\(\int \frac {(e x)^{5/2}}{(c+d x)^3 (a+b x^2)^2} \, dx\) [427]

Optimal result

Integrand size = 24, antiderivative size = 671 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^2 d e^2 \sqrt {e x}}{2 \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {c d \left (7 b c^2-9 a d^2\right ) e^2 \sqrt {e x}}{4 \left (b c^2+a d^2\right )^3 (c+d x)}+\frac {e^2 \sqrt {e x} \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{2 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )}-\frac {3 \sqrt {c} \sqrt {d} \left (5 b^2 c^4-22 a b c^2 d^2+5 a^2 d^4\right ) e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 \left (b c^2+a d^2\right )^4}-\frac {3 \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b^2 c^4+4 \sqrt {a} b^{3/2} c^3 d-14 a b c^2 d^2+4 a^{3/2} \sqrt {b} c d^3+a^2 d^4\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}+\frac {3 \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b^2 c^4+4 \sqrt {a} b^{3/2} c^3 d-14 a b c^2 d^2+4 a^{3/2} \sqrt {b} c d^3+a^2 d^4\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {3 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b^2 c^4-4 \sqrt {a} b^{3/2} c^3 d-14 a b c^2 d^2-4 a^{3/2} \sqrt {b} c d^3+a^2 d^4\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^4} \] Output:

-1/2*c^2*d*e^2*(e*x)^(1/2)/(a*d^2+b*c^2)^2/(d*x+c)^2-1/4*c*d*(-9*a*d^2+7*b 
*c^2)*e^2*(e*x)^(1/2)/(a*d^2+b*c^2)^3/(d*x+c)+1/2*e^2*(e*x)^(1/2)*(a^2*d^3 
-b^2*c^3*x-3*a*b*c*d*(-d*x+c))/(a*d^2+b*c^2)^3/(b*x^2+a)-3/4*c^(1/2)*d^(1/ 
2)*(5*a^2*d^4-22*a*b*c^2*d^2+5*b^2*c^4)*e^(5/2)*arctan(d^(1/2)*(e*x)^(1/2) 
/c^(1/2)/e^(1/2))/(a*d^2+b*c^2)^4-3/8*(b^(1/2)*c+a^(1/2)*d)*(b^2*c^4+4*a^( 
1/2)*b^(3/2)*c^3*d-14*a*b*c^2*d^2+4*a^(3/2)*b^(1/2)*c*d^3+a^2*d^4)*e^(5/2) 
*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(1/4)/b^( 
1/4)/(a*d^2+b*c^2)^4+3/8*(b^(1/2)*c+a^(1/2)*d)*(b^2*c^4+4*a^(1/2)*b^(3/2)* 
c^3*d-14*a*b*c^2*d^2+4*a^(3/2)*b^(1/2)*c*d^3+a^2*d^4)*e^(5/2)*arctan(1+2^( 
1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(1/4)/b^(1/4)/(a*d^2+b 
*c^2)^4-3/8*(b^(1/2)*c-a^(1/2)*d)*(b^2*c^4-4*a^(1/2)*b^(3/2)*c^3*d-14*a*b* 
c^2*d^2-4*a^(3/2)*b^(1/2)*c*d^3+a^2*d^4)*e^(5/2)*arctanh(2^(1/2)*a^(1/4)*b 
^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(1/4)/b^(1/4)/(a 
*d^2+b*c^2)^4
 

Mathematica [A] (verified)

Time = 2.37 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.70 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {(e x)^{5/2} \left (-\frac {2 \left (b c^2+a d^2\right ) \sqrt {x} \left (-a^2 d^3 \left (9 c^2+13 c d x+2 d^2 x^2\right )+b^2 c^3 x \left (2 c^2+13 c d x+9 d^2 x^2\right )+a b c d \left (15 c^3+13 c^2 d x-13 c d^2 x^2-15 d^3 x^3\right )\right )}{(c+d x)^2 \left (a+b x^2\right )}-6 \sqrt {c} \sqrt {d} \left (5 b^2 c^4-22 a b c^2 d^2+5 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )-\frac {3 \sqrt {2} \left (b^{5/2} c^5+5 \sqrt {a} b^2 c^4 d-10 a b^{3/2} c^3 d^2-10 a^{3/2} b c^2 d^3+5 a^2 \sqrt {b} c d^4+a^{5/2} d^5\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a} \sqrt [4]{b}}+\frac {3 \sqrt {2} \left (-b^{5/2} c^5+5 \sqrt {a} b^2 c^4 d+10 a b^{3/2} c^3 d^2-10 a^{3/2} b c^2 d^3-5 a^2 \sqrt {b} c d^4+a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a} \sqrt [4]{b}}\right )}{8 \left (b c^2+a d^2\right )^4 x^{5/2}} \] Input:

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

Output:

((e*x)^(5/2)*((-2*(b*c^2 + a*d^2)*Sqrt[x]*(-(a^2*d^3*(9*c^2 + 13*c*d*x + 2 
*d^2*x^2)) + b^2*c^3*x*(2*c^2 + 13*c*d*x + 9*d^2*x^2) + a*b*c*d*(15*c^3 + 
13*c^2*d*x - 13*c*d^2*x^2 - 15*d^3*x^3)))/((c + d*x)^2*(a + b*x^2)) - 6*Sq 
rt[c]*Sqrt[d]*(5*b^2*c^4 - 22*a*b*c^2*d^2 + 5*a^2*d^4)*ArcTan[(Sqrt[d]*Sqr 
t[x])/Sqrt[c]] - (3*Sqrt[2]*(b^(5/2)*c^5 + 5*Sqrt[a]*b^2*c^4*d - 10*a*b^(3 
/2)*c^3*d^2 - 10*a^(3/2)*b*c^2*d^3 + 5*a^2*Sqrt[b]*c*d^4 + a^(5/2)*d^5)*Ar 
cTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(1/4)*b^ 
(1/4)) + (3*Sqrt[2]*(-(b^(5/2)*c^5) + 5*Sqrt[a]*b^2*c^4*d + 10*a*b^(3/2)*c 
^3*d^2 - 10*a^(3/2)*b*c^2*d^3 - 5*a^2*Sqrt[b]*c*d^4 + a^(5/2)*d^5)*ArcTanh 
[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(1/4)*b^(1/4 
))))/(8*(b*c^2 + a*d^2)^4*x^(5/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1623\) vs. \(2(671)=1342\).

Time = 3.83 (sec) , antiderivative size = 1623, normalized size of antiderivative = 2.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 a \left (5 b c^2-a d^2\right ) e^2 \sqrt {e x} d^3}{\left (b c^2+a d^2\right )^4}-\frac {4 b c (e x)^{5/2} d^3}{\left (b c^2+a d^2\right )^3 (c+d x)}-\frac {(e x)^{5/2} d^3}{2 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {20 b c e (e x)^{3/2} d^2}{3 \left (b c^2+a d^2\right )^3}+\frac {4 b c \left (b c^2-a d^2\right ) e (e x)^{3/2} d^2}{\left (b c^2+a d^2\right )^4}-\frac {4 b c \left (5 b c^2-a d^2\right ) e (e x)^{3/2} d^2}{3 \left (b c^2+a d^2\right )^4}+\frac {\sqrt {2} a^{3/4} \left (3 b^{3/2} c^3+5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{\sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {\sqrt {2} a^{3/4} \left (3 b^{3/2} c^3+5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c-a^{3/2} d^3\right ) e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{\sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {a^{3/4} \left (3 b^{3/2} c^3-5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) e^{5/2} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}+\frac {a^{3/4} \left (3 b^{3/2} c^3-5 \sqrt {a} b d c^2-3 a \sqrt {b} d^2 c+a^{3/2} d^3\right ) e^{5/2} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^4}-\frac {5 e (e x)^{3/2} d^2}{4 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {15 e^2 \sqrt {e x} d}{4 \left (b c^2+a d^2\right )^2}-\frac {20 b c^2 e^2 \sqrt {e x} d}{\left (b c^2+a d^2\right )^3}-\frac {5 \left (3 b c^2-a d^2\right ) e^2 \sqrt {e x} d}{2 \left (b c^2+a d^2\right )^3}+\frac {4 b c^2 \left (5 b c^2-a d^2\right ) e^2 \sqrt {e x} d}{\left (b c^2+a d^2\right )^4}-\frac {15 \sqrt {c} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) \sqrt {d}}{4 \left (b c^2+a d^2\right )^2}+\frac {20 b c^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) \sqrt {d}}{\left (b c^2+a d^2\right )^3}-\frac {4 b c^{5/2} \left (5 b c^2-a d^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) \sqrt {d}}{\left (b c^2+a d^2\right )^4}-\frac {\left (3 b^{3/2} c^3+15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (3 b^{3/2} c^3+15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c-5 a^{3/2} d^3\right ) e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (3 b^{3/2} c^3-15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{8 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\left (3 b^{3/2} c^3-15 \sqrt {a} b d c^2-9 a \sqrt {b} d^2 c+5 a^{3/2} d^3\right ) e^{5/2} \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{8 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {b e (e x)^{3/2} \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{2 \left (b c^2+a d^2\right )^3 \left (b x^2+a\right )}\)

Input:

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

Output:

(4*b*c^2*d*(5*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^2 + a*d^2)^4 + (4*a*d^3*( 
5*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^2 + a*d^2)^4 - (20*b*c^2*d*e^2*Sqrt[e 
*x])/(b*c^2 + a*d^2)^3 - (5*d*(3*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(2*(b*c^2 + 
 a*d^2)^3) + (15*d*e^2*Sqrt[e*x])/(4*(b*c^2 + a*d^2)^2) + (4*b*c*d^2*(b*c^ 
2 - a*d^2)*e*(e*x)^(3/2))/(b*c^2 + a*d^2)^4 - (4*b*c*d^2*(5*b*c^2 - a*d^2) 
*e*(e*x)^(3/2))/(3*(b*c^2 + a*d^2)^4) + (20*b*c*d^2*e*(e*x)^(3/2))/(3*(b*c 
^2 + a*d^2)^3) - (d^3*(e*x)^(5/2))/(2*(b*c^2 + a*d^2)^2*(c + d*x)^2) - (5* 
d^2*e*(e*x)^(3/2))/(4*(b*c^2 + a*d^2)^2*(c + d*x)) - (4*b*c*d^3*(e*x)^(5/2 
))/((b*c^2 + a*d^2)^3*(c + d*x)) - (b*e*(e*x)^(3/2)*(c*(b*c^2 - 3*a*d^2) - 
 d*(3*b*c^2 - a*d^2)*x))/(2*(b*c^2 + a*d^2)^3*(a + b*x^2)) - (4*b*c^(5/2)* 
Sqrt[d]*(5*b*c^2 - a*d^2)*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt 
[e])])/(b*c^2 + a*d^2)^4 + (20*b*c^(5/2)*Sqrt[d]*e^(5/2)*ArcTan[(Sqrt[d]*S 
qrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^3 - (15*Sqrt[c]*Sqrt[d]*e^(5 
/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(4*(b*c^2 + a*d^2)^2) - 
 ((3*b^(3/2)*c^3 + 15*Sqrt[a]*b*c^2*d - 9*a*Sqrt[b]*c*d^2 - 5*a^(3/2)*d^3) 
*e^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqr 
t[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)^3) + (Sqrt[2]*a^(3/4)*d^2*(3*b^(3/2)* 
c^3 + 5*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*e^(5/2)*ArcTan[ 
1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(b^(1/4)*(b*c^2 + a*d^ 
2)^4) + ((3*b^(3/2)*c^3 + 15*Sqrt[a]*b*c^2*d - 9*a*Sqrt[b]*c*d^2 - 5*a^...
 

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 [A] (verified)

Time = 1.67 (sec) , antiderivative size = 614, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(2 e^{6} \left (\frac {\frac {\left (\frac {3}{4} a^{2} c \,d^{4} b +\frac {1}{2} a \,b^{2} c^{3} d^{2}-\frac {1}{4} b^{3} c^{5}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{5} e \,a^{3}-\frac {1}{2} a^{2} c^{2} e \,d^{3} b -\frac {3}{4} a \,c^{4} e d \,b^{2}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {3 \left (d^{5} e \,a^{3}-10 a^{2} c^{2} e \,d^{3} b +5 a \,c^{4} e d \,b^{2}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {3 \left (5 a^{2} c \,d^{4} b -10 a \,b^{2} c^{3} d^{2}+b^{3} c^{5}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4}}-\frac {c d \left (\frac {\left (-\frac {9}{8} a^{2} d^{5}-\frac {1}{4} d^{3} a \,c^{2} b +\frac {7}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}-2 b \,c^{2} d^{2} a -9 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (5 a^{2} d^{4}-22 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(614\)
default	\(2 e^{6} \left (\frac {\frac {\left (\frac {3}{4} a^{2} c \,d^{4} b +\frac {1}{2} a \,b^{2} c^{3} d^{2}-\frac {1}{4} b^{3} c^{5}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{5} e \,a^{3}-\frac {1}{2} a^{2} c^{2} e \,d^{3} b -\frac {3}{4} a \,c^{4} e d \,b^{2}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {3 \left (d^{5} e \,a^{3}-10 a^{2} c^{2} e \,d^{3} b +5 a \,c^{4} e d \,b^{2}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {3 \left (5 a^{2} c \,d^{4} b -10 a \,b^{2} c^{3} d^{2}+b^{3} c^{5}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4}}-\frac {c d \left (\frac {\left (-\frac {9}{8} a^{2} d^{5}-\frac {1}{4} d^{3} a \,c^{2} b +\frac {7}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}-2 b \,c^{2} d^{2} a -9 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {3 \left (5 a^{2} d^{4}-22 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(614\)
pseudoelliptic	\(\frac {15 e \left (\frac {d \left (a^{2} d^{4}-10 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \left (d x +c \right )^{2} e \sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right ) \sqrt {\frac {a \,e^{2}}{b}}}{5}+\frac {12 \left (\sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \left (-\frac {2 b^{2} c^{4}}{9}-\frac {4 b^{2} c^{3} d x}{9}-\frac {b \,d^{2} \left (9 b \,x^{2}+a \right ) c^{2}}{9}+\frac {4 a b c \,d^{3} x}{3}+a \,d^{4} \left (\frac {5 b \,x^{2}}{3}+a \right )\right ) c \left (e x \right )^{\frac {3}{2}}+d e \left (\left (\frac {2 a^{2} d^{4} x^{2}}{9}+\frac {4 a^{2} c x \,d^{3}}{9}+a \,d^{2} \left (\frac {b \,x^{2}}{9}+a \right ) c^{2}-\frac {4 a b \,c^{3} d x}{3}-\frac {5 b \left (\frac {3 b \,x^{2}}{5}+a \right ) c^{4}}{3}\right ) \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {e x}-\frac {5 \left (d x +c \right )^{2} e \left (b \,x^{2}+a \right ) c \left (a^{2} d^{4}-\frac {22}{5} b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{3}\right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{5}+\left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \left (d x +c \right )^{2} \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +\frac {1}{5} b^{2} c^{4}\right ) e^{2} \sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right ) c \right )}{16 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right )^{4} \left (d x +c \right )^{2} \left (b \,x^{2}+a \right )}\)	\(647\)

Input:

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

Output:

2*e^6*(1/e^3/(a*d^2+b*c^2)^4*(((3/4*a^2*c*d^4*b+1/2*a*b^2*c^3*d^2-1/4*b^3* 
c^5)*(e*x)^(3/2)+(1/4*d^5*e*a^3-1/2*a^2*c^2*e*d^3*b-3/4*a*c^4*e*d*b^2)*(e* 
x)^(1/2))/(b*e^2*x^2+a*e^2)+3/32*(a^3*d^5*e-10*a^2*b*c^2*d^3*e+5*a*b^2*c^4 
*d*e)*(a*e^2/b)^(1/4)/a/e^2*2^(1/2)*(ln((e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2 
^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b) 
^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/ 
(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+3/32*(5*a^2*b*c*d^4-10*a*b^2*c^3*d^2+b^3*c 
^5)/b/(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2) 
+(a*e^2/b)^(1/2))/(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2) 
))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2 
/b)^(1/4)*(e*x)^(1/2)-1)))-c*d/e^3/(a*d^2+b*c^2)^4*(((-9/8*a^2*d^5-1/4*d^3 
*a*c^2*b+7/8*b^2*c^4*d)*(e*x)^(3/2)-1/8*c*e*(7*a^2*d^4-2*a*b*c^2*d^2-9*b^2 
*c^4)*(e*x)^(1/2))/(d*e*x+c*e)^2+3/8*(5*a^2*d^4-22*a*b*c^2*d^2+5*b^2*c^4)/ 
(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1335 vs. \(2 (544) = 1088\).

Time = 0.26 (sec) , antiderivative size = 1335, normalized size of antiderivative = 1.99 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*e^2*(6*(5*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e - 10*(a*b^3*e^2)^(1/4)*a^2*b 
^2*c^2*d^3*e + (a*b^3*e^2)^(1/4)*a^3*b*d^5*e + (a*b^3*e^2)^(3/4)*b^2*c^5 - 
 10*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 + 5*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*arctan( 
1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt 
(2)*a*b^6*c^8*e + 4*sqrt(2)*a^2*b^5*c^6*d^2*e + 6*sqrt(2)*a^3*b^4*c^4*d^4* 
e + 4*sqrt(2)*a^4*b^3*c^2*d^6*e + sqrt(2)*a^5*b^2*d^8*e) + 6*(5*(a*b^3*e^2 
)^(1/4)*a*b^3*c^4*d*e - 10*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e + (a*b^3*e^ 
2)^(1/4)*a^3*b*d^5*e + (a*b^3*e^2)^(3/4)*b^2*c^5 - 10*(a*b^3*e^2)^(3/4)*a* 
b*c^3*d^2 + 5*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a 
*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a*b^6*c^8*e + 4*sqr 
t(2)*a^2*b^5*c^6*d^2*e + 6*sqrt(2)*a^3*b^4*c^4*d^4*e + 4*sqrt(2)*a^4*b^3*c 
^2*d^6*e + sqrt(2)*a^5*b^2*d^8*e) + 3*(5*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e - 
 10*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e + (a*b^3*e^2)^(1/4)*a^3*b*d^5*e - 
(a*b^3*e^2)^(3/4)*b^2*c^5 + 10*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 - 5*(a*b^3*e^ 
2)^(3/4)*a^2*c*d^4)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e 
^2/b))/(sqrt(2)*a*b^6*c^8*e + 4*sqrt(2)*a^2*b^5*c^6*d^2*e + 6*sqrt(2)*a^3* 
b^4*c^4*d^4*e + 4*sqrt(2)*a^4*b^3*c^2*d^6*e + sqrt(2)*a^5*b^2*d^8*e) - 3*( 
5*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e - 10*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e 
 + (a*b^3*e^2)^(1/4)*a^3*b*d^5*e - (a*b^3*e^2)^(3/4)*b^2*c^5 + 10*(a*b^3*e 
^2)^(3/4)*a*b*c^3*d^2 - 5*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*log(e*x - sqrt(2...
 

Mupad [B] (verification not implemented)

Time = 14.31 (sec) , antiderivative size = 35748, normalized size of antiderivative = 53.28 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- (((e*x)^(3/2)*(2*b^2*c^5*e^5 - 13*a^2*c*d^4*e^5 + 13*a*b*c^3*d^2*e^5))/( 
4*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - (3*(e*x)^(1/2)*(3 
*a^2*c^2*d^3*e^6 - 5*a*b*c^4*d*e^6))/(4*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 
 + 2*a*b*c^2*d^2)) + (3*b*(3*b*c^3*d^2*e^3 - 5*a*c*d^4*e^3)*(e*x)^(7/2))/( 
4*(a*d^2 + b*c^2)*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)) - (e*(e*x)^(5/2)*(2 
*a^2*d^5*e^3 - 13*b^2*c^4*d*e^3 + 13*a*b*c^2*d^3*e^3))/(4*(a*d^2 + b*c^2)* 
(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2)))/(a*c^2*e^4 + e^2*x^2*(a*d^2*e^2 + b* 
c^2*e^2) + b*d^2*e^4*x^4 + 2*b*c*d*e^4*x^3 + 2*a*c*d*e^4*x) - (atan(((((3* 
(e*x)^(1/2)*(162*a^10*b^3*d^21*e^20 + 74682*a^2*b^11*c^16*d^5*e^20 - 68266 
8*a^3*b^10*c^14*d^7*e^20 + 2537568*a^4*b^9*c^12*d^9*e^20 - 4020678*a^5*b^8 
*c^10*d^11*e^20 + 2544372*a^6*b^7*c^8*d^13*e^20 - 674892*a^7*b^6*c^6*d^15* 
e^20 + 79056*a^8*b^5*c^4*d^17*e^20 - 729*a^9*b^4*c^2*d^19*e^20 - 2025*a*b^ 
12*c^18*d^3*e^20))/(128*(a^12*d^24 + b^12*c^24 + 12*a*b^11*c^22*d^2 + 12*a 
^11*b*c^2*d^22 + 66*a^2*b^10*c^20*d^4 + 220*a^3*b^9*c^18*d^6 + 495*a^4*b^8 
*c^16*d^8 + 792*a^5*b^7*c^14*d^10 + 924*a^6*b^6*c^12*d^12 + 792*a^7*b^5*c^ 
10*d^14 + 495*a^8*b^4*c^8*d^16 + 220*a^9*b^3*c^6*d^18 + 66*a^10*b^2*c^4*d^ 
20)) + (3*((3*(2430*a^2*b^12*c^18*d^4*e^18 - 27486*a^3*b^11*c^16*d^6*e^18 
+ 86940*a^4*b^10*c^14*d^8*e^18 - 6615*a^5*b^9*c^12*d^10*e^18 - 228096*a^6* 
b^8*c^10*d^12*e^18 + 26514*a^7*b^7*c^8*d^14*e^18 + 96660*a^8*b^6*c^6*d^16* 
e^18 - (64341*a^9*b^5*c^4*d^18*e^18)/2 + 2322*a^10*b^4*c^2*d^20*e^18 + ...
 

Reduce [F]

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

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

Output:

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