\(\int \frac {(e x)^{3/2}}{(c+d x)^2 (a+b x^2)^2} \, dx\) [422]

Optimal result

Integrand size = 24, antiderivative size = 542 \[ \int \frac {(e x)^{3/2}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {c d^2 e \sqrt {e x}}{\left (b c^2+a d^2\right )^2 (c+d x)}+\frac {e \sqrt {e x} \left (a d^2-b c (c-2 d x)\right )}{2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}+\frac {\sqrt {c} d^{3/2} \left (5 b c^2-3 a d^2\right ) e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\left (b c^2+a d^2\right )^3}-\frac {\left (b^2 c^4-6 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2+10 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (b^2 c^4-6 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2+10 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\left (b^2 c^4+6 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2-10 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^3} \] Output:

c*d^2*e*(e*x)^(1/2)/(a*d^2+b*c^2)^2/(d*x+c)+1/2*e*(e*x)^(1/2)*(a*d^2-b*c*( 
-2*d*x+c))/(a*d^2+b*c^2)^2/(b*x^2+a)+c^(1/2)*d^(3/2)*(-3*a*d^2+5*b*c^2)*e^ 
(3/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/(a*d^2+b*c^2)^3-1/8*(b^2 
*c^4-6*a^(1/2)*b^(3/2)*c^3*d-12*a*b*c^2*d^2+10*a^(3/2)*b^(1/2)*c*d^3+3*a^2 
*d^4)*e^(3/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2 
)/a^(3/4)/b^(1/4)/(a*d^2+b*c^2)^3+1/8*(b^2*c^4-6*a^(1/2)*b^(3/2)*c^3*d-12* 
a*b*c^2*d^2+10*a^(3/2)*b^(1/2)*c*d^3+3*a^2*d^4)*e^(3/2)*arctan(1+2^(1/2)*b 
^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(3/4)/b^(1/4)/(a*d^2+b*c^2)^ 
3+1/8*(b^2*c^4+6*a^(1/2)*b^(3/2)*c^3*d-12*a*b*c^2*d^2-10*a^(3/2)*b^(1/2)*c 
*d^3+3*a^2*d^4)*e^(3/2)*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^(3/4)/b^(1/4)/(a*d^2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.67 \[ \int \frac {(e x)^{3/2}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \left (\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (a d^2 (3 c+d x)+b c \left (-c^2+c d x+4 d^2 x^2\right )\right )}{(c+d x) \left (a+b x^2\right )}+8 \sqrt {c} d^{3/2} \left (5 b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )-\frac {\sqrt {2} \left (b^2 c^4-6 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2+10 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4} \sqrt [4]{b}}+\frac {\sqrt {2} \left (b^2 c^4+6 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2-10 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4} \sqrt [4]{b}}\right )}{8 \left (b c^2+a d^2\right )^3 x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*((4*(b*c^2 + a*d^2)*Sqrt[x]*(a*d^2*(3*c + d*x) + b*c*(-c^2 + 
c*d*x + 4*d^2*x^2)))/((c + d*x)*(a + b*x^2)) + 8*Sqrt[c]*d^(3/2)*(5*b*c^2 
- 3*a*d^2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] - (Sqrt[2]*(b^2*c^4 - 6*Sqrt[ 
a]*b^(3/2)*c^3*d - 12*a*b*c^2*d^2 + 10*a^(3/2)*Sqrt[b]*c*d^3 + 3*a^2*d^4)* 
ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(3/4)* 
b^(1/4)) + (Sqrt[2]*(b^2*c^4 + 6*Sqrt[a]*b^(3/2)*c^3*d - 12*a*b*c^2*d^2 - 
10*a^(3/2)*Sqrt[b]*c*d^3 + 3*a^2*d^4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr 
t[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(3/4)*b^(1/4))))/(8*(b*c^2 + a*d^2)^3*x^( 
3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1124\) vs. \(2(542)=1084\).

Time = 2.83 (sec) , antiderivative size = 1124, normalized size of antiderivative = 2.07, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(e x)^{3/2} d^3}{\left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\sqrt [4]{a} \left (3 b c^2-4 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{a} \left (3 b c^2-4 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {\sqrt [4]{a} \left (3 b c^2+4 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \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}{2 \sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{a} \left (3 b c^2+4 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \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}{2 \sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {3 e \sqrt {e x} d^2}{\left (b c^2+a d^2\right )^2}-\frac {8 b c^2 e \sqrt {e x} d^2}{\left (b c^2+a d^2\right )^3}+\frac {2 \left (3 b c^2-a d^2\right ) e \sqrt {e x} d^2}{\left (b c^2+a d^2\right )^3}-\frac {3 \sqrt {c} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{3/2}}{\left (b c^2+a d^2\right )^2}+\frac {8 b c^{5/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{3/2}}{\left (b c^2+a d^2\right )^3}-\frac {\left (b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}+\frac {\left (b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}-\frac {\left (b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \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} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}+\frac {\left (b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) e^{3/2} \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} a^{3/4} \sqrt [4]{b} \left (b c^2+a d^2\right )^2}-\frac {e \sqrt {e x} \left (b c^2-2 b d x c-a d^2\right )}{2 \left (b c^2+a d^2\right )^2 \left (b x^2+a\right )}\)

Input:

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

Output:

(-8*b*c^2*d^2*e*Sqrt[e*x])/(b*c^2 + a*d^2)^3 + (2*d^2*(3*b*c^2 - a*d^2)*e* 
Sqrt[e*x])/(b*c^2 + a*d^2)^3 + (3*d^2*e*Sqrt[e*x])/(b*c^2 + a*d^2)^2 - (d^ 
3*(e*x)^(3/2))/((b*c^2 + a*d^2)^2*(c + d*x)) - (e*Sqrt[e*x]*(b*c^2 - a*d^2 
 - 2*b*c*d*x))/(2*(b*c^2 + a*d^2)^2*(a + b*x^2)) + (8*b*c^(5/2)*d^(3/2)*e^ 
(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^2)^3 - ( 
3*Sqrt[c]*d^(3/2)*e^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/( 
b*c^2 + a*d^2)^2 + (a^(1/4)*d^2*(3*b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d - a*d^2)* 
e^(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2 
]*b^(1/4)*(b*c^2 + a*d^2)^3) - ((b*c^2 - 6*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*e^ 
(3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2 
]*a^(3/4)*b^(1/4)*(b*c^2 + a*d^2)^2) - (a^(1/4)*d^2*(3*b*c^2 - 4*Sqrt[a]*S 
qrt[b]*c*d - a*d^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4 
)*Sqrt[e])])/(Sqrt[2]*b^(1/4)*(b*c^2 + a*d^2)^3) + ((b*c^2 - 6*Sqrt[a]*Sqr 
t[b]*c*d - a*d^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)* 
Sqrt[e])])/(4*Sqrt[2]*a^(3/4)*b^(1/4)*(b*c^2 + a*d^2)^2) + (a^(1/4)*d^2*(3 
*b*c^2 + 4*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*e^(3/2)*Log[Sqrt[a]*Sqrt[e] + Sqrt 
[b]*Sqrt[e]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*b^(1/4)*(b* 
c^2 + a*d^2)^3) - ((b*c^2 + 6*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*e^(3/2)*Log[Sqr 
t[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(8* 
Sqrt[2]*a^(3/4)*b^(1/4)*(b*c^2 + a*d^2)^2) - (a^(1/4)*d^2*(3*b*c^2 + 4*...
 

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 = 0.77 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {c \,d^{2} \left (\frac {\left (-\frac {a \,d^{2}}{2}-\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (3 a \,d^{2}-5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {\frac {\left (\frac {1}{2} a b c \,d^{3}+\frac {1}{2} b^{2} c^{3} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} e \,d^{4}-\frac {1}{4} b^{2} c^{4} e \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (3 a^{2} e \,d^{4}-12 a b \,c^{2} d^{2} e +b^{2} c^{4} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (10 a b c \,d^{3}-6 b^{2} c^{3} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\)	\(505\)
default	\(2 e^{5} \left (-\frac {c \,d^{2} \left (\frac {\left (-\frac {a \,d^{2}}{2}-\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (3 a \,d^{2}-5 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {\frac {\left (\frac {1}{2} a b c \,d^{3}+\frac {1}{2} b^{2} c^{3} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} e \,d^{4}-\frac {1}{4} b^{2} c^{4} e \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (3 a^{2} e \,d^{4}-12 a b \,c^{2} d^{2} e +b^{2} c^{4} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (10 a b c \,d^{3}-6 b^{2} c^{3} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{3} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\)	\(505\)
pseudoelliptic	\(\frac {\frac {3 \left (d x +c \right ) e \left (a^{2} d^{4}-4 b \,c^{2} d^{2} a +\frac {1}{3} b^{2} c^{4}\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 ) \sqrt {d e c}\, \left (b \,x^{2}+a \right ) \sqrt {\frac {a \,e^{2}}{b}}}{16}+\frac {5 \left (\frac {4 \left (2 d \left (d x +c \right ) b \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, c \left (e x \right )^{\frac {3}{2}}+3 e \left (\left (-\frac {b \,c^{3}}{3}-\frac {b \,c^{2} d x}{3}+d^{2} \left (\frac {2 b \,x^{2}}{3}+a \right ) c +\frac {a x \,d^{3}}{3}\right ) \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, \sqrt {e x}-2 d^{2} \left (d x +c \right ) e \left (a \,d^{2}-\frac {5 b \,c^{2}}{3}\right ) \left (b \,x^{2}+a \right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) c \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 ) d \left (d x +c \right ) \left (a \,d^{2}-\frac {3 b \,c^{2}}{5}\right ) e^{2} \sqrt {2}\, \sqrt {d e c}\, \left (b \,x^{2}+a \right ) c \right ) a}{8}}{\left (b \,x^{2}+a \right ) \left (a \,d^{2}+b \,c^{2}\right )^{3} a \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (d x +c \right ) \sqrt {d e c}}\)	\(546\)

Input:

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

Output:

2*e^5*(-c*d^2/e^3/(a*d^2+b*c^2)^3*((-1/2*a*d^2-1/2*b*c^2)*(e*x)^(1/2)/(d*e 
*x+c*e)+1/2*(3*a*d^2-5*b*c^2)/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^( 
1/2)))+1/e^3/(a*d^2+b*c^2)^3*(((1/2*a*b*c*d^3+1/2*b^2*c^3*d)*(e*x)^(3/2)+( 
1/4*a^2*e*d^4-1/4*b^2*c^4*e)*(e*x)^(1/2))/(b*e^2*x^2+a*e^2)+1/32*(3*a^2*d^ 
4*e-12*a*b*c^2*d^2*e+b^2*c^4*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))+1/32*(10*a*b*c*d 
^3-6*b^2*c^3*d)/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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7542 vs. \(2 (435) = 870\).

Time = 51.83 (sec) , antiderivative size = 15096, normalized size of antiderivative = 27.85 \[ \int \frac {(e x)^{3/2}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)/(d*x+c)^2/(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. 1044 vs. \(2 (435) = 870\).

Time = 0.21 (sec) , antiderivative size = 1044, normalized size of antiderivative = 1.93 \[ \int \frac {(e x)^{3/2}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(2*((a*b^3*e^2)^(1/4)*b^3*c^4*e^2 - 12*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d^2 
*e^2 + 3*(a*b^3*e^2)^(1/4)*a^2*b*d^4*e^2 - 6*(a*b^3*e^2)^(3/4)*b*c^3*d*e + 
 10*(a*b^3*e^2)^(3/4)*a*c*d^3*e)*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^5*c^6 + 3*sqrt(2)*a^2*b^4* 
c^4*d^2 + 3*sqrt(2)*a^3*b^3*c^2*d^4 + sqrt(2)*a^4*b^2*d^6) + 2*((a*b^3*e^2 
)^(1/4)*b^3*c^4*e^2 - 12*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d^2*e^2 + 3*(a*b^3*e^ 
2)^(1/4)*a^2*b*d^4*e^2 - 6*(a*b^3*e^2)^(3/4)*b*c^3*d*e + 10*(a*b^3*e^2)^(3 
/4)*a*c*d^3*e)*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^5*c^6 + 3*sqrt(2)*a^2*b^4*c^4*d^2 + 3*sqrt( 
2)*a^3*b^3*c^2*d^4 + sqrt(2)*a^4*b^2*d^6) + ((a*b^3*e^2)^(1/4)*b^3*c^4*e^2 
 - 12*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d^2*e^2 + 3*(a*b^3*e^2)^(1/4)*a^2*b*d^4* 
e^2 + 6*(a*b^3*e^2)^(3/4)*b*c^3*d*e - 10*(a*b^3*e^2)^(3/4)*a*c*d^3*e)*log( 
e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a*b^5*c^ 
6 + 3*sqrt(2)*a^2*b^4*c^4*d^2 + 3*sqrt(2)*a^3*b^3*c^2*d^4 + sqrt(2)*a^4*b^ 
2*d^6) - ((a*b^3*e^2)^(1/4)*b^3*c^4*e^2 - 12*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d 
^2*e^2 + 3*(a*b^3*e^2)^(1/4)*a^2*b*d^4*e^2 + 6*(a*b^3*e^2)^(3/4)*b*c^3*d*e 
 - 10*(a*b^3*e^2)^(3/4)*a*c*d^3*e)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt( 
e*x) + sqrt(a*e^2/b))/(sqrt(2)*a*b^5*c^6 + 3*sqrt(2)*a^2*b^4*c^4*d^2 + 3*s 
qrt(2)*a^3*b^3*c^2*d^4 + sqrt(2)*a^4*b^2*d^6) + 8*(5*b*c^3*d^2*e^3 - 3*a*c 
*d^4*e^3)*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^3*c^6 + 3*a*b^2*c^4*d^2 +...
 

Mupad [B] (verification not implemented)

Time = 12.02 (sec) , antiderivative size = 30042, normalized size of antiderivative = 55.43 \[ \int \frac {(e x)^{3/2}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((32*b^11*c^16*d^3*e^15 + 75168*a^2*b^9*c^12*d^7*e^15 - 18464*a^3*b 
^8*c^10*d^9*e^15 - 160160*a^4*b^7*c^8*d^11*e^15 + 30240*a^5*b^6*c^6*d^13*e 
^15 + 74720*a^6*b^5*c^4*d^15*e^15 - 19296*a^7*b^4*c^2*d^17*e^15 - 2720*a*b 
^10*c^14*d^5*e^15)/(16*(a^8*d^16 + b^8*c^16 + 8*a*b^7*c^14*d^2 + 8*a^7*b*c 
^2*d^14 + 28*a^2*b^6*c^12*d^4 + 56*a^3*b^5*c^10*d^6 + 70*a^4*b^4*c^8*d^8 + 
 56*a^5*b^3*c^6*d^10 + 28*a^6*b^2*c^4*d^12)) - (((21504*a^7*b^8*c^9*d^14*e 
^12 - 81408*a^3*b^12*c^17*d^6*e^12 - 258048*a^4*b^11*c^15*d^8*e^12 - 40857 
6*a^5*b^10*c^13*d^10*e^12 - 301056*a^6*b^9*c^11*d^12*e^12 - 9216*a^2*b^13* 
c^19*d^4*e^12 + 233472*a^8*b^7*c^7*d^16*e^12 + 195072*a^9*b^6*c^5*d^18*e^1 
2 + 72704*a^10*b^5*c^3*d^20*e^12 + 512*a*b^14*c^21*d^2*e^12 + 10752*a^11*b 
^4*c*d^22*e^12)/(16*(a^8*d^16 + b^8*c^16 + 8*a*b^7*c^14*d^2 + 8*a^7*b*c^2* 
d^14 + 28*a^2*b^6*c^12*d^4 + 56*a^3*b^5*c^10*d^6 + 70*a^4*b^4*c^8*d^8 + 56 
*a^5*b^3*c^6*d^10 + 28*a^6*b^2*c^4*d^12)) - ((e*x)^(1/2)*((9*a^4*d^8*e^3*( 
-a^3*b)^(1/2) + b^4*c^8*e^3*(-a^3*b)^(1/2) - 164*a^3*b^3*c^5*d^3*e^3 + 276 
*a^4*b^2*c^3*d^5*e^3 - 60*a^5*b*c*d^7*e^3 + 12*a^2*b^4*c^7*d*e^3 + 270*a^2 
*b^2*c^4*d^4*e^3*(-a^3*b)^(1/2) - 60*a*b^3*c^6*d^2*e^3*(-a^3*b)^(1/2) - 17 
2*a^3*b*c^2*d^6*e^3*(-a^3*b)^(1/2))/(64*(a^9*b*d^12 + a^3*b^7*c^12 + 6*a^4 
*b^6*c^10*d^2 + 15*a^5*b^5*c^8*d^4 + 20*a^6*b^4*c^6*d^6 + 15*a^7*b^3*c^4*d 
^8 + 6*a^8*b^2*c^2*d^10)))^(1/2)*(4096*a^13*b^4*d^25*e^10 - 4096*a^2*b^15* 
c^22*d^3*e^10 - 36864*a^3*b^14*c^20*d^5*e^10 - 143360*a^4*b^13*c^18*d^7...
 

Reduce [F]

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

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

Output:

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