\(\int \frac {1}{(e x)^{3/2} (c+d x)^3 (a+b x^2)^2} \, dx\) [431]

Optimal result

Integrand size = 24, antiderivative size = 715 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^3 \left (a+b x^2\right )^2} \, dx=-\frac {2}{a^2 c^3 e \sqrt {e x}}-\frac {d^5 \sqrt {e x}}{2 c^2 \left (b c^2+a d^2\right )^2 e^2 (c+d x)^2}-\frac {d^5 \left (23 b c^2+7 a d^2\right ) \sqrt {e x}}{4 c^3 \left (b c^2+a d^2\right )^3 e^2 (c+d x)}+\frac {b^2 \sqrt {e x} \left (a^2 d^3-b^2 c^3 x-3 a b c d (c-d x)\right )}{2 a^2 \left (b c^2+a d^2\right )^3 e^2 \left (a+b x^2\right )}-\frac {d^{9/2} \left (143 b^2 c^4+62 a b c^2 d^2+15 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 c^{7/2} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {b^{7/4} \left (5 b^{5/2} c^5+9 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2+46 a^{3/2} b c^2 d^3-39 a^2 \sqrt {b} c d^4-11 a^{5/2} d^5\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {b^{7/4} \left (5 b^{5/2} c^5+9 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2+46 a^{3/2} b c^2 d^3-39 a^2 \sqrt {b} c d^4-11 a^{5/2} d^5\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {b^{7/4} \left (5 b^{5/2} c^5-9 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2-46 a^{3/2} b c^2 d^3-39 a^2 \sqrt {b} c d^4+11 a^{5/2} d^5\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{9/4} \left (b c^2+a d^2\right )^4 e^{3/2}} \] Output:

-2/a^2/c^3/e/(e*x)^(1/2)-1/2*d^5*(e*x)^(1/2)/c^2/(a*d^2+b*c^2)^2/e^2/(d*x+ 
c)^2-1/4*d^5*(7*a*d^2+23*b*c^2)*(e*x)^(1/2)/c^3/(a*d^2+b*c^2)^3/e^2/(d*x+c 
)+1/2*b^2*(e*x)^(1/2)*(a^2*d^3-b^2*c^3*x-3*a*b*c*d*(-d*x+c))/a^2/(a*d^2+b* 
c^2)^3/e^2/(b*x^2+a)-1/4*d^(9/2)*(15*a^2*d^4+62*a*b*c^2*d^2+143*b^2*c^4)*a 
rctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(7/2)/(a*d^2+b*c^2)^4/e^(3/2) 
+1/8*b^(7/4)*(5*b^(5/2)*c^5+9*a^(1/2)*b^2*c^4*d+14*a*b^(3/2)*c^3*d^2+46*a^ 
(3/2)*b*c^2*d^3-39*a^2*b^(1/2)*c*d^4-11*a^(5/2)*d^5)*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)^4/e^(3/2)-1 
/8*b^(7/4)*(5*b^(5/2)*c^5+9*a^(1/2)*b^2*c^4*d+14*a*b^(3/2)*c^3*d^2+46*a^(3 
/2)*b*c^2*d^3-39*a^2*b^(1/2)*c*d^4-11*a^(5/2)*d^5)*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)^4/e^(3/2)+1/8 
*b^(7/4)*(5*b^(5/2)*c^5-9*a^(1/2)*b^2*c^4*d+14*a*b^(3/2)*c^3*d^2-46*a^(3/2 
)*b*c^2*d^3-39*a^2*b^(1/2)*c*d^4+11*a^(5/2)*d^5)*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)^4/e^(3/2)
 

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 574, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\frac {2 \left (b c^2+a d^2\right ) \left (10 b^4 c^6 x^2 (c+d x)^2+2 a b^3 c^4 (c+d x)^2 \left (4 c^2+3 c d x+9 d^2 x^2\right )+a^4 d^6 \left (8 c^2+25 c d x+15 d^2 x^2\right )+a^3 b d^4 \left (24 c^4+73 c^3 d x+55 c^2 d^2 x^2+25 c d^3 x^3+15 d^4 x^4\right )+a^2 b^2 c^2 d^2 \left (24 c^4+46 c^3 d x+44 c^2 d^2 x^2+71 c d^3 x^3+47 d^4 x^4\right )\right )}{a^2 c^3 (c+d x)^2 \left (a+b x^2\right )}-\frac {2 d^{9/2} \left (143 b^2 c^4+62 a b c^2 d^2+15 a^2 d^4\right ) \sqrt {x} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{7/2}}+\frac {\sqrt {2} b^{7/4} \left (5 b^{5/2} c^5+9 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2+46 a^{3/2} b c^2 d^3-39 a^2 \sqrt {b} c d^4-11 a^{5/2} d^5\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4}}+\frac {\sqrt {2} b^{7/4} \left (5 b^{5/2} c^5-9 \sqrt {a} b^2 c^4 d+14 a b^{3/2} c^3 d^2-46 a^{3/2} b c^2 d^3-39 a^2 \sqrt {b} c d^4+11 a^{5/2} d^5\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4}}\right )}{8 \left (b c^2+a d^2\right )^4 (e x)^{3/2}} \] Input:

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

Output:

(x*((-2*(b*c^2 + a*d^2)*(10*b^4*c^6*x^2*(c + d*x)^2 + 2*a*b^3*c^4*(c + d*x 
)^2*(4*c^2 + 3*c*d*x + 9*d^2*x^2) + a^4*d^6*(8*c^2 + 25*c*d*x + 15*d^2*x^2 
) + a^3*b*d^4*(24*c^4 + 73*c^3*d*x + 55*c^2*d^2*x^2 + 25*c*d^3*x^3 + 15*d^ 
4*x^4) + a^2*b^2*c^2*d^2*(24*c^4 + 46*c^3*d*x + 44*c^2*d^2*x^2 + 71*c*d^3* 
x^3 + 47*d^4*x^4)))/(a^2*c^3*(c + d*x)^2*(a + b*x^2)) - (2*d^(9/2)*(143*b^ 
2*c^4 + 62*a*b*c^2*d^2 + 15*a^2*d^4)*Sqrt[x]*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt 
[c]])/c^(7/2) + (Sqrt[2]*b^(7/4)*(5*b^(5/2)*c^5 + 9*Sqrt[a]*b^2*c^4*d + 14 
*a*b^(3/2)*c^3*d^2 + 46*a^(3/2)*b*c^2*d^3 - 39*a^2*Sqrt[b]*c*d^4 - 11*a^(5 
/2)*d^5)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr 
t[x])])/a^(9/4) + (Sqrt[2]*b^(7/4)*(5*b^(5/2)*c^5 - 9*Sqrt[a]*b^2*c^4*d + 
14*a*b^(3/2)*c^3*d^2 - 46*a^(3/2)*b*c^2*d^3 - 39*a^2*Sqrt[b]*c*d^4 + 11*a^ 
(5/2)*d^5)*Sqrt[x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sq 
rt[b]*x)])/a^(9/4)))/(8*(b*c^2 + a*d^2)^4*(e*x)^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1549\) vs. \(2(715)=1430\).

Time = 3.35 (sec) , antiderivative size = 1549, normalized size of antiderivative = 2.17, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {15 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{9/2}}{4 c^{7/2} \left (b c^2+a d^2\right )^2 e^{3/2}}-\frac {12 b \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 )^3 e^{3/2}}-\frac {4 b \left (5 b c^2-a d^2\right ) \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 )^4 e^{3/2}}-\frac {15 d^4}{4 c^3 \left (b c^2+a d^2\right )^2 e \sqrt {e x}}-\frac {12 b d^4}{c \left (b c^2+a d^2\right )^3 e \sqrt {e x}}-\frac {4 b \left (5 b c^2-a d^2\right ) d^4}{c \left (b c^2+a d^2\right )^4 e \sqrt {e x}}+\frac {5 d^4}{4 c^2 \left (b c^2+a d^2\right )^2 e \sqrt {e x} (c+d x)}+\frac {4 b d^4}{\left (b c^2+a d^2\right )^3 e \sqrt {e x} (c+d x)}+\frac {d^4}{2 c \left (b c^2+a d^2\right )^2 e \sqrt {e x} (c+d x)^2}+\frac {\sqrt {2} b^{7/4} \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 ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {\sqrt {2} b^{7/4} \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 ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {b^{7/4} \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 ) \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} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}+\frac {b^{7/4} \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 ) \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} a^{5/4} \left (b c^2+a d^2\right )^4 e^{3/2}}-\frac {12 b^2 c \left (b c^2-a d^2\right ) d^2}{a \left (b c^2+a d^2\right )^4 e \sqrt {e x}}+\frac {b^{7/4} \left (5 b^{3/2} c^3+9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c-3 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 )^3 e^{3/2}}-\frac {b^{7/4} \left (5 b^{3/2} c^3+9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c-3 a^{3/2} d^3\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 )^3 e^{3/2}}-\frac {b^{7/4} \left (5 b^{3/2} c^3-9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c+3 a^{3/2} d^3\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 )^3 e^{3/2}}+\frac {b^{7/4} \left (5 b^{3/2} c^3-9 \sqrt {a} b d c^2-15 a \sqrt {b} d^2 c+3 a^{3/2} d^3\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 )^3 e^{3/2}}-\frac {5 b^2 c \left (b c^2-3 a d^2\right )}{2 a^2 \left (b c^2+a d^2\right )^3 e \sqrt {e x}}+\frac {b^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 a \left (b c^2+a d^2\right )^3 e \sqrt {e x} \left (b x^2+a\right )}\)

Input:

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

Output:

(-12*b^2*c*d^2*(b*c^2 - a*d^2))/(a*(b*c^2 + a*d^2)^4*e*Sqrt[e*x]) - (4*b*d 
^4*(5*b*c^2 - a*d^2))/(c*(b*c^2 + a*d^2)^4*e*Sqrt[e*x]) - (12*b*d^4)/(c*(b 
*c^2 + a*d^2)^3*e*Sqrt[e*x]) - (5*b^2*c*(b*c^2 - 3*a*d^2))/(2*a^2*(b*c^2 + 
 a*d^2)^3*e*Sqrt[e*x]) - (15*d^4)/(4*c^3*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]) + 
d^4/(2*c*(b*c^2 + a*d^2)^2*e*Sqrt[e*x]*(c + d*x)^2) + (4*b*d^4)/((b*c^2 + 
a*d^2)^3*e*Sqrt[e*x]*(c + d*x)) + (5*d^4)/(4*c^2*(b*c^2 + a*d^2)^2*e*Sqrt[ 
e*x]*(c + d*x)) + (b^2*(c*(b*c^2 - 3*a*d^2) - d*(3*b*c^2 - a*d^2)*x))/(2*a 
*(b*c^2 + a*d^2)^3*e*Sqrt[e*x]*(a + b*x^2)) - (4*b*d^(9/2)*(5*b*c^2 - a*d^ 
2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(c^(3/2)*(b*c^2 + a*d^2) 
^4*e^(3/2)) - (12*b*d^(9/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])]) 
/(c^(3/2)*(b*c^2 + a*d^2)^3*e^(3/2)) - (15*d^(9/2)*ArcTan[(Sqrt[d]*Sqrt[e* 
x])/(Sqrt[c]*Sqrt[e])])/(4*c^(7/2)*(b*c^2 + a*d^2)^2*e^(3/2)) + (b^(7/4)*( 
5*b^(3/2)*c^3 + 9*Sqrt[a]*b*c^2*d - 15*a*Sqrt[b]*c*d^2 - 3*a^(3/2)*d^3)*Ar 
cTan[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)^3*e^(3/2)) + (Sqrt[2]*b^(7/4)*d^2*(3*b^(3/2)*c^3 + 5*Sqr 
t[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - a^(3/2)*d^3)*ArcTan[1 - (Sqrt[2]*b^(1/4 
)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(a^(5/4)*(b*c^2 + a*d^2)^4*e^(3/2)) - (b^ 
(7/4)*(5*b^(3/2)*c^3 + 9*Sqrt[a]*b*c^2*d - 15*a*Sqrt[b]*c*d^2 - 3*a^(3/2)* 
d^3)*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)^3*e^(3/2)) - (Sqrt[2]*b^(7/4)*d^2*(3*b^(3/2)*c...
 

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 = 1.31 (sec) , antiderivative size = 642, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(2 e^{6} \left (-\frac {d^{5} \left (\frac {\left (\frac {7}{8} a^{2} d^{5}+\frac {15}{4} d^{3} a \,c^{2} b +\frac {23}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+34 b \,c^{2} d^{2} a +25 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}+62 b \,c^{2} d^{2} a +143 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{c^{3} e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {b^{2} \left (\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 {\left (11 d^{5} e \,a^{3}-46 a^{2} c^{2} e \,d^{3} b -9 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 {\left (39 a^{2} c \,d^{4} b -14 a \,b^{2} c^{3} d^{2}-5 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}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} e^{7} a^{2}}-\frac {1}{c^{3} e^{7} a^{2} \sqrt {e x}}\right )\)	\(642\)
default	\(2 e^{6} \left (-\frac {d^{5} \left (\frac {\left (\frac {7}{8} a^{2} d^{5}+\frac {15}{4} d^{3} a \,c^{2} b +\frac {23}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+34 b \,c^{2} d^{2} a +25 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}+62 b \,c^{2} d^{2} a +143 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{c^{3} e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {b^{2} \left (\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 {\left (11 d^{5} e \,a^{3}-46 a^{2} c^{2} e \,d^{3} b -9 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 {\left (39 a^{2} c \,d^{4} b -14 a \,b^{2} c^{3} d^{2}-5 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}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4} e^{7} a^{2}}-\frac {1}{c^{3} e^{7} a^{2} \sqrt {e x}}\right )\)	\(642\)
risch	\(-\frac {2}{a^{2} c^{3} e \sqrt {e x}}-\frac {-\frac {2 c^{3} b^{2} \left (\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 {\left (11 d^{5} e \,a^{3}-46 a^{2} c^{2} e \,d^{3} b -9 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 {\left (39 a^{2} c \,d^{4} b -14 a \,b^{2} c^{3} d^{2}-5 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}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}+\frac {2 a^{2} d^{5} \left (\frac {\left (\frac {7}{8} a^{2} d^{5}+\frac {15}{4} d^{3} a \,c^{2} b +\frac {23}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+34 b \,c^{2} d^{2} a +25 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}+62 b \,c^{2} d^{2} a +143 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{4}}}{a^{2} c^{3} e}\)	\(644\)
pseudoelliptic	\(\frac {\frac {11 d \left (d x +c \right )^{2} \left (a^{2} d^{4}-\frac {46}{11} b \,c^{2} d^{2} a -\frac {9}{11} 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 ) \sqrt {d e c}\, b^{2} \sqrt {2}\, \left (b \,x^{2}+a \right ) c^{3} \sqrt {e x}\, \sqrt {\frac {a \,e^{2}}{b}}}{16}+\frac {39 \left (\left (-\frac {20 d^{5} \left (a^{2} d^{4}+\frac {62}{15} b \,c^{2} d^{2} a +\frac {143}{15} b^{2} c^{4}\right ) \left (d x +c \right )^{2} \left (b \,x^{2}+a \right ) a^{2} \sqrt {e x}\, \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{13}-\frac {32 \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {5 c^{6} x^{2} \left (d x +c \right )^{2} b^{4}}{4}+\left (d x +c \right )^{2} \left (\frac {9}{4} d^{2} x^{2}+\frac {3}{4} c d x +c^{2}\right ) a \,c^{4} b^{3}+3 d^{2} \left (\frac {47}{24} d^{4} x^{4}+\frac {71}{24} c \,d^{3} x^{3}+\frac {11}{6} d^{2} c^{2} x^{2}+\frac {23}{12} c^{3} d x +c^{4}\right ) a^{2} c^{2} b^{2}+3 d^{4} a^{3} \left (\frac {5}{8} d^{4} x^{4}+\frac {25}{24} c \,d^{3} x^{3}+\frac {55}{24} d^{2} c^{2} x^{2}+\frac {73}{24} c^{3} d x +c^{4}\right ) b +a^{4} d^{6} \left (\frac {15}{8} d^{2} x^{2}+\frac {25}{8} c d x +c^{2}\right )\right )}{39}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\left (d x +c \right )^{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}\, b^{2} \sqrt {2}\, \left (b \,x^{2}+a \right ) \left (a^{2} d^{4}-\frac {14}{39} b \,c^{2} d^{2} a -\frac {5}{39} b^{2} c^{4}\right ) c^{4} \sqrt {e x}\right ) e}{16}}{e^{2} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (a \,d^{2}+b \,c^{2}\right )^{4} a^{2} \left (b \,x^{2}+a \right ) \left (d x +c \right )^{2} c^{3} \sqrt {d e c}\, \sqrt {e x}}\)	\(714\)

Input:

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

Output:

2*e^6*(-d^5/c^3/e^7/(a*d^2+b*c^2)^4*(((7/8*a^2*d^5+15/4*d^3*a*c^2*b+23/8*b 
^2*c^4*d)*(e*x)^(3/2)+1/8*c*e*(9*a^2*d^4+34*a*b*c^2*d^2+25*b^2*c^4)*(e*x)^ 
(1/2))/(d*e*x+c*e)^2+1/8*(15*a^2*d^4+62*a*b*c^2*d^2+143*b^2*c^4)/(d*e*c)^( 
1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))+b^2/(a*d^2+b*c^2)^4/e^7/a^2*(((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)+1/32* 
(11*a^3*d^5*e-46*a^2*b*c^2*d^3*e-9*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) 
)+1/32*(39*a^2*b*c*d^4-14*a*b^2*c^3*d^2-5*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)))- 
1/c^3/e^7/a^2/(e*x)^(1/2))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(e*x)^(3/2)/(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. 1470 vs. \(2 (586) = 1172\).

Time = 0.25 (sec) , antiderivative size = 1470, normalized size of antiderivative = 2.06 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(2*(9*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 46*(a*b^3*e^2)^(1/4)*a^2*b^2* 
c^2*d^3*e - 11*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e + 5*(a*b^3*e^2)^(3/4)*b^2*c^5 
 + 14*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 - 39*(a*b^3*e^2)^(3/4)*a^2*c*d^4)*arct 
an(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(s 
qrt(2)*a^3*b^4*c^8*e^2 + 4*sqrt(2)*a^4*b^3*c^6*d^2*e^2 + 6*sqrt(2)*a^5*b^2 
*c^4*d^4*e^2 + 4*sqrt(2)*a^6*b*c^2*d^6*e^2 + sqrt(2)*a^7*d^8*e^2) + 2*(9*( 
a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 46*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e - 
11*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e + 5*(a*b^3*e^2)^(3/4)*b^2*c^5 + 14*(a*b^3 
*e^2)^(3/4)*a*b*c^3*d^2 - 39*(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^3* 
b^4*c^8*e^2 + 4*sqrt(2)*a^4*b^3*c^6*d^2*e^2 + 6*sqrt(2)*a^5*b^2*c^4*d^4*e^ 
2 + 4*sqrt(2)*a^6*b*c^2*d^6*e^2 + sqrt(2)*a^7*d^8*e^2) + (9*(a*b^3*e^2)^(1 
/4)*a*b^3*c^4*d*e + 46*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^3*e - 11*(a*b^3*e^2 
)^(1/4)*a^3*b*d^5*e - 5*(a*b^3*e^2)^(3/4)*b^2*c^5 - 14*(a*b^3*e^2)^(3/4)*a 
*b*c^3*d^2 + 39*(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^3*b^4*c^8*e^2 + 4*sqrt(2)*a^4*b 
^3*c^6*d^2*e^2 + 6*sqrt(2)*a^5*b^2*c^4*d^4*e^2 + 4*sqrt(2)*a^6*b*c^2*d^6*e 
^2 + sqrt(2)*a^7*d^8*e^2) - (9*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d*e + 46*(a*b^3 
*e^2)^(1/4)*a^2*b^2*c^2*d^3*e - 11*(a*b^3*e^2)^(1/4)*a^3*b*d^5*e - 5*(a*b^ 
3*e^2)^(3/4)*b^2*c^5 - 14*(a*b^3*e^2)^(3/4)*a*b*c^3*d^2 + 39*(a*b^3*e^2...
 

Mupad [B] (verification not implemented)

Time = 15.83 (sec) , antiderivative size = 9622, normalized size of antiderivative = 13.46 \[ \int \frac {1}{(e x)^{3/2} (c+d x)^3 \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(root(146800640*a^22*b^3*c^13*d^26*e^9*g^6 + 146800640*a^12*b^13 
*c^33*d^6*e^9*g^6 + 31457280*a^23*b^2*c^11*d^28*e^9*g^6 + 31457280*a^11*b^ 
14*c^35*d^4*e^9*g^6 + 4194304*a^10*b^15*c^37*d^2*e^9*g^6 + 3373793280*a^17 
*b^8*c^23*d^16*e^9*g^6 + 2998927360*a^18*b^7*c^21*d^18*e^9*g^6 + 299892736 
0*a^16*b^9*c^25*d^14*e^9*g^6 + 4194304*a^24*b*c^9*d^30*e^9*g^6 + 209924915 
2*a^19*b^6*c^19*d^20*e^9*g^6 + 2099249152*a^15*b^10*c^27*d^12*e^9*g^6 + 11 
45044992*a^20*b^5*c^17*d^22*e^9*g^6 + 1145044992*a^14*b^11*c^29*d^10*e^9*g 
^6 + 477102080*a^21*b^4*c^15*d^24*e^9*g^6 + 477102080*a^13*b^12*c^31*d^8*e 
^9*g^6 + 262144*a^25*c^7*d^32*e^9*g^6 + 262144*a^9*b^16*c^39*e^9*g^6 + 203 
882496*a^8*b^13*c^26*d^7*e^6*g^4 + 120070144*a^19*b^2*c^4*d^29*e^6*g^4 + 7 
1204864*a^7*b^14*c^28*d^5*e^6*g^4 + 11730944*a^6*b^15*c^30*d^3*e^6*g^4 + 1 
1398053888*a^14*b^7*c^14*d^19*e^6*g^4 + 14991360*a^20*b*c^2*d^31*e^6*g^4 + 
 737280*a^5*b^16*c^32*d*e^6*g^4 + 2095853568*a^17*b^4*c^8*d^25*e^6*g^4 + 1 
989353472*a^11*b^10*c^20*d^13*e^6*g^4 + 5545377792*a^12*b^9*c^18*d^15*e^6* 
g^4 + 9786470400*a^13*b^8*c^16*d^17*e^6*g^4 + 5178687488*a^16*b^5*c^10*d^2 
3*e^6*g^4 + 604094464*a^18*b^3*c^6*d^27*e^6*g^4 + 573227008*a^10*b^11*c^22 
*d^11*e^6*g^4 + 9142550528*a^15*b^6*c^12*d^21*e^6*g^4 + 322883584*a^9*b^12 
*c^24*d^9*e^6*g^4 + 921600*a^21*d^33*e^6*g^4 + 92067840*a^12*b^5*c^3*d^24* 
e^3*g^2 + 50719872*a^4*b^13*c^19*d^8*e^3*g^2 + 20577792*a^3*b^14*c^21*d^6* 
e^3*g^2 - 20542464*a^11*b^6*c^5*d^22*e^3*g^2 + 5154624*a^2*b^15*c^23*d^...
 

Reduce [F]

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

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

Output:

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