\(\int \frac {(e x)^{5/2} (c+d x)}{(a+b x^2)^3} \, dx\) [434]

Optimal result

Integrand size = 22, antiderivative size = 305 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {e^2 \sqrt {e x} (a d-b c x)}{4 b^2 \left (a+b x^2\right )^2}-\frac {3 e^2 \sqrt {e x} (3 a d-b c x)}{16 a b^2 \left (a+b x^2\right )}-\frac {\left (3 \sqrt {b} c+5 \sqrt {a} d\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{5/4} b^{9/4}}+\frac {\left (3 \sqrt {b} c+5 \sqrt {a} d\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{5/4} b^{9/4}}-\frac {\left (3 \sqrt {b} c-5 \sqrt {a} d\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{5/4} b^{9/4}} \] Output:

1/4*e^2*(e*x)^(1/2)*(-b*c*x+a*d)/b^2/(b*x^2+a)^2-3/16*e^2*(e*x)^(1/2)*(-b* 
c*x+3*a*d)/a/b^2/(b*x^2+a)-1/64*(3*b^(1/2)*c+5*a^(1/2)*d)*e^(5/2)*arctan(1 
-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/b^(9/4)+1/64 
*(3*b^(1/2)*c+5*a^(1/2)*d)*e^(5/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^ 
(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/b^(9/4)-1/64*(3*b^(1/2)*c-5*a^(1/2)*d)*e^(5 
/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)/b^(9/4)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.65 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {(e x)^{5/2} \left (-\frac {4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (5 a^2 d-3 b^2 c x^3+a b x (c+9 d x)\right )}{\left (a+b x^2\right )^2}-\sqrt {2} \left (3 \sqrt {b} c+5 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (-3 \sqrt {b} c+5 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{64 a^{5/4} b^{9/4} x^{5/2}} \] Input:

Integrate[((e*x)^(5/2)*(c + d*x))/(a + b*x^2)^3,x]
 

Output:

((e*x)^(5/2)*((-4*a^(1/4)*b^(1/4)*Sqrt[x]*(5*a^2*d - 3*b^2*c*x^3 + a*b*x*( 
c + 9*d*x)))/(a + b*x^2)^2 - Sqrt[2]*(3*Sqrt[b]*c + 5*Sqrt[a]*d)*ArcTan[(S 
qrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + Sqrt[2]*(-3*Sqrt[ 
b]*c + 5*Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + S 
qrt[b]*x)]))/(64*a^(5/4)*b^(9/4)*x^(5/2))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {549, 27, 550, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 549

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 550

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[((e*x)^(5/2)*(c + d*x))/(a + b*x^2)^3,x]
 

Output:

-1/4*(e*(e*x)^(3/2)*(c + d*x))/(b*(a + b*x^2)^2) + (e^2*(-1/2*(Sqrt[e*x]*( 
5*a*d - 3*b*c*x))/(a*b*(a + b*x^2)) + (e*((Sqrt[b]*(3*c + (5*Sqrt[a]*d)/Sq 
rt[b])*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[ 
2]*a^(1/4)*b^(1/4)*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^( 
1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/2 - (Sqrt[b]*(3*c - (5* 
Sqrt[a]*d)/Sqrt[b])*(-1/2*Log[Sqrt[a]*e + Sqrt[b]*e*x - Sqrt[2]*a^(1/4)*b^ 
(1/4)*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]) + Log[Sqrt[a]*e 
 + Sqrt[b]*e*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*a^( 
1/4)*b^(1/4)*Sqrt[e])))/2))/(2*a*b)))/(8*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 549

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[e*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), 
 x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b 
*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 
1]
 

rule 550

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(e*x)^m*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x 
] - Simp[e/(2*a*b*(p + 1))   Int[(e*x)^(m - 1)*(a*d*m - b*c*(m + 2*p + 3)*x 
)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && 
 LtQ[0, m, 1]
 

rule 554

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] 
 :> Simp[2   Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] 
 /; FreeQ[{a, b, c, d, e}, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\frac {3 c \left (e x \right )^{\frac {7}{2}}}{32 a e}-\frac {9 d \left (e x \right )^{\frac {5}{2}}}{32 b}-\frac {c e \left (e x \right )^{\frac {3}{2}}}{32 b}-\frac {5 a d \,e^{2} \sqrt {e x}}{32 b^{2}}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {5 d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {3 c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a e \,b^{2}}\right )\)	\(364\)
default	\(2 e^{4} \left (\frac {\frac {3 c \left (e x \right )^{\frac {7}{2}}}{32 a e}-\frac {9 d \left (e x \right )^{\frac {5}{2}}}{32 b}-\frac {c e \left (e x \right )^{\frac {3}{2}}}{32 b}-\frac {5 a d \,e^{2} \sqrt {e x}}{32 b^{2}}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {5 d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {3 c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{32 a e \,b^{2}}\right )\)	\(364\)
pseudoelliptic	\(\frac {5 e^{2} \left (\frac {3 \sqrt {2}\, c e \left (b \,x^{2}+a \right )^{2} \ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{10}+\frac {\sqrt {2}\, d \sqrt {\frac {a \,e^{2}}{b}}\, \left (b \,x^{2}+a \right )^{2} \ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +\frac {3 c e}{5}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+\sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +\frac {3 c e}{5}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \sqrt {e x}\, \left (a^{2} d +\frac {b x \left (9 d x +c \right ) a}{5}-\frac {3 b^{2} c \,x^{3}}{5}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}\right )}{64 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a \,b^{2} \left (b \,x^{2}+a \right )^{2}}\)	\(384\)

Input:

int((e*x)^(5/2)*(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^4*((3/32*c/a/e*(e*x)^(7/2)-9/32*d*(e*x)^(5/2)/b-1/32*c*e*(e*x)^(3/2)/b 
-5/32*a*d*e^2*(e*x)^(1/2)/b^2)/(b*e^2*x^2+a*e^2)^2+1/32/a/e/b^2*(5/8*d/e*( 
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*ar 
ctan(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))+3/8*c/(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*ar 
ctan(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. 1157 vs. \(2 (220) = 440\).

Time = 0.14 (sec) , antiderivative size = 1157, normalized size of antiderivative = 3.79 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

1/64*((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(-(30*c*d*e^5 + a^2*b^4*sq 
rt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^10/(a^5*b^9)))/(a^2*b^4 
))*log(-(81*b^2*c^4 - 625*a^2*d^4)*sqrt(e*x)*e^7 + (3*a^4*b^7*c*sqrt(-(81* 
b^2*c^4 - 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^10/(a^5*b^9)) - 5*(9*a^2*b^3*c^ 
2*d - 25*a^3*b^2*d^3)*e^5)*sqrt(-(30*c*d*e^5 + a^2*b^4*sqrt(-(81*b^2*c^4 - 
 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^10/(a^5*b^9)))/(a^2*b^4))) - (a*b^4*x^4 
+ 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(-(30*c*d*e^5 + a^2*b^4*sqrt(-(81*b^2*c^4 - 
 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^10/(a^5*b^9)))/(a^2*b^4))*log(-(81*b^2*c 
^4 - 625*a^2*d^4)*sqrt(e*x)*e^7 - (3*a^4*b^7*c*sqrt(-(81*b^2*c^4 - 450*a*b 
*c^2*d^2 + 625*a^2*d^4)*e^10/(a^5*b^9)) - 5*(9*a^2*b^3*c^2*d - 25*a^3*b^2* 
d^3)*e^5)*sqrt(-(30*c*d*e^5 + a^2*b^4*sqrt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 
+ 625*a^2*d^4)*e^10/(a^5*b^9)))/(a^2*b^4))) - (a*b^4*x^4 + 2*a^2*b^3*x^2 + 
 a^3*b^2)*sqrt(-(30*c*d*e^5 - a^2*b^4*sqrt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 
+ 625*a^2*d^4)*e^10/(a^5*b^9)))/(a^2*b^4))*log(-(81*b^2*c^4 - 625*a^2*d^4) 
*sqrt(e*x)*e^7 + (3*a^4*b^7*c*sqrt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 + 625*a^ 
2*d^4)*e^10/(a^5*b^9)) + 5*(9*a^2*b^3*c^2*d - 25*a^3*b^2*d^3)*e^5)*sqrt(-( 
30*c*d*e^5 - a^2*b^4*sqrt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^ 
10/(a^5*b^9)))/(a^2*b^4))) + (a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(-( 
30*c*d*e^5 - a^2*b^4*sqrt(-(81*b^2*c^4 - 450*a*b*c^2*d^2 + 625*a^2*d^4)*e^ 
10/(a^5*b^9)))/(a^2*b^4))*log(-(81*b^2*c^4 - 625*a^2*d^4)*sqrt(e*x)*e^7...
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 83.83 (sec) , antiderivative size = 6147, normalized size of antiderivative = 20.15 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**(5/2)*(d*x+c)/(b*x**2+a)**3,x)
 

Output:

c*(-21*a**(23/4)*b*e**(5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I 
*pi/4)/a**(1/4))*gamma(7/4)/(256*a**7*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*ga 
mma(11/4) + 768*a**6*b**(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4) + 768*a 
**5*b**(19/4)*x**(19/2)*exp(3*I*pi/4)*gamma(11/4) + 256*a**4*b**(23/4)*x** 
(23/2)*exp(3*I*pi/4)*gamma(11/4)) - 21*I*a**(23/4)*b*e**(5/2)*x**(11/2)*lo 
g(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(7/4)/(256*a**7* 
b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 768*a**6*b**(15/4)*x**(15/ 
2)*exp(3*I*pi/4)*gamma(11/4) + 768*a**5*b**(19/4)*x**(19/2)*exp(3*I*pi/4)* 
gamma(11/4) + 256*a**4*b**(23/4)*x**(23/2)*exp(3*I*pi/4)*gamma(11/4)) + 21 
*a**(23/4)*b*e**(5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/ 
4)/a**(1/4))*gamma(7/4)/(256*a**7*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma( 
11/4) + 768*a**6*b**(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4) + 768*a**5* 
b**(19/4)*x**(19/2)*exp(3*I*pi/4)*gamma(11/4) + 256*a**4*b**(23/4)*x**(23/ 
2)*exp(3*I*pi/4)*gamma(11/4)) + 21*I*a**(23/4)*b*e**(5/2)*x**(11/2)*log(1 
- b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(7/4)/(256*a**7*b**( 
11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 768*a**6*b**(15/4)*x**(15/2)*e 
xp(3*I*pi/4)*gamma(11/4) + 768*a**5*b**(19/4)*x**(19/2)*exp(3*I*pi/4)*gamm 
a(11/4) + 256*a**4*b**(23/4)*x**(23/2)*exp(3*I*pi/4)*gamma(11/4)) - 63*a** 
(19/4)*b**2*e**(5/2)*x**(15/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/ 
a**(1/4))*gamma(7/4)/(256*a**7*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a)^3,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 [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.34 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} + 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} + 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} - 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a^{2} b^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e^{2} - 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c e\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a^{2} b^{4}} + \frac {3 \, \sqrt {e x} b^{2} c e^{6} x^{3} - 9 \, \sqrt {e x} a b d e^{6} x^{2} - \sqrt {e x} a b c e^{6} x - 5 \, \sqrt {e x} a^{2} d e^{6}}{16 \, {\left (b e^{2} x^{2} + a e^{2}\right )}^{2} a b^{2}} \] Input:

integrate((e*x)^(5/2)*(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

1/64*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*a*b*d*e^2 + 3*(a*b^3*e^2)^(3/4)*c*e)*arc 
tan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/( 
a^2*b^4) + 1/64*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*a*b*d*e^2 + 3*(a*b^3*e^2)^(3/ 
4)*c*e)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2 
/b)^(1/4))/(a^2*b^4) + 1/128*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*a*b*d*e^2 - 3*(a 
*b^3*e^2)^(3/4)*c*e)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a* 
e^2/b))/(a^2*b^4) - 1/128*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*a*b*d*e^2 - 3*(a*b^ 
3*e^2)^(3/4)*c*e)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2 
/b))/(a^2*b^4) + 1/16*(3*sqrt(e*x)*b^2*c*e^6*x^3 - 9*sqrt(e*x)*a*b*d*e^6*x 
^2 - sqrt(e*x)*a*b*c*e^6*x - 5*sqrt(e*x)*a^2*d*e^6)/((b*e^2*x^2 + a*e^2)^2 
*a*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 849, normalized size of antiderivative = 2.78 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(((e*x)^(5/2)*(c + d*x))/(a + b*x^2)^3,x)
 

Output:

2*atanh((25*d^2*e^8*(e*x)^(1/2)*((25*d^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^4*b 
^9) - (9*c^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^5*b^8) - (15*c*d*e^5)/(2048*a^2 
*b^4))^(1/2))/(32*((27*c^3*e^11)/(2048*a^2*b) + (125*d^3*e^11*(-a^5*b^9)^( 
1/2))/(2048*a^3*b^7) - (75*c*d^2*e^11)/(2048*a*b^2) - (45*c^2*d*e^11*(-a^5 
*b^9)^(1/2))/(2048*a^4*b^6))) + (9*c^2*e^8*(e*x)^(1/2)*((25*d^2*e^5*(-a^5* 
b^9)^(1/2))/(4096*a^4*b^9) - (9*c^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^5*b^8) - 
 (15*c*d*e^5)/(2048*a^2*b^4))^(1/2))/(32*((75*c*d^2*e^11)/(2048*b^3) - (27 
*c^3*e^11)/(2048*a*b^2) - (125*d^3*e^11*(-a^5*b^9)^(1/2))/(2048*a^2*b^8) + 
 (45*c^2*d*e^11*(-a^5*b^9)^(1/2))/(2048*a^3*b^7))))*(-(9*b*c^2*e^5*(-a^5*b 
^9)^(1/2) - 25*a*d^2*e^5*(-a^5*b^9)^(1/2) + 30*a^3*b^5*c*d*e^5)/(4096*a^5* 
b^9))^(1/2) - ((c*e^5*(e*x)^(3/2))/(16*b) - (3*c*e^3*(e*x)^(7/2))/(16*a) + 
 (9*d*e^4*(e*x)^(5/2))/(16*b) + (5*a*d*e^6*(e*x)^(1/2))/(16*b^2))/(a^2*e^4 
 + b^2*e^4*x^4 + 2*a*b*e^4*x^2) + 2*atanh((25*d^2*e^8*(e*x)^(1/2)*((9*c^2* 
e^5*(-a^5*b^9)^(1/2))/(4096*a^5*b^8) - (15*c*d*e^5)/(2048*a^2*b^4) - (25*d 
^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^4*b^9))^(1/2))/(32*((27*c^3*e^11)/(2048*a 
^2*b) - (125*d^3*e^11*(-a^5*b^9)^(1/2))/(2048*a^3*b^7) - (75*c*d^2*e^11)/( 
2048*a*b^2) + (45*c^2*d*e^11*(-a^5*b^9)^(1/2))/(2048*a^4*b^6))) + (9*c^2*e 
^8*(e*x)^(1/2)*((9*c^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^5*b^8) - (15*c*d*e^5) 
/(2048*a^2*b^4) - (25*d^2*e^5*(-a^5*b^9)^(1/2))/(4096*a^4*b^9))^(1/2))/(32 
*((75*c*d^2*e^11)/(2048*b^3) - (27*c^3*e^11)/(2048*a*b^2) + (125*d^3*e^...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 963, normalized size of antiderivative = 3.16 \[ \int \frac {(e x)^{5/2} (c+d x)}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((e*x)^(5/2)*(d*x+c)/(b*x^2+a)^3,x)
                                                                                    
                                                                                    
 

Output:

(sqrt(e)*e**2*( - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt 
(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c - 12*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*x**2 - 6*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)))*b**3*c*x**4 - 10*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*d - 20*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*d*x**2 - 10*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)*sq 
rt(2)))*a*b**2*d*x**4 + 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4 
)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*c + 12* 
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*x**2 + 6*b**(1/4)*a**(3/4)*sqrt 
(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4 
)*sqrt(2)))*b**3*c*x**4 + 10*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**3*d + 20 
*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqr 
t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*d*x**2 + 10*b**(3/4)*a**(1/4)*sq 
rt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a*...