\(\int \frac {(d+e x)^3}{(a+b x^2+c x^4)^2} \, dx\) [241]

Optimal result

Integrand size = 22, antiderivative size = 446 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {d x \left (\left (b^2-2 a c\right ) d^2-3 a b e^2+c \left (b d^2-6 a e^2\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {e \left (3 b d^2-2 a e^2+\left (6 c d^2-b e^2\right ) x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} d \left (b^2 d^2+b \left (\sqrt {b^2-4 a c} d^2+12 a e^2\right )-6 a \left (2 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} d \left (b d^2-6 a e^2-\frac {b^2 d^2-12 a c d^2+12 a b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {e \left (6 c d^2-b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:

1/2*d*x*((-2*a*c+b^2)*d^2-3*a*b*e^2+c*(-6*a*e^2+b*d^2)*x^2)/a/(-4*a*c+b^2) 
/(c*x^4+b*x^2+a)-1/2*e*(3*b*d^2-2*a*e^2+(-b*e^2+6*c*d^2)*x^2)/(-4*a*c+b^2) 
/(c*x^4+b*x^2+a)+1/4*c^(1/2)*d*(b^2*d^2+b*((-4*a*c+b^2)^(1/2)*d^2+12*a*e^2 
)-6*a*(2*c*d^2+(-4*a*c+b^2)^(1/2)*e^2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a* 
c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^( 
1/2)+1/4*c^(1/2)*d*(b*d^2-6*a*e^2-(12*a*b*e^2-12*a*c*d^2+b^2*d^2)/(-4*a*c+ 
b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2) 
/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)+e*(-b*e^2+6*c*d^2)*arctanh((2 
*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {-4 a^2 e^3-2 b d^3 x \left (b+c x^2\right )+2 a b e \left (3 d^2+3 d e x-e^2 x^2\right )+4 a c d x \left (d^2+3 d e x+3 e^2 x^2\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} d \left (b^2 d^2+b \left (\sqrt {b^2-4 a c} d^2+12 a e^2\right )-6 a \left (2 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \left (b^2 d^2-12 a c d^2-b \sqrt {b^2-4 a c} d^2+12 a b e^2+6 a \sqrt {b^2-4 a c} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {2 \left (-6 c d^2 e+b e^3\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {2 \left (-6 c d^2 e+b e^3\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \] Input:

Integrate[(d + e*x)^3/(a + b*x^2 + c*x^4)^2,x]
 

Output:

((-4*a^2*e^3 - 2*b*d^3*x*(b + c*x^2) + 2*a*b*e*(3*d^2 + 3*d*e*x - e^2*x^2) 
 + 4*a*c*d*x*(d^2 + 3*d*e*x + 3*e^2*x^2))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c 
*x^4)) + (Sqrt[2]*Sqrt[c]*d*(b^2*d^2 + b*(Sqrt[b^2 - 4*a*c]*d^2 + 12*a*e^2 
) - 6*a*(2*c*d^2 + Sqrt[b^2 - 4*a*c]*e^2))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt 
[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c 
]]) - (Sqrt[2]*Sqrt[c]*d*(b^2*d^2 - 12*a*c*d^2 - b*Sqrt[b^2 - 4*a*c]*d^2 + 
 12*a*b*e^2 + 6*a*Sqrt[b^2 - 4*a*c]*e^2)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b 
 + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]] 
) + (2*(-6*c*d^2*e + b*e^3)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 
4*a*c)^(3/2) - (2*(-6*c*d^2*e + b*e^3)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2 
])/(b^2 - 4*a*c)^(3/2))/4
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 420, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2202, 1492, 25, 27, 1480, 218, 1576, 27, 1159, 1083, 219}

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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {\int -\frac {d \left (b^2 d^2-6 a c d^2+3 a b e^2+c \left (b d^2-6 a e^2\right ) x^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^2}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d \left (b^2 d^2-6 a c d^2+3 a b e^2+c \left (b d^2-6 a e^2\right ) x^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^2}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {b^2 d^2-6 a c d^2+3 a b e^2+c \left (b d^2-6 a e^2\right ) x^2}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^2}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {d \left (\frac {1}{2} c \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx\right )}{2 a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^2}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^2}dx+\frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {1}{2} \int \frac {e \left (3 d^2+e^2 x^2\right )}{\left (c x^4+b x^2+a\right )^2}dx^2+\frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} e \int \frac {3 d^2+e^2 x^2}{\left (c x^4+b x^2+a\right )^2}dx^2+\frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1159

\(\displaystyle \frac {1}{2} e \left (-\frac {\left (6 c d^2-b e^2\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {-2 a e^2+x^2 \left (6 c d^2-b e^2\right )+3 b d^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} e \left (\frac {2 \left (6 c d^2-b e^2\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}-\frac {-2 a e^2+x^2 \left (6 c d^2-b e^2\right )+3 b d^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {12 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-6 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (\frac {2 \left (6 c d^2-b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {-2 a e^2+x^2 \left (6 c d^2-b e^2\right )+3 b d^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

Input:

Int[(d + e*x)^3/(a + b*x^2 + c*x^4)^2,x]
 

Output:

(d*x*((b^2 - 2*a*c)*d^2 - 3*a*b*e^2 + c*(b*d^2 - 6*a*e^2)*x^2))/(2*a*(b^2 
- 4*a*c)*(a + b*x^2 + c*x^4)) + (d*((Sqrt[c]*(b*d^2 - 6*a*e^2 + (b^2*d^2 - 
 12*a*c*d^2 + 12*a*b*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sq 
rt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[ 
c]*(b*d^2 - 6*a*e^2 - (b^2*d^2 - 12*a*c*d^2 + 12*a*b*e^2)/Sqrt[b^2 - 4*a*c 
])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[ 
b + Sqrt[b^2 - 4*a*c]])))/(2*a*(b^2 - 4*a*c)) + (e*(-((3*b*d^2 - 2*a*e^2 + 
 (6*c*d^2 - b*e^2)*x^2)/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (2*(6*c*d^2 
 - b*e^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/ 
2
 

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 218

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

rule 219

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1159

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* 
x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* 
c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & 
& LtQ[p, -1] && NeQ[p, -3/2]
 

rule 1480

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

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {\frac {c d \left (6 a \,e^{2}-b \,d^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right ) a}-\frac {e \left (b \,e^{2}-6 c \,d^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}+\frac {d \left (3 a b \,e^{2}+2 a c \,d^{2}-b^{2} d^{2}\right ) x}{2 \left (4 a c -b^{2}\right ) a}-\frac {e \left (2 a \,e^{2}-3 b \,d^{2}\right )}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c d \left (6 a \,e^{2}-b \,d^{2}\right ) \textit {\_R}^{2}}{\left (4 a c -b^{2}\right ) a}-\frac {2 e \left (b \,e^{2}-6 c \,d^{2}\right ) \textit {\_R}}{4 a c -b^{2}}-\frac {d \left (3 a b \,e^{2}-6 a c \,d^{2}+b^{2} d^{2}\right )}{\left (4 a c -b^{2}\right ) a}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +b \textit {\_R}}\right )}{4}\)	\(293\)
default	\(16 c^{2} \left (-\frac {\frac {-\frac {d \left (-4 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}+\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+24 a^{2} c \,e^{2}-6 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) x}{16 a c}-\frac {e \left (4 \sqrt {-4 a c +b^{2}}\, a c \,e^{2}-\sqrt {-4 a c +b^{2}}\, b^{2} e^{2}-4 a b c \,e^{2}+24 a \,c^{2} d^{2}+b^{3} e^{2}-6 b^{2} c \,d^{2}\right )}{16 c^{2}}}{x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}}+\frac {-\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a b \,e^{3}-24 \sqrt {-4 a c +b^{2}}\, a c \,d^{2} e \right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (12 \sqrt {-4 a c +b^{2}}\, a b d \,e^{2}-12 \sqrt {-4 a c +b^{2}}\, a c \,d^{3}+\sqrt {-4 a c +b^{2}}\, b^{2} d^{3}+24 a^{2} c d \,e^{2}-6 a \,b^{2} d \,e^{2}-4 a b c \,d^{3}+b^{3} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {\frac {d \left (4 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}-\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+24 a^{2} c \,e^{2}-6 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) x}{16 a c}+\frac {e \left (-4 \sqrt {-4 a c +b^{2}}\, a c \,e^{2}+\sqrt {-4 a c +b^{2}}\, b^{2} e^{2}-4 a b c \,e^{2}+24 a \,c^{2} d^{2}+b^{3} e^{2}-6 b^{2} c \,d^{2}\right )}{16 c^{2}}}{x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}+\frac {\frac {\left (-4 \sqrt {-4 a c +b^{2}}\, a b \,e^{3}+24 \sqrt {-4 a c +b^{2}}\, a c \,d^{2} e \right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (-12 \sqrt {-4 a c +b^{2}}\, a b d \,e^{2}+12 \sqrt {-4 a c +b^{2}}\, a c \,d^{3}-\sqrt {-4 a c +b^{2}}\, b^{2} d^{3}+24 a^{2} c d \,e^{2}-6 a \,b^{2} d \,e^{2}-4 a b c \,d^{3}+b^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}\right )\)	\(810\)

Input:

int((e*x+d)^3/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

(1/2*c*d*(6*a*e^2-b*d^2)/(4*a*c-b^2)/a*x^3-1/2*e*(b*e^2-6*c*d^2)/(4*a*c-b^ 
2)*x^2+1/2*d*(3*a*b*e^2+2*a*c*d^2-b^2*d^2)/(4*a*c-b^2)/a*x-1/2*e*(2*a*e^2- 
3*b*d^2)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/4*sum((c*d*(6*a*e^2-b*d^2)/(4*a*c- 
b^2)/a*_R^2-2*e*(b*e^2-6*c*d^2)/(4*a*c-b^2)*_R-d*(3*a*b*e^2-6*a*c*d^2+b^2* 
d^2)/(4*a*c-b^2)/a)/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**3/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate((e*x+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
 

Output:

-1/2*(3*a*b*d^2*e - 2*a^2*e^3 - (b*c*d^3 - 6*a*c*d*e^2)*x^3 + (6*a*c*d^2*e 
 - a*b*e^3)*x^2 + (3*a*b*d*e^2 - (b^2 - 2*a*c)*d^3)*x)/((a*b^2*c - 4*a^2*c 
^2)*x^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*x^2) - 1/2*integrate(-(3 
*a*b*d*e^2 + (b^2 - 6*a*c)*d^3 + (b*c*d^3 - 6*a*c*d*e^2)*x^2 - 2*(6*a*c*d^ 
2*e - a*b*e^3)*x)/(c*x^4 + b*x^2 + a), x)/(a*b^2 - 4*a^2*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5647 vs. \(2 (396) = 792\).

Time = 1.17 (sec) , antiderivative size = 5647, normalized size of antiderivative = 12.66 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
 

Output:

1/2*(b*c*d^3*x^3 - 6*a*c*d*e^2*x^3 - 6*a*c*d^2*e*x^2 + a*b*e^3*x^2 + b^2*d 
^3*x - 2*a*c*d^3*x - 3*a*b*d*e^2*x - 3*a*b*d^2*e + 2*a^2*e^3)/((c*x^4 + b* 
x^2 + a)*(a*b^2 - 4*a^2*c)) - 1/16*((2*b^3*c^2 - 8*a*b*c^3 - sqrt(2)*sqrt( 
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4* 
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*s 
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
+ sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*(a*b^2 - 4*a^2*c)^2* 
d^3 - 6*(2*a*b^2*c^2 - 8*a^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*a*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
 - 4*a*c)*c)*a^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a 
*c)*c)*a*b*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a 
*c^2 - 2*(b^2 - 4*a*c)*a*c^2)*(a*b^2 - 4*a^2*c)^2*d*e^2 - 2*(sqrt(2)*sqrt( 
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*a*b^ 
6*c + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^2 + 20*sqrt(2)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a*b^4*c^2 + 28*a^2*b^4*c^2 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 
4*a*c)*c)*a^4*c^3 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 - 
 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 128*a^3*b^2*c^3 
+ 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 192*a^4*c^4 + 2*...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 22.48 (sec) , antiderivative size = 5441, normalized size of antiderivative = 12.20 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^3/(a + b*x^2 + c*x^4)^2,x)
 

Output:

symsum(log(root(1572864*a^8*b^2*c^5*z^4 - 983040*a^7*b^4*c^4*z^4 + 327680* 
a^6*b^6*c^3*z^4 - 61440*a^5*b^8*c^2*z^4 + 6144*a^4*b^10*c*z^4 - 1048576*a^ 
9*c^6*z^4 - 256*a^3*b^12*z^4 + 208896*a^6*b*c^4*d^2*e^4*z^2 + 1728*a^2*b^8 
*c*d^4*e^2*z^2 - 1536*a^3*b^7*c*d^2*e^4*z^2 + 294912*a^5*b^2*c^4*d^4*e^2*z 
^2 - 147456*a^5*b^3*c^3*d^2*e^4*z^2 - 49152*a^4*b^4*c^3*d^4*e^2*z^2 + 3225 
6*a^4*b^5*c^2*d^2*e^4*z^2 - 4608*a^3*b^6*c^2*d^4*e^2*z^2 - 96*a*b^10*d^4*e 
^2*z^2 - 1536*a^4*b^6*c*e^6*z^2 + 61440*a^5*b*c^5*d^6*z^2 + 432*a*b^9*c*d^ 
6*z^2 - 442368*a^6*c^5*d^4*e^2*z^2 - 144*a^2*b^9*d^2*e^4*z^2 - 8192*a^6*b^ 
2*c^3*e^6*z^2 + 6144*a^5*b^4*c^2*e^6*z^2 - 61440*a^4*b^3*c^4*d^6*z^2 + 240 
64*a^3*b^5*c^3*d^6*z^2 - 4608*a^2*b^7*c^2*d^6*z^2 + 128*a^3*b^8*e^6*z^2 - 
16*b^11*d^6*z^2 - 2016*a*b^6*c^2*d^8*e*z + 912*a*b^7*c*d^6*e^3*z + 47616*a 
^4*b^2*c^3*d^4*e^5*z + 35584*a^3*b^3*c^3*d^6*e^3*z - 14976*a^3*b^4*c^2*d^4 
*e^5*z - 9408*a^2*b^5*c^2*d^6*e^3*z - 6912*a^4*b^3*c^2*d^2*e^7*z - 47616*a 
^3*b^2*c^4*d^8*e*z - 46080*a^4*b*c^4*d^6*e^3*z + 14976*a^2*b^4*c^3*d^8*e*z 
 + 9216*a^5*b*c^3*d^2*e^7*z + 2016*a^2*b^6*c*d^4*e^5*z + 1728*a^3*b^5*c*d^ 
2*e^7*z - 55296*a^5*c^4*d^4*e^5*z - 144*a^2*b^7*d^2*e^7*z + 55296*a^4*c^5* 
d^8*e*z - 96*a*b^8*d^4*e^5*z + 96*b^8*c*d^8*e*z - 16*b^9*d^6*e^3*z - 2592* 
a^3*b*c^3*d^6*e^6 - 2592*a^2*b*c^4*d^10*e^2 + 672*a*b^3*c^3*d^10*e^2 + 198 
*a*b^4*c^2*d^8*e^4 - 153*a^2*b^4*c*d^4*e^8 - 48*a^3*b^3*c*d^2*e^10 - 138*a 
*b^5*c*d^6*e^6 + 624*a^2*b^3*c^2*d^6*e^6 - 576*a^2*b^2*c^3*d^8*e^4 + 36...
 

Reduce [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 5447, normalized size of antiderivative = 12.21 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3/(c*x^4+b*x^2+a)^2,x)
 

Output:

(8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sq 
rt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*e 
**3 - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqr 
t(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3* 
b*c*d**2*e + 8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*ata 
n((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b)) 
*a**2*b**3*e**3*x**2 - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt( 
a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sq 
rt(a) + b))*a**2*b**2*c*d**2*e*x**2 + 8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2 
*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqr 
t(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c*e**3*x**4 - 48*sqrt(2*sqrt(c)*sqrt(a 
) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*s 
qrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b*c**2*d**2*e*x**4 - 24*sqrt(a 
)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c 
)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c*d*e**2 - 6*sqrt(a)*sqrt(2*sqrt( 
c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sq 
rt(c)*sqrt(a) + b))*a**2*b**3*d*e**2 + 16*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + 
 b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a 
) + b))*a**2*b**2*c*d**3 - 24*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sq 
rt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a...