abs(d) - sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt 
((d*x + c)*d*e - c*d*e))^2*a*b*c^3*d^4*e^3*abs(d) + 3*sqrt(d*e)*(sqrt(d*e) 
*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b^2*c^4*d*e^2*abs(d) + 5*s 
qrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a*b*c^2 
*d^3*e^2*abs(d) - 8*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d* 
e - c*d*e))^4*a^2*d^5*e^2*abs(d) - sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sq 
rt((d*x + c)*d*e - c*d*e))^6*b^2*c^3*e*abs(d) + 5*sqrt(d*e)*(sqrt(d*e)*sqr 
t(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^6*a*b*c*d^2*e*abs(d))/((b*c^4*d^ 
4*e^4 - 4*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b*c^3* 
d^3*e^3 + 6*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b*c^ 
2*d^2*e^2 + 16*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a 
*d^4*e^2 - 4*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^6*b*c 
*d*e + (sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^8*b)*a*b^2) 
 + 1/8*(sqrt(d*e)*b^3*c^8*d^4*e^4*abs(d) + 11*sqrt(d*e)*a*b^2*c^6*d^6*e^4* 
abs(d) - 8*sqrt(d*e)*a^2*b*c^4*d^8*e^4*abs(d) - 4*sqrt(d*e)*(sqrt(d*e)*sqr 
t(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b^3*c^6*d^2*e^2*abs(d) - 20*sq 
rt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a*b^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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