\(\int \frac {\sqrt {e x}}{(c+d x)^2 (a+b x^2)^2} \, dx\) [423]

Optimal result

Integrand size = 24, antiderivative size = 547 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=-\frac {d^3 \sqrt {e x}}{\left (b c^2+a d^2\right )^2 (c+d x)}+\frac {b \sqrt {e x} \left (b c^2 x+a d (2 c-d x)\right )}{2 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )}-\frac {d^{5/2} \left (7 b c^2-a d^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{b} \left (b^2 c^4-2 \sqrt {a} b^{3/2} c^3 d+12 a b c^2 d^2+14 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^3}+\frac {\sqrt [4]{b} \left (b^2 c^4-2 \sqrt {a} b^{3/2} c^3 d+12 a b c^2 d^2+14 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^3}-\frac {\sqrt [4]{b} \left (b^2 c^4+2 \sqrt {a} b^{3/2} c^3 d+12 a b c^2 d^2-14 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{5/4} \left (b c^2+a d^2\right )^3} \] Output:

-d^3*(e*x)^(1/2)/(a*d^2+b*c^2)^2/(d*x+c)+1/2*b*(e*x)^(1/2)*(b*c^2*x+a*d*(- 
d*x+2*c))/a/(a*d^2+b*c^2)^2/(b*x^2+a)-d^(5/2)*(-a*d^2+7*b*c^2)*e^(1/2)*arc 
tan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(1/2)/(a*d^2+b*c^2)^3-1/8*b^(1/ 
4)*(b^2*c^4-2*a^(1/2)*b^(3/2)*c^3*d+12*a*b*c^2*d^2+14*a^(3/2)*b^(1/2)*c*d^ 
3-5*a^2*d^4)*e^(1/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)) 
*2^(1/2)/a^(5/4)/(a*d^2+b*c^2)^3+1/8*b^(1/4)*(b^2*c^4-2*a^(1/2)*b^(3/2)*c^ 
3*d+12*a*b*c^2*d^2+14*a^(3/2)*b^(1/2)*c*d^3-5*a^2*d^4)*e^(1/2)*arctan(1+2^ 
(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/(a*d^2+b*c^2)^3 
-1/8*b^(1/4)*(b^2*c^4+2*a^(1/2)*b^(3/2)*c^3*d+12*a*b*c^2*d^2-14*a^(3/2)*b^ 
(1/2)*c*d^3-5*a^2*d^4)*e^(1/2)*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^(5/4)/(a*d^2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 1.90 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (-2 a^2 d^3+b^2 c^2 x (c+d x)+a b d \left (2 c^2+c d x-3 d^2 x^2\right )\right )}{a (c+d x) \left (a+b x^2\right )}+\frac {8 d^{5/2} \left (-7 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} \left (-b^2 c^4+2 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2-14 a^{3/2} \sqrt {b} c d^3+5 a^2 d^4\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4}}+\frac {\sqrt {2} \sqrt [4]{b} \left (-b^2 c^4-2 \sqrt {a} b^{3/2} c^3 d-12 a b c^2 d^2+14 a^{3/2} \sqrt {b} c d^3+5 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4}}\right )}{8 \left (b c^2+a d^2\right )^3 \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*((4*(b*c^2 + a*d^2)*Sqrt[x]*(-2*a^2*d^3 + b^2*c^2*x*(c + d*x) + 
 a*b*d*(2*c^2 + c*d*x - 3*d^2*x^2)))/(a*(c + d*x)*(a + b*x^2)) + (8*d^(5/2 
)*(-7*b*c^2 + a*d^2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/Sqrt[c] + (Sqrt[2] 
*b^(1/4)*(-(b^2*c^4) + 2*Sqrt[a]*b^(3/2)*c^3*d - 12*a*b*c^2*d^2 - 14*a^(3/ 
2)*Sqrt[b]*c*d^3 + 5*a^2*d^4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4 
)*b^(1/4)*Sqrt[x])])/a^(5/4) + (Sqrt[2]*b^(1/4)*(-(b^2*c^4) - 2*Sqrt[a]*b^ 
(3/2)*c^3*d - 12*a*b*c^2*d^2 + 14*a^(3/2)*Sqrt[b]*c*d^3 + 5*a^2*d^4)*ArcTa 
nh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(5/4)))/(8* 
(b*c^2 + a*d^2)^3*Sqrt[x])
 

Rubi [A] (verified)

Time = 2.71 (sec) , antiderivative size = 1033, normalized size of antiderivative = 1.89, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d^3*Sqrt[e*x])/((b*c^2 + a*d^2)^2*(c + d*x))) + (b*Sqrt[e*x]*(2*a*c*d + 
 (b*c^2 - a*d^2)*x))/(2*a*(b*c^2 + a*d^2)^2*(a + b*x^2)) - (8*b*c^(3/2)*d^ 
(5/2)*Sqrt[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(b*c^2 + a*d^ 
2)^3 + (d^(5/2)*Sqrt[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(Sq 
rt[c]*(b*c^2 + a*d^2)^2) - (b^(1/4)*d^2*(3*b*c^2 + 4*Sqrt[a]*Sqrt[b]*c*d - 
 a*d^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]) 
/(Sqrt[2]*a^(1/4)*(b*c^2 + a*d^2)^3) - (b^(1/4)*(b*c^2 - 2*Sqrt[a]*Sqrt[b] 
*c*d - a*d^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt 
[e])])/(4*Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2) + (b^(1/4)*d^2*(3*b*c^2 + 4*S 
qrt[a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x]) 
/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(1/4)*(b*c^2 + a*d^2)^3) + (b^(1/4)*(b*c^2 
 - 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt 
[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2) + (b^(1/4 
)*d^2*(3*b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*Log[Sqrt[a]*Sqrt[e 
] + Sqrt[b]*Sqrt[e]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*a^( 
1/4)*(b*c^2 + a*d^2)^3) + (b^(1/4)*(b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2) 
*Sqrt[e]*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^(5/4)*(b*c^2 + a*d^2)^2) - (b^(1/4)*d^2*(3*b*c^2 
 - 4*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*Sqrt[e]*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sq 
rt[e]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*a^(1/4)*(b*c^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 = 0.76 (sec) , antiderivative size = 508, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {d^{3} \left (\frac {\left (-\frac {a \,d^{2}}{2}-\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (a \,d^{2}-7 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {b \left (\frac {-\frac {\left (a^{2} d^{4}-b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{4 a}+\left (\frac {1}{2} a c \,d^{3} e +\frac {1}{2} c^{3} d e b \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {\left (14 a^{2} c \,d^{3} e -2 c^{3} a d e b \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (-5 a^{2} d^{4}+12 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\)	\(508\)
default	\(2 e^{5} \left (\frac {d^{3} \left (\frac {\left (-\frac {a \,d^{2}}{2}-\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (a \,d^{2}-7 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3}}+\frac {b \left (\frac {-\frac {\left (a^{2} d^{4}-b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{4 a}+\left (\frac {1}{2} a c \,d^{3} e +\frac {1}{2} c^{3} d e b \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {\left (14 a^{2} c \,d^{3} e -2 c^{3} a d e b \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (-5 a^{2} d^{4}+12 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a}\right )}{e^{4} \left (a \,d^{2}+b \,c^{2}\right )^{3}}\right )\)	\(508\)
pseudoelliptic	\(\frac {\frac {7 \sqrt {d e c}\, d \left (d x +c \right ) 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 ) b \sqrt {2}\, \left (b \,x^{2}+a \right ) c \left (a \,d^{2}-\frac {b \,c^{2}}{7}\right ) \sqrt {\frac {a \,e^{2}}{b}}}{8}+\frac {\left (-\sqrt {d e c}\, \left (d x +c \right ) b \left (a \,d^{2}+b \,c^{2}\right ) \left (a \,d^{2}-b \,c^{2}\right ) \left (e x \right )^{\frac {3}{2}}+2 d e \left (-\left (\left (b \,x^{2}+a \right ) d^{2}-b c d x -b \,c^{2}\right ) \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {e x}+d^{2} \left (d x +c \right ) e \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) \left (a \,d^{2}-7 b \,c^{2}\right ) \left (b \,x^{2}+a \right )\right ) a \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{2}-\frac {5 \sqrt {d e c}\, \left (d x +c \right ) e^{2} \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\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}\right ) \left (a^{2} d^{4}-\frac {12}{5} b \,c^{2} d^{2} a -\frac {1}{5} b^{2} c^{4}\right ) \sqrt {2}\, \left (b \,x^{2}+a \right )}{8}}{e \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}}\)	\(545\)

Input:

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

Output:

2*e^5*(d^3/e^4/(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*(a*d^2-7*b*c^2)/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)) 
)+b/e^4/(a*d^2+b*c^2)^3*((-1/4*(a^2*d^4-b^2*c^4)/a*(e*x)^(3/2)+(1/2*a*c*d^ 
3*e+1/2*c^3*d*e*b)*(e*x)^(1/2))/(b*e^2*x^2+a*e^2)+1/4/a*(1/8*(14*a^2*c*d^3 
*e-2*a*b*c^3*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*arc 
tan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+1/8*(-5*a^2*d^4+12*a*b*c^2*d^2 
+b^2*c^4)/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. 7639 vs. \(2 (439) = 878\).

Time = 71.43 (sec) , antiderivative size = 15290, normalized size of antiderivative = 27.95 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(1/2)/(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. 1065 vs. \(2 (439) = 878\).

Time = 0.22 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(2*(2*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d*e - 14*(a*b^3*e^2)^(1/4)*a^2*b^2* 
c*d^3*e - (a*b^3*e^2)^(3/4)*b^2*c^4 - 12*(a*b^3*e^2)^(3/4)*a*b*c^2*d^2 + 5 
*(a*b^3*e^2)^(3/4)*a^2*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^2*b^5*c^6 + 3*sqrt(2)*a^3*b^4*c^4 
*d^2 + 3*sqrt(2)*a^4*b^3*c^2*d^4 + sqrt(2)*a^5*b^2*d^6) + 2*(2*(a*b^3*e^2) 
^(1/4)*a*b^3*c^3*d*e - 14*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^3*e - (a*b^3*e^2)^ 
(3/4)*b^2*c^4 - 12*(a*b^3*e^2)^(3/4)*a*b*c^2*d^2 + 5*(a*b^3*e^2)^(3/4)*a^2 
*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^2*b^5*c^6 + 3*sqrt(2)*a^3*b^4*c^4*d^2 + 3*sqrt(2)*a^4* 
b^3*c^2*d^4 + sqrt(2)*a^5*b^2*d^6) + (2*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d*e - 
14*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^3*e + (a*b^3*e^2)^(3/4)*b^2*c^4 + 12*(a*b 
^3*e^2)^(3/4)*a*b*c^2*d^2 - 5*(a*b^3*e^2)^(3/4)*a^2*d^4)*log(e*x + sqrt(2) 
*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^2*b^5*c^6 + 3*sqrt( 
2)*a^3*b^4*c^4*d^2 + 3*sqrt(2)*a^4*b^3*c^2*d^4 + sqrt(2)*a^5*b^2*d^6) - (2 
*(a*b^3*e^2)^(1/4)*a*b^3*c^3*d*e - 14*(a*b^3*e^2)^(1/4)*a^2*b^2*c*d^3*e + 
(a*b^3*e^2)^(3/4)*b^2*c^4 + 12*(a*b^3*e^2)^(3/4)*a*b*c^2*d^2 - 5*(a*b^3*e^ 
2)^(3/4)*a^2*d^4)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2 
/b))/(sqrt(2)*a^2*b^5*c^6 + 3*sqrt(2)*a^3*b^4*c^4*d^2 + 3*sqrt(2)*a^4*b^3* 
c^2*d^4 + sqrt(2)*a^5*b^2*d^6) + 8*(7*b*c^2*d^3*e^2 - a*d^5*e^2)*arctan(sq 
rt(e*x)*d/sqrt(c*d*e))/((b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + ...
 

Mupad [B] (verification not implemented)

Time = 10.96 (sec) , antiderivative size = 5655, normalized size of antiderivative = 10.34 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(- root(196608*a^16*b*c^3*d^22*g^6 + 15138816*a^11*b^6*c^13*d^12 
*g^6 + 12976128*a^12*b^5*c^11*d^14*g^6 + 12976128*a^10*b^7*c^15*d^10*g^6 + 
 8110080*a^13*b^4*c^9*d^16*g^6 + 8110080*a^9*b^8*c^17*d^8*g^6 + 3604480*a^ 
14*b^3*c^7*d^18*g^6 + 3604480*a^8*b^9*c^19*d^6*g^6 + 1081344*a^15*b^2*c^5* 
d^20*g^6 + 1081344*a^7*b^10*c^21*d^4*g^6 + 196608*a^6*b^11*c^23*d^2*g^6 + 
16384*a^5*b^12*c^25*g^6 + 16384*a^17*c*d^24*g^6 + 5582848*a^8*b^5*c^10*d^1 
1*e*g^4 + 4739072*a^7*b^6*c^12*d^9*e*g^4 + 3162112*a^9*b^4*c^8*d^13*e*g^4 
+ 1974272*a^6*b^7*c^14*d^7*e*g^4 + 434176*a^10*b^3*c^6*d^15*e*g^4 - 329728 
*a^11*b^2*c^4*d^17*e*g^4 + 290816*a^5*b^8*c^16*d^5*e*g^4 - 22528*a^4*b^9*c 
^18*d^3*e*g^4 - 104448*a^12*b*c^2*d^19*e*g^4 - 2048*a^3*b^10*c^20*d*e*g^4 
+ 4096*a^13*d^21*e*g^4 + 2321184*a^5*b^4*c^7*d^10*e^2*g^2 - 1479888*a^6*b^ 
3*c^5*d^12*e^2*g^2 + 311648*a^7*b^2*c^3*d^14*e^2*g^2 - 76712*a^4*b^5*c^9*d 
^8*e^2*g^2 - 7008*a^3*b^6*c^11*d^6*e^2*g^2 + 3760*a^2*b^7*c^13*d^4*e^2*g^2 
 + 224*a*b^8*c^15*d^2*e^2*g^2 - 15420*a^8*b*c*d^16*e^2*g^2 + 4*b^9*c^17*e^ 
2*g^2 + 2436*a*b^4*c^6*d^7*e^3 + 29926*a^2*b^3*c^4*d^9*e^3 - 8700*a^3*b^2* 
c^2*d^11*e^3 + 49*b^5*c^8*d^5*e^3 + 625*a^4*b*d^13*e^3, g, k)*(root(196608 
*a^16*b*c^3*d^22*g^6 + 15138816*a^11*b^6*c^13*d^12*g^6 + 12976128*a^12*b^5 
*c^11*d^14*g^6 + 12976128*a^10*b^7*c^15*d^10*g^6 + 8110080*a^13*b^4*c^9*d^ 
16*g^6 + 8110080*a^9*b^8*c^17*d^8*g^6 + 3604480*a^14*b^3*c^7*d^18*g^6 + 36 
04480*a^8*b^9*c^19*d^6*g^6 + 1081344*a^15*b^2*c^5*d^20*g^6 + 1081344*a^...
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 3810, normalized size of antiderivative = 6.97 \[ \int \frac {\sqrt {e x}}{(c+d x)^2 \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*(10*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**3*c**2*d**4 + 10*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**3*c*d**5*x - 24*b**(1/4)*a**(3/4)*sqrt(2)*at 
an((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt 
(2)))*a**2*b*c**4*d**2 - 24*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*b*c**3* 
d**3*x + 10*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*b*c**2*d**4*x**2 + 10*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*b*c*d**5*x**3 - 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**2*c**6 - 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**2*c 
**5*d*x - 24*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**2*c**4*d**2*x**2 - 24* 
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**2*c**3*d**3*x**3 - 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)))*b**3*c**6*x**2 - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((...