\(\int \frac {1}{(e x)^{3/2} (c+d x) (a+b x^2)^2} \, dx\) [418]

Optimal result

Integrand size = 24, antiderivative size = 474 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {2}{a^2 c e \sqrt {e x}}-\frac {b \sqrt {e x} (a d+b c x)}{2 a^2 \left (b c^2+a d^2\right ) e^2 \left (a+b x^2\right )}-\frac {2 d^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{3/2} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {b^{3/4} \left (5 b^{3/2} c^3+3 \sqrt {a} b c^2 d+9 a \sqrt {b} c d^2+7 a^{3/2} d^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {b^{3/4} \left (5 b^{3/2} c^3+3 \sqrt {a} b c^2 d+9 a \sqrt {b} c d^2+7 a^{3/2} d^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^2 e^{3/2}}+\frac {b^{3/4} \left (5 b^{3/2} c^3-3 \sqrt {a} b c^2 d+9 a \sqrt {b} c d^2-7 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^2 e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\frac {4 \left (b c^2+a d^2\right ) \left (4 a^2 d^2+5 b^2 c^2 x^2+a b \left (4 c^2+c d x+4 d^2 x^2\right )\right )}{a^2 c \left (a+b x^2\right )}-\frac {16 d^{9/2} \sqrt {x} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt {2} b^{3/4} \left (5 b^{3/2} c^3+3 \sqrt {a} b c^2 d+9 a \sqrt {b} c d^2+7 a^{3/2} d^3\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4}}+\frac {\sqrt {2} b^{3/4} \left (5 b^{3/2} c^3-3 \sqrt {a} b c^2 d+9 a \sqrt {b} c d^2-7 a^{3/2} d^3\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4}}\right )}{8 \left (b c^2+a d^2\right )^2 (e x)^{3/2}} \] Input:

Integrate[1/((e*x)^(3/2)*(c + d*x)*(a + b*x^2)^2),x]
 

Output:

(x*((-4*(b*c^2 + a*d^2)*(4*a^2*d^2 + 5*b^2*c^2*x^2 + a*b*(4*c^2 + c*d*x + 
4*d^2*x^2)))/(a^2*c*(a + b*x^2)) - (16*d^(9/2)*Sqrt[x]*ArcTan[(Sqrt[d]*Sqr 
t[x])/Sqrt[c]])/c^(3/2) + (Sqrt[2]*b^(3/4)*(5*b^(3/2)*c^3 + 3*Sqrt[a]*b*c^ 
2*d + 9*a*Sqrt[b]*c*d^2 + 7*a^(3/2)*d^3)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b] 
*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(9/4) + (Sqrt[2]*b^(3/4)*(5*b^(3 
/2)*c^3 - 3*Sqrt[a]*b*c^2*d + 9*a*Sqrt[b]*c*d^2 - 7*a^(3/2)*d^3)*Sqrt[x]*A 
rcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(9/4))) 
/(8*(b*c^2 + a*d^2)^2*(e*x)^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(950\) vs. \(2(474)=948\).

Time = 2.37 (sec) , antiderivative size = 950, normalized size of antiderivative = 2.00, 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.

\(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^2 (c+d x)} \, dx\)

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((e*x)^(3/2)*(c + d*x)*(a + b*x^2)^2),x]
 

Output:

(-2*b*c*d^2)/(a*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]) - (2*d^4)/(c*(b*c^2 + a*d^2 
)^2*e*Sqrt[e*x]) - (5*b*c)/(2*a^2*(b*c^2 + a*d^2)*e*Sqrt[e*x]) + (b*(c - d 
*x))/(2*a*(b*c^2 + a*d^2)*e*Sqrt[e*x]*(a + b*x^2)) - (2*d^(9/2)*ArcTan[(Sq 
rt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(c^(3/2)*(b*c^2 + a*d^2)^2*e^(3/2)) + 
 (b^(3/4)*d^2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x 
])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2*e^(3/2)) + (b^(3 
/4)*(5*Sqrt[b]*c + 3*Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^ 
(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(9/4)*(b*c^2 + a*d^2)*e^(3/2)) - (b^(3/4)*d^ 
2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)* 
Sqrt[e])])/(Sqrt[2]*a^(5/4)*(b*c^2 + a*d^2)^2*e^(3/2)) - (b^(3/4)*(5*Sqrt[ 
b]*c + 3*Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e 
])])/(4*Sqrt[2]*a^(9/4)*(b*c^2 + a*d^2)*e^(3/2)) - (b^(3/4)*d^2*(Sqrt[b]*c 
 - Sqrt[a]*d)*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^(5/4)*(b*c^2 + a*d^2)^2*e^(3/2)) - (b^(3/4) 
*(5*Sqrt[b]*c - 3*Sqrt[a]*d)*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^(9/4)*(b*c^2 + a*d^2)*e^(3/2 
)) + (b^(3/4)*d^2*(Sqrt[b]*c - Sqrt[a]*d)*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^(5/4)*(b*c^2 + 
a*d^2)^2*e^(3/2)) + (b^(3/4)*(5*Sqrt[b]*c - 3*Sqrt[a]*d)*Log[Sqrt[a]*Sqrt[ 
e] + Sqrt[b]*Sqrt[e]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(8*Sqrt[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 = 0.68 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {2}{a^{2} c e \sqrt {e x}}-\frac {\frac {2 b c \left (\frac {\left (\frac {1}{4} a b c \,d^{2}+\frac {1}{4} c^{3} b^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} d^{3} e +\frac {1}{4} a b \,c^{2} d e \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (7 a^{2} d^{3} e +3 a b \,c^{2} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (9 a b c \,d^{2}+5 c^{3} b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {2 d^{5} a^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}}{a^{2} c e}\)	\(470\)
derivativedivides	\(2 e^{4} \left (-\frac {b \left (\frac {\left (\frac {1}{4} a b c \,d^{2}+\frac {1}{4} c^{3} b^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} d^{3} e +\frac {1}{4} a b \,c^{2} d e \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (7 a^{2} d^{3} e +3 a b \,c^{2} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (9 a b c \,d^{2}+5 c^{3} b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} e^{5} a^{2}}-\frac {1}{c \,e^{5} a^{2} \sqrt {e x}}-\frac {d^{5} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{c \,e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}\right )\)	\(471\)
default	\(2 e^{4} \left (-\frac {b \left (\frac {\left (\frac {1}{4} a b c \,d^{2}+\frac {1}{4} c^{3} b^{2}\right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} d^{3} e +\frac {1}{4} a b \,c^{2} d e \right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (7 a^{2} d^{3} e +3 a b \,c^{2} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (9 a b c \,d^{2}+5 c^{3} b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} e^{5} a^{2}}-\frac {1}{c \,e^{5} a^{2} \sqrt {e x}}-\frac {d^{5} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{c \,e^{5} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}\right )\)	\(471\)
pseudoelliptic	\(-\frac {2 \left (\frac {7 d \left (a \,d^{2}+\frac {3 b \,c^{2}}{7}\right ) b \sqrt {2}\, \left (b \,x^{2}+a \right ) \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {d e c}\, c \sqrt {e x}\, \sqrt {\frac {a \,e^{2}}{b}}}{32}+e \left (\left (a^{2} d^{5} \sqrt {e x}\, \left (b \,x^{2}+a \right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )+\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, \left (a^{2} d^{2}+b \left (d^{2} x^{2}+\frac {1}{4} c d x +c^{2}\right ) a +\frac {5 b^{2} c^{2} x^{2}}{4}\right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\frac {9 \left (a \,d^{2}+\frac {5 b \,c^{2}}{9}\right ) \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}\right ) b \sqrt {2}\, \left (b \,x^{2}+a \right ) \sqrt {d e c}\, c^{2} \sqrt {e x}}{16}\right )\right )}{\sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} e^{2} \sqrt {d e c}\, c \,a^{2} \left (b \,x^{2}+a \right ) \left (a \,d^{2}+b \,c^{2}\right )^{2}}\)	\(485\)

Input:

int(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5260 vs. \(2 (366) = 732\).

Time = 30.85 (sec) , antiderivative size = 10540, normalized size of antiderivative = 22.24 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 163.48 (sec) , antiderivative size = 123299, normalized size of antiderivative = 260.12 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)**(3/2)/(d*x+c)/(b*x**2+a)**2,x)
 

Output:

-7*pi*a**(79/4)*b**(5/4)*c**2*d**(7/2)*x**(23/2)*exp(-I*pi/4)*log(1 - b**( 
1/4)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(-1/4)*gamma(5/4)*gamma(9/4) 
/(32*pi*a**(45/2)*sqrt(b)*c**2*d**(9/2)*e**(3/2)*x**(23/2)*gamma(3/4)*gamm 
a(5/4)*gamma(9/4) + 64*pi*a**(43/2)*b**(3/2)*c**4*d**(5/2)*e**(3/2)*x**(23 
/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 128*pi*a**(43/2)*b**(3/2)*c**2*d**( 
9/2)*e**(3/2)*x**(27/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 32*pi*a**(41/2) 
*b**(5/2)*c**6*sqrt(d)*e**(3/2)*x**(23/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) 
 + 256*pi*a**(41/2)*b**(5/2)*c**4*d**(5/2)*e**(3/2)*x**(27/2)*gamma(3/4)*g 
amma(5/4)*gamma(9/4) + 192*pi*a**(41/2)*b**(5/2)*c**2*d**(9/2)*e**(3/2)*x* 
*(31/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 128*pi*a**(39/2)*b**(7/2)*c**6* 
sqrt(d)*e**(3/2)*x**(27/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 384*pi*a**(3 
9/2)*b**(7/2)*c**4*d**(5/2)*e**(3/2)*x**(31/2)*gamma(3/4)*gamma(5/4)*gamma 
(9/4) + 128*pi*a**(39/2)*b**(7/2)*c**2*d**(9/2)*e**(3/2)*x**(35/2)*gamma(3 
/4)*gamma(5/4)*gamma(9/4) + 192*pi*a**(37/2)*b**(9/2)*c**6*sqrt(d)*e**(3/2 
)*x**(31/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 256*pi*a**(37/2)*b**(9/2)*c 
**4*d**(5/2)*e**(3/2)*x**(35/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 32*pi*a 
**(37/2)*b**(9/2)*c**2*d**(9/2)*e**(3/2)*x**(39/2)*gamma(3/4)*gamma(5/4)*g 
amma(9/4) + 128*pi*a**(35/2)*b**(11/2)*c**6*sqrt(d)*e**(3/2)*x**(35/2)*gam 
ma(3/4)*gamma(5/4)*gamma(9/4) + 64*pi*a**(35/2)*b**(11/2)*c**4*d**(5/2)*e* 
*(3/2)*x**(39/2)*gamma(3/4)*gamma(5/4)*gamma(9/4) + 32*pi*a**(33/2)*b**...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(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. 828 vs. \(2 (366) = 732\).

Time = 0.17 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/8*(16*d^5*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^2*c^5 + 2*a*b*c^3*d^2 + a 
^2*c*d^4)*sqrt(c*d*e)) + 2*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e + 7*(a*b^3*e 
^2)^(1/4)*a^2*b*d^3*e + 5*(a*b^3*e^2)^(3/4)*b*c^3 + 9*(a*b^3*e^2)^(3/4)*a* 
c*d^2)*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^3*b^3*c^4*e^2 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^2 + sqrt(2 
)*a^5*b*d^4*e^2) + 2*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e + 7*(a*b^3*e^2)^(1 
/4)*a^2*b*d^3*e + 5*(a*b^3*e^2)^(3/4)*b*c^3 + 9*(a*b^3*e^2)^(3/4)*a*c*d^2) 
*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^3*b^3*c^4*e^2 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^2 + sqrt(2)*a^5 
*b*d^4*e^2) + (3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e + 7*(a*b^3*e^2)^(1/4)*a^2 
*b*d^3*e - 5*(a*b^3*e^2)^(3/4)*b*c^3 - 9*(a*b^3*e^2)^(3/4)*a*c*d^2)*log(e* 
x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^3*b^3*c^ 
4*e^2 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^2 + sqrt(2)*a^5*b*d^4*e^2) - (3*(a*b^3 
*e^2)^(1/4)*a*b^2*c^2*d*e + 7*(a*b^3*e^2)^(1/4)*a^2*b*d^3*e - 5*(a*b^3*e^2 
)^(3/4)*b*c^3 - 9*(a*b^3*e^2)^(3/4)*a*c*d^2)*log(e*x - sqrt(2)*(a*e^2/b)^( 
1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^3*b^3*c^4*e^2 + 2*sqrt(2)*a^4*b 
^2*c^2*d^2*e^2 + sqrt(2)*a^5*b*d^4*e^2) + 4*(5*b^2*c^2*e^2*x^2 + 4*a*b*d^2 
*e^2*x^2 + a*b*c*d*e^2*x + 4*a*b*c^2*e^2 + 4*a^2*d^2*e^2)/((a^2*b*c^3 + a^ 
3*c*d^2)*(sqrt(e*x)*b*e^2*x^2 + sqrt(e*x)*a*e^2)))/e
 

Mupad [B] (verification not implemented)

Time = 13.09 (sec) , antiderivative size = 26582, normalized size of antiderivative = 56.08 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)),x)
 

Output:

- ((2*e)/(a*c) + (b*d*e*x)/(2*a*(a*d^2 + b*c^2)) + (b*e*x^2*(4*a*d^2 + 5*b 
*c^2))/(2*a^2*c*(a*d^2 + b*c^2)))/(b*(e*x)^(5/2) + a*e^2*(e*x)^(1/2)) - at 
an(((11875*a^5*b^10*c^15*d - a^9*b^3*(72128*a^3*c*d^15 + 265655*b^3*c^7*d^ 
9 - 76440*a*b^2*c^5*d^11 - 178585*a^2*b*c^3*d^13) + 68800*a^6*b^9*c^13*d^3 
 + 89403*a^7*b^8*c^11*d^5 - 126488*a^8*b^7*c^9*d^7)*(a^25*c^2*d^19*e^7*(-( 
49*a^3*d^6*(-a^9*b^3)^(1/2) - 25*b^3*c^6*(-a^9*b^3)^(1/2) + 30*a^5*b^4*c^5 
*d + 126*a^7*b^2*c*d^5 + 124*a^6*b^3*c^3*d^3 - 81*a*b^2*c^4*d^2*(-a^9*b^3) 
^(1/2) - 39*a^2*b*c^2*d^4*(-a^9*b^3)^(1/2))/(a^13*d^8*e^3 + a^9*b^4*c^8*e^ 
3 + 4*a^10*b^3*c^6*d^2*e^3 + 6*a^11*b^2*c^4*d^4*e^3 + 4*a^12*b*c^2*d^6*e^3 
))^(5/2)*(e*x)^(1/2)*2i - a^11*b^10*c^19*e^4*(-(49*a^3*d^6*(-a^9*b^3)^(1/2 
) - 25*b^3*c^6*(-a^9*b^3)^(1/2) + 30*a^5*b^4*c^5*d + 126*a^7*b^2*c*d^5 + 1 
24*a^6*b^3*c^3*d^3 - 81*a*b^2*c^4*d^2*(-a^9*b^3)^(1/2) - 39*a^2*b*c^2*d^4* 
(-a^9*b^3)^(1/2))/(a^13*d^8*e^3 + a^9*b^4*c^8*e^3 + 4*a^10*b^3*c^6*d^2*e^3 
 + 6*a^11*b^2*c^4*d^4*e^3 + 4*a^12*b*c^2*d^6*e^3))^(3/2)*(e*x)^(1/2)*25i - 
 a^15*b^2*d^17*e*(-(49*a^3*d^6*(-a^9*b^3)^(1/2) - 25*b^3*c^6*(-a^9*b^3)^(1 
/2) + 30*a^5*b^4*c^5*d + 126*a^7*b^2*c*d^5 + 124*a^6*b^3*c^3*d^3 - 81*a*b^ 
2*c^4*d^2*(-a^9*b^3)^(1/2) - 39*a^2*b*c^2*d^4*(-a^9*b^3)^(1/2))/(a^13*d^8* 
e^3 + a^9*b^4*c^8*e^3 + 4*a^10*b^3*c^6*d^2*e^3 + 6*a^11*b^2*c^4*d^4*e^3 + 
4*a^12*b*c^2*d^6*e^3))^(1/2)*(e*x)^(1/2)*3136i - a^12*b^9*c^17*d^2*e^4*(-( 
49*a^3*d^6*(-a^9*b^3)^(1/2) - 25*b^3*c^6*(-a^9*b^3)^(1/2) + 30*a^5*b^4*...
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 1586, normalized size of antiderivative = 3.35 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*(18*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c**3*d**2 + 
10*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*s 
qrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c**5 + 18*sqrt(x)*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**2*x**2 + 10*sqrt(x)*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**5*x**2 + 14*sqrt(x)*b**(3/4)*a**(1/4)*sqr 
t(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**3 + 6*sqrt(x)*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*b*c**4*d + 14*sqrt(x)*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*b*c**2*d 
**3*x**2 + 6*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c**4*d*x**2 
- 18*sqrt(x)*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**2 - 10*sqrt( 
x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*s 
qrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*c**5 - 18*sqrt(x)*b**(1/4)*a** 
(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**...