\(\int \frac {1}{(e x)^{5/2} (c+d x) (a+b x^2)^2} \, dx\) [419]

Optimal result

Integrand size = 24, antiderivative size = 495 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=-\frac {2}{3 a^2 c e (e x)^{3/2}}+\frac {2 d}{a^2 c^2 e^2 \sqrt {e x}}-\frac {b^2 \sqrt {e x} (c-d x)}{2 a^2 \left (b c^2+a d^2\right ) e^3 \left (a+b x^2\right )}+\frac {2 d^{11/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{5/2} \left (b c^2+a d^2\right )^2 e^{5/2}}+\frac {b^{5/4} \left (7 b^{3/2} c^3-5 \sqrt {a} b c^2 d+11 a \sqrt {b} c d^2-9 a^{3/2} d^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 e^{5/2}}-\frac {b^{5/4} \left (7 b^{3/2} c^3-5 \sqrt {a} b c^2 d+11 a \sqrt {b} c d^2-9 a^{3/2} d^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 e^{5/2}}-\frac {b^{5/4} \left (7 b^{3/2} c^3+5 \sqrt {a} b c^2 d+11 a \sqrt {b} c d^2+9 a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 e^{5/2}} \] Output:

-2/3/a^2/c/e/(e*x)^(3/2)+2*d/a^2/c^2/e^2/(e*x)^(1/2)-1/2*b^2*(e*x)^(1/2)*( 
-d*x+c)/a^2/(a*d^2+b*c^2)/e^3/(b*x^2+a)+2*d^(11/2)*arctan(d^(1/2)*(e*x)^(1 
/2)/c^(1/2)/e^(1/2))/c^(5/2)/(a*d^2+b*c^2)^2/e^(5/2)+1/8*b^(5/4)*(7*b^(3/2 
)*c^3-5*a^(1/2)*b*c^2*d+11*a*b^(1/2)*c*d^2-9*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^(11/4)/(a*d^2+b*c^2)^2/e^( 
5/2)-1/8*b^(5/4)*(7*b^(3/2)*c^3-5*a^(1/2)*b*c^2*d+11*a*b^(1/2)*c*d^2-9*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^ 
(11/4)/(a*d^2+b*c^2)^2/e^(5/2)-1/8*b^(5/4)*(7*b^(3/2)*c^3+5*a^(1/2)*b*c^2* 
d+11*a*b^(1/2)*c*d^2+9*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^(11/4)/(a*d^2+b*c^2)^2/e^(5/2 
)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\frac {4 \left (b c^2+a d^2\right ) \left (b^2 c^2 x^2 (7 c-15 d x)+4 a^2 d^2 (c-3 d x)+4 a b \left (c^3-3 c^2 d x+c d^2 x^2-3 d^3 x^3\right )\right )}{a^2 c^2 \left (a+b x^2\right )}+\frac {48 d^{11/2} x^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{5/2}}+\frac {3 \sqrt {2} b^{5/4} \left (7 b^{3/2} c^3-5 \sqrt {a} b c^2 d+11 a \sqrt {b} c d^2-9 a^{3/2} d^3\right ) x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4}}-\frac {3 \sqrt {2} b^{5/4} \left (7 b^{3/2} c^3+5 \sqrt {a} b c^2 d+11 a \sqrt {b} c d^2+9 a^{3/2} d^3\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4}}\right )}{24 \left (b c^2+a d^2\right )^2 (e x)^{5/2}} \] Input:

Integrate[1/((e*x)^(5/2)*(c + d*x)*(a + b*x^2)^2),x]
 

Output:

(x*((-4*(b*c^2 + a*d^2)*(b^2*c^2*x^2*(7*c - 15*d*x) + 4*a^2*d^2*(c - 3*d*x 
) + 4*a*b*(c^3 - 3*c^2*d*x + c*d^2*x^2 - 3*d^3*x^3)))/(a^2*c^2*(a + b*x^2) 
) + (48*d^(11/2)*x^(3/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/c^(5/2) + (3*S 
qrt[2]*b^(5/4)*(7*b^(3/2)*c^3 - 5*Sqrt[a]*b*c^2*d + 11*a*Sqrt[b]*c*d^2 - 9 
*a^(3/2)*d^3)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4 
)*Sqrt[x])])/a^(11/4) - (3*Sqrt[2]*b^(5/4)*(7*b^(3/2)*c^3 + 5*Sqrt[a]*b*c^ 
2*d + 11*a*Sqrt[b]*c*d^2 + 9*a^(3/2)*d^3)*x^(3/2)*ArcTanh[(Sqrt[2]*a^(1/4) 
*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(11/4)))/(24*(b*c^2 + a*d^2)^2 
*(e*x)^(5/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1049\) vs. \(2(495)=990\).

Time = 2.63 (sec) , antiderivative size = 1049, normalized size of antiderivative = 2.12, 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.

\(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^2 (c+d x)} \, dx\)

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{11/2}}{c^{5/2} \left (b c^2+a d^2\right )^2 e^{5/2}}+\frac {2 d^5}{c^2 \left (b c^2+a d^2\right )^2 e^2 \sqrt {e x}}-\frac {2 d^4}{3 c \left (b c^2+a d^2\right )^2 e (e x)^{3/2}}+\frac {2 b d^3}{a \left (b c^2+a d^2\right )^2 e^2 \sqrt {e x}}+\frac {b^{5/4} \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^{7/4} \left (b c^2+a d^2\right )^2 e^{5/2}}-\frac {b^{5/4} \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^{7/4} \left (b c^2+a d^2\right )^2 e^{5/2}}+\frac {b^{5/4} \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^{7/4} \left (b c^2+a d^2\right )^2 e^{5/2}}-\frac {b^{5/4} \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^{7/4} \left (b c^2+a d^2\right )^2 e^{5/2}}-\frac {2 b c d^2}{3 a \left (b c^2+a d^2\right )^2 e (e x)^{3/2}}+\frac {5 b d}{2 a^2 \left (b c^2+a d^2\right ) e^2 \sqrt {e x}}+\frac {b^{5/4} \left (7 \sqrt {b} c-5 \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^{11/4} \left (b c^2+a d^2\right ) e^{5/2}}-\frac {b^{5/4} \left (7 \sqrt {b} c-5 \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^{11/4} \left (b c^2+a d^2\right ) e^{5/2}}+\frac {b^{5/4} \left (7 \sqrt {b} c+5 \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^{11/4} \left (b c^2+a d^2\right ) e^{5/2}}-\frac {b^{5/4} \left (7 \sqrt {b} c+5 \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^{11/4} \left (b c^2+a d^2\right ) e^{5/2}}+\frac {b (c-d x)}{2 a \left (b c^2+a d^2\right ) e (e x)^{3/2} \left (b x^2+a\right )}-\frac {7 b c}{6 a^2 \left (b c^2+a d^2\right ) e (e x)^{3/2}}\)

Input:

Int[1/((e*x)^(5/2)*(c + d*x)*(a + b*x^2)^2),x]
 

Output:

(-2*b*c*d^2)/(3*a*(b*c^2 + a*d^2)^2*e*(e*x)^(3/2)) - (2*d^4)/(3*c*(b*c^2 + 
 a*d^2)^2*e*(e*x)^(3/2)) - (7*b*c)/(6*a^2*(b*c^2 + a*d^2)*e*(e*x)^(3/2)) + 
 (2*b*d^3)/(a*(b*c^2 + a*d^2)^2*e^2*Sqrt[e*x]) + (2*d^5)/(c^2*(b*c^2 + a*d 
^2)^2*e^2*Sqrt[e*x]) + (5*b*d)/(2*a^2*(b*c^2 + a*d^2)*e^2*Sqrt[e*x]) + (b* 
(c - d*x))/(2*a*(b*c^2 + a*d^2)*e*(e*x)^(3/2)*(a + b*x^2)) + (2*d^(11/2)*A 
rcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(c^(5/2)*(b*c^2 + a*d^2)^2*e 
^(5/2)) + (b^(5/4)*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^(7/4)*(b*c^2 + a*d^2)^2*e^(5/2) 
) + (b^(5/4)*(7*Sqrt[b]*c - 5*Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[ 
e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(11/4)*(b*c^2 + a*d^2)*e^(5/2)) - ( 
b^(5/4)*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^(7/4)*(b*c^2 + a*d^2)^2*e^(5/2)) - (b^(5/4 
)*(7*Sqrt[b]*c - 5*Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1 
/4)*Sqrt[e])])/(4*Sqrt[2]*a^(11/4)*(b*c^2 + a*d^2)*e^(5/2)) + (b^(5/4)*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^(7/4)*(b*c^2 + a*d^2)^2*e^(5/2)) 
 + (b^(5/4)*(7*Sqrt[b]*c + 5*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]*a^(11/4)*(b*c^2 + a 
*d^2)*e^(5/2)) - (b^(5/4)*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^...
 

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 = 0.95

method	result	size
risch	\(-\frac {2 \left (-3 d x +c \right )}{3 a^{2} c^{2} \sqrt {e x}\, x \,e^{2}}+\frac {\frac {2 b^{2} c^{2} \left (\frac {\left (\frac {1}{4} a \,d^{3}+\frac {1}{4} b \,c^{2} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (-\frac {1}{4} d^{2} e a c -\frac {1}{4} b e \,c^{3}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (-11 d^{2} e a c -7 b e \,c^{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (9 a \,d^{3}+5 b \,c^{2} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {2 a^{2} d^{6} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}}{a^{2} c^{2} e^{2}}\)	\(470\)
derivativedivides	\(2 e^{4} \left (-\frac {b^{2} \left (\frac {\left (-\frac {1}{4} a \,d^{3}-\frac {1}{4} b \,c^{2} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{2} e a c +\frac {1}{4} b e \,c^{3}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (11 d^{2} e a c +7 b e \,c^{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (-9 a \,d^{3}-5 b \,c^{2} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} a^{2} e^{6}}+\frac {d^{6} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{6} c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}-\frac {1}{3 a^{2} e^{5} c \left (e x \right )^{\frac {3}{2}}}+\frac {d}{a^{2} e^{6} c^{2} \sqrt {e x}}\right )\)	\(476\)
default	\(2 e^{4} \left (-\frac {b^{2} \left (\frac {\left (-\frac {1}{4} a \,d^{3}-\frac {1}{4} b \,c^{2} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (\frac {1}{4} d^{2} e a c +\frac {1}{4} b e \,c^{3}\right ) \sqrt {e x}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\left (11 d^{2} e a c +7 b e \,c^{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,e^{2}}+\frac {\left (-9 a \,d^{3}-5 b \,c^{2} d \right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{2} a^{2} e^{6}}+\frac {d^{6} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e^{6} c^{2} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d e c}}-\frac {1}{3 a^{2} e^{5} c \left (e x \right )^{\frac {3}{2}}}+\frac {d}{a^{2} e^{6} c^{2} \sqrt {e x}}\right )\)	\(476\)
pseudoelliptic	\(\frac {2 \left (a^{2} \left (e x \right )^{\frac {3}{2}} d^{6} \left (b \,x^{2}+a \right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )-\frac {e \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, \left (b \left (\frac {7 b \,x^{2}}{4}+a \right ) c^{3}-3 d x \left (\frac {5 b \,x^{2}}{4}+a \right ) b \,c^{2}+a \,d^{2} \left (b \,x^{2}+a \right ) c -3 a \,d^{3} x \left (b \,x^{2}+a \right )\right )}{3}\right ) e a \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\frac {9 \left (-\frac {11 \left (a \,d^{2}+\frac {7 b \,c^{2}}{11}\right ) b \left (\frac {\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )\right ) c \sqrt {\frac {a \,e^{2}}{b}}}{9}+\left (a \,d^{2}+\frac {5 b \,c^{2}}{9}\right ) d e a \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}\right )\right ) b \sqrt {2}\, \left (b \,x^{2}+a \right ) \sqrt {d e c}\, c^{2} \left (e x \right )^{\frac {3}{2}}}{8}}{e^{3} a^{3} c^{2} \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}}\)	\(486\)

Input:

int(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-2/3*(-3*d*x+c)/a^2/c^2/(e*x)^(1/2)/x/e^2+1/a^2/c^2*(2*b^2*c^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*(-11*a*c*d^2*e-7*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*(9*a*d^3+5*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)))+2*a^2*d^6/(a*d 
^2+b*c^2)^2/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))/e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5343 vs. \(2 (384) = 768\).

Time = 100.05 (sec) , antiderivative size = 10706, normalized size of antiderivative = 21.63 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(5/2)/(d*x+c)/(b*x**2+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x)^(5/2)/(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. 822 vs. \(2 (384) = 768\).

Time = 0.18 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

2*d^6*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b^2*c^6*e^2 + 2*a*b*c^4*d^2*e^2 + 
a^2*c^2*d^4*e^2)*sqrt(c*d*e)) - 1/4*(7*(a*b^3*e^2)^(1/4)*b^3*c^3*e + 11*(a 
*b^3*e^2)^(1/4)*a*b^2*c*d^2*e - 5*(a*b^3*e^2)^(3/4)*b*c^2*d - 9*(a*b^3*e^2 
)^(3/4)*a*d^3)*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^4 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^4 
+ sqrt(2)*a^5*b*d^4*e^4) - 1/4*(7*(a*b^3*e^2)^(1/4)*b^3*c^3*e + 11*(a*b^3* 
e^2)^(1/4)*a*b^2*c*d^2*e - 5*(a*b^3*e^2)^(3/4)*b*c^2*d - 9*(a*b^3*e^2)^(3/ 
4)*a*d^3)*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^4 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^4 + sq 
rt(2)*a^5*b*d^4*e^4) - 1/8*(7*(a*b^3*e^2)^(1/4)*b^3*c^3*e + 11*(a*b^3*e^2) 
^(1/4)*a*b^2*c*d^2*e + 5*(a*b^3*e^2)^(3/4)*b*c^2*d + 9*(a*b^3*e^2)^(3/4)*a 
*d^3)*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^4 + 2*sqrt(2)*a^4*b^2*c^2*d^2*e^4 + sqrt(2)*a^5*b*d^4*e^4) 
 + 1/8*(7*(a*b^3*e^2)^(1/4)*b^3*c^3*e + 11*(a*b^3*e^2)^(1/4)*a*b^2*c*d^2*e 
 + 5*(a*b^3*e^2)^(3/4)*b*c^2*d + 9*(a*b^3*e^2)^(3/4)*a*d^3)*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^4 + 
2*sqrt(2)*a^4*b^2*c^2*d^2*e^4 + sqrt(2)*a^5*b*d^4*e^4) + 1/2*(sqrt(e*x)*b^ 
2*d*e*x - sqrt(e*x)*b^2*c*e)/((a^2*b*c^2*e^2 + a^3*d^2*e^2)*(b*e^2*x^2 + a 
*e^2)) + 2/3*(3*d*e*x - c*e)/(sqrt(e*x)*a^2*c^2*e^3*x)
 

Mupad [B] (verification not implemented)

Time = 17.12 (sec) , antiderivative size = 18465, normalized size of antiderivative = 37.30 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((e*x)^(5/2)*(a + b*x^2)^2*(c + d*x)),x)
 

Output:

atan((((e*x)^(1/2)*(153664*a^10*b^18*c^36*d^5*e^18 + 1583680*a^11*b^17*c^3 
4*d^7*e^18 + 7203072*a^12*b^16*c^32*d^9*e^18 + 18986240*a^13*b^15*c^30*d^1 
1*e^18 + 32077696*a^14*b^14*c^28*d^13*e^18 + 36456320*a^15*b^13*c^26*d^15* 
e^18 + 28859648*a^16*b^12*c^24*d^17*e^18 + 16859392*a^17*b^11*c^22*d^19*e^ 
18 + 8106560*a^18*b^10*c^20*d^21*e^18 + 3428416*a^19*b^9*c^18*d^23*e^18 + 
1079296*a^20*b^8*c^16*d^25*e^18 + 165888*a^21*b^7*c^14*d^27*e^18) - ((49*b 
^3*c^6*(-a^11*b^5)^(1/2) - 81*a^3*d^6*(-a^11*b^5)^(1/2) + 70*a^6*b^5*c^5*d 
 + 198*a^8*b^3*c*d^5 + 236*a^7*b^4*c^3*d^3 + 129*a*b^2*c^4*d^2*(-a^11*b^5) 
^(1/2) + 31*a^2*b*c^2*d^4*(-a^11*b^5)^(1/2))/(64*(a^15*d^8*e^5 + a^11*b^4* 
c^8*e^5 + 4*a^12*b^3*c^6*d^2*e^5 + 6*a^13*b^2*c^4*d^4*e^5 + 4*a^14*b*c^2*d 
^6*e^5)))^(1/2)*((((49*b^3*c^6*(-a^11*b^5)^(1/2) - 81*a^3*d^6*(-a^11*b^5)^ 
(1/2) + 70*a^6*b^5*c^5*d + 198*a^8*b^3*c*d^5 + 236*a^7*b^4*c^3*d^3 + 129*a 
*b^2*c^4*d^2*(-a^11*b^5)^(1/2) + 31*a^2*b*c^2*d^4*(-a^11*b^5)^(1/2))/(64*( 
a^15*d^8*e^5 + a^11*b^4*c^8*e^5 + 4*a^12*b^3*c^6*d^2*e^5 + 6*a^13*b^2*c^4* 
d^4*e^5 + 4*a^14*b*c^2*d^6*e^5)))^(1/2)*((e*x)^(1/2)*((49*b^3*c^6*(-a^11*b 
^5)^(1/2) - 81*a^3*d^6*(-a^11*b^5)^(1/2) + 70*a^6*b^5*c^5*d + 198*a^8*b^3* 
c*d^5 + 236*a^7*b^4*c^3*d^3 + 129*a*b^2*c^4*d^2*(-a^11*b^5)^(1/2) + 31*a^2 
*b*c^2*d^4*(-a^11*b^5)^(1/2))/(64*(a^15*d^8*e^5 + a^11*b^4*c^8*e^5 + 4*a^1 
2*b^3*c^6*d^2*e^5 + 6*a^13*b^2*c^4*d^4*e^5 + 4*a^14*b*c^2*d^6*e^5)))^(1/2) 
*(262144*a^20*b^16*c^42*d^3*e^28 + 2621440*a^21*b^15*c^40*d^5*e^28 + 11...
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1663, normalized size of antiderivative = 3.36 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*( - 54*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**3 
*x - 30*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**5*d*x - 54*sqr 
t(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**3*x**3 - 30*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*d*x**3 + 66*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**2*x + 42*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*b**2*c**6*x + 66*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*b**2*c**4*d**2*x**3 + 42*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)))*b* 
*3*c**6*x**3 + 54*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* 
*3*x + 30*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**5*d*x + 54*s 
qrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sq...