\(\int \frac {(e x)^{3/2}}{(c+d x) (a+b x^2)^2} \, dx\) [415]

Optimal result

Integrand size = 24, antiderivative size = 433 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {e \sqrt {e x} (c-d x)}{2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )}+\frac {2 c^{3/2} d^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\left (b c^2+a d^2\right )^2}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} c d+a d^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} c d+a d^2\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2}+\frac {\left (\sqrt {b} c \left (b c^2-3 a d^2\right )+\sqrt {a} d \left (3 b c^2-a d^2\right )\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{3/4} b^{3/4} \left (b c^2+a d^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.66 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \left (\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} (-c+d x)}{a+b x^2}+16 c^{3/2} d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )-\frac {\sqrt {2} \left (b^{3/2} c^3-3 \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2+a^{3/2} d^3\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4} b^{3/4}}+\frac {\sqrt {2} \left (b^{3/2} c^3+3 \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2-a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4} b^{3/4}}\right )}{8 \left (b c^2+a d^2\right )^2 x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*((4*(b*c^2 + a*d^2)*Sqrt[x]*(-c + d*x))/(a + b*x^2) + 16*c^(3 
/2)*d^(3/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] - (Sqrt[2]*(b^(3/2)*c^3 - 3* 
Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + a^(3/2)*d^3)*ArcTan[(Sqrt[a] - Sqrt[ 
b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(3/4)*b^(3/4)) + (Sqrt[2]*(b^ 
(3/2)*c^3 + 3*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*ArcTanh[( 
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(3/4)*b^(3/4)) 
))/(8*(b*c^2 + a*d^2)^2*x^(3/2))
 

Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.95, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(e*Sqrt[e*x]*(c - d*x))/((b*c^2 + a*d^2)*(a + b*x^2)) + (2*c^(3/2)*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*(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)*ArcTan[1 - (Sqrt[2]*b^ 
(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2)^2) - 
 ((Sqrt[b]*c - 3*Sqrt[a]*d)*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^(3/4)*(b*c^2 + a*d^2)) - (a^(1/4 
)*d^2*(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x 
])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2)^2) + ((Sqrt[b]*c - 
 3*Sqrt[a]*d)*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^(3/4)*(b*c^2 + a*d^2)) + (a^(1/4)*d^2*(Sqrt[b] 
*c + Sqrt[a]*d)*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^(3/4)*(b*c^2 + a*d^2)^2) - ((Sqrt 
[b]*c + 3*Sqrt[a]*d)*e^(3/2)*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x - Sqr 
t[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(8*Sqrt[2]*a^(3/4)*b^(3/4)*(b*c^2 + a*d^2 
)) - (a^(1/4)*d^2*(Sqrt[b]*c + Sqrt[a]*d)*e^(3/2)*Log[Sqrt[a]*Sqrt[e] + Sq 
rt[b]*Sqrt[e]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*b^(3/4)*( 
b*c^2 + a*d^2)^2) + ((Sqrt[b]*c + 3*Sqrt[a]*d)*e^(3/2)*Log[Sqrt[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^(3/4)*(b*c^2 + a*d^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 [A] (verified)

Time = 0.62 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(2 e^{4} \left (-\frac {\frac {\left (-\frac {1}{4} a \,d^{3}-\frac {1}{4} b \,c^{2} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{2} e a c +\frac {1}{4} b e \,c^{3}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (3 d^{2} e a c -b e \,c^{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (-a \,d^{3}+3 b \,c^{2} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {c^{2} d^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}\right )\)	\(438\)
default	\(2 e^{4} \left (-\frac {\frac {\left (-\frac {1}{4} a \,d^{3}-\frac {1}{4} b \,c^{2} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{2} e a c +\frac {1}{4} b e \,c^{3}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (3 d^{2} e a c -b e \,c^{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (-a \,d^{3}+3 b \,c^{2} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {c^{2} d^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}\right )\)	\(438\)
pseudoelliptic	\(\frac {-3 e \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \left (a \,d^{2}-\frac {b \,c^{2}}{3}\right ) b \sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right ) c \sqrt {\frac {a \,e^{2}}{b}}+\left (-8 \left (-d \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\sqrt {e x}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}-4 d^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) e \left (b \,x^{2}+a \right ) c \right ) e c \right ) b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+d \,e^{2} \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {d e c}\, \sqrt {2}\, \left (b \,x^{2}+a \right ) \left (a \,d^{2}-3 b \,c^{2}\right )\right ) a}{16 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, \left (b \,x^{2}+a \right ) \left (a \,d^{2}+b \,c^{2}\right )^{2} a b}\)	\(461\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4981 vs. \(2 (332) = 664\).

Time = 5.21 (sec) , antiderivative size = 9974, normalized size of antiderivative = 23.03 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)/(d*x+c)/(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. 760 vs. \(2 (332) = 664\).

Time = 0.19 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.76 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(16*c^2*d^2*e^3*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^2*c^4 + 2*a*b*c^2* 
d^2 + a^2*d^4)*sqrt(c*d*e)) + 2*((a*b^3*e^2)^(1/4)*b^3*c^3*e^2 - 3*(a*b^3* 
e^2)^(1/4)*a*b^2*c*d^2*e^2 - 3*(a*b^3*e^2)^(3/4)*b*c^2*d*e + (a*b^3*e^2)^( 
3/4)*a*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^4 + 2*sqrt(2)*a^2*b^4*c^2*d^2 + sqrt(2)*a 
^3*b^3*d^4) + 2*((a*b^3*e^2)^(1/4)*b^3*c^3*e^2 - 3*(a*b^3*e^2)^(1/4)*a*b^2 
*c*d^2*e^2 - 3*(a*b^3*e^2)^(3/4)*b*c^2*d*e + (a*b^3*e^2)^(3/4)*a*d^3*e)*ar 
ctan(-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^4 + 2*sqrt(2)*a^2*b^4*c^2*d^2 + sqrt(2)*a^3*b^3*d^4) + ( 
(a*b^3*e^2)^(1/4)*b^3*c^3*e^2 - 3*(a*b^3*e^2)^(1/4)*a*b^2*c*d^2*e^2 + 3*(a 
*b^3*e^2)^(3/4)*b*c^2*d*e - (a*b^3*e^2)^(3/4)*a*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^4 + 2*sqrt(2)*a 
^2*b^4*c^2*d^2 + sqrt(2)*a^3*b^3*d^4) - ((a*b^3*e^2)^(1/4)*b^3*c^3*e^2 - 3 
*(a*b^3*e^2)^(1/4)*a*b^2*c*d^2*e^2 + 3*(a*b^3*e^2)^(3/4)*b*c^2*d*e - (a*b^ 
3*e^2)^(3/4)*a*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^4 + 2*sqrt(2)*a^2*b^4*c^2*d^2 + sqrt(2)*a^3*b^3* 
d^4) + 4*(sqrt(e*x)*d*e^4*x - sqrt(e*x)*c*e^4)/((b*e^2*x^2 + a*e^2)*(b*c^2 
 + a*d^2)))/e
 

Mupad [B] (verification not implemented)

Time = 12.08 (sec) , antiderivative size = 19089, normalized size of antiderivative = 44.09 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan(-(((((((b^6*c^9*d^3*e^15)/2 + 259*a^2*b^4*c^5*d^7*e^15 - 94*a^3*b^3* 
c^3*d^9*e^15 - 30*a*b^5*c^7*d^5*e^15 + (a^4*b^2*c*d^11*e^15)/2)/(a^3*d^6 + 
 b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4) + ((((e*x)^(1/2)*(5408*a^2*b 
^7*c^9*d^6*e^13 - 32*b^9*c^13*d^2*e^13 + 8064*a^3*b^6*c^7*d^8*e^13 + 1056* 
a^4*b^5*c^5*d^10*e^13 - 960*a^5*b^4*c^3*d^12*e^13 + 64*a*b^8*c^11*d^4*e^13 
 + 736*a^6*b^3*c*d^14*e^13))/(8*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a 
^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)) + (((224*a^2*b^8*c^10*d^4*e^12 + 1216*a 
^3*b^7*c^8*d^6*e^12 + 1984*a^4*b^6*c^6*d^8*e^12 + 1376*a^5*b^5*c^4*d^10*e^ 
12 + 352*a^6*b^4*c^2*d^12*e^12 - 32*a*b^9*c^12*d^2*e^12)/(a^3*d^6 + b^3*c^ 
6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4) - ((e*x)^(1/2)*(-c^3*d^3*e^3)^(1/2) 
*(4096*a^9*b^4*d^17*e^10 - 4096*a^2*b^11*c^14*d^3*e^10 - 20480*a^3*b^10*c^ 
12*d^5*e^10 - 36864*a^4*b^9*c^10*d^7*e^10 - 20480*a^5*b^8*c^8*d^9*e^10 + 2 
0480*a^6*b^7*c^6*d^11*e^10 + 36864*a^7*b^6*c^4*d^13*e^10 + 20480*a^8*b^5*c 
^2*d^15*e^10))/(8*(a*d^2 + b*c^2)^2*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 
 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)))*(-c^3*d^3*e^3)^(1/2))/(a*d^2 + b*c 
^2)^2)*(-c^3*d^3*e^3)^(1/2))/(a*d^2 + b*c^2)^2)*(-c^3*d^3*e^3)^(1/2))/(a*d 
^2 + b*c^2)^2 - ((e*x)^(1/2)*(a^4*b*d^13*e^16 + 33*b^5*c^8*d^5*e^16 + 38*a 
^2*b^3*c^4*d^9*e^16 + 4*a^3*b^2*c^2*d^11*e^16 - 188*a*b^4*c^6*d^7*e^16))/( 
8*(a^4*d^8 + b^4*c^8 + 4*a*b^3*c^6*d^2 + 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d 
^4)))*(-c^3*d^3*e^3)^(1/2)*1i)/(a*d^2 + b*c^2)^2 - ((((((b^6*c^9*d^3*e^...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1415, normalized size of antiderivative = 3.27 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*e*( - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) 
 - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*d**3 + 6*b**(1/4)* 
a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b** 
(1/4)*a**(1/4)*sqrt(2)))*a*b*c**2*d - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b* 
*(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))* 
a*b*d**3*x**2 + 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 
) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c**2*d*x**2 + 6*b 
**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt( 
b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c*d**2 - 2*b**(3/4)*a**(1/4)*sqrt(2) 
*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*s 
qrt(2)))*a*b*c**3 + 6*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq 
rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 - 
2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sq 
rt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c**3*x**2 + 2*b**(1/4)*a**(3/4)*s 
qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 
1/4)*sqrt(2)))*a**2*d**3 - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**( 
1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c**2*d 
+ 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d**3*x**2 - 6*b**(1/4)*a**(3/4)* 
sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*...