\(\int \frac {(d+e x^2)^3}{(a+c x^4)^2} \, dx\) [318]

Optimal result

Integrand size = 19, antiderivative size = 268 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\frac {x \left (c d \left (d^2-\frac {3 a e^2}{c}\right )+e \left (3 c d^2-a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}} \] Output:

1/4*x*(c*d*(d^2-3*a*e^2/c)+e*(-a*e^2+3*c*d^2)*x^2)/a/c/(c*x^4+a)+3/16*(c^( 
1/2)*d+a^(1/2)*e)*(a*e^2+c*d^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/ 
2)/a^(7/4)/c^(7/4)+3/16*(c^(1/2)*d+a^(1/2)*e)*(a*e^2+c*d^2)*arctan(1+2^(1/ 
2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(7/4)/c^(7/4)+3/16*(c^(1/2)*d-a^(1/2)*e)*( 
a*e^2+c*d^2)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1 
/2)/a^(7/4)/c^(7/4)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} c^{3/4} \left (a e^2 x \left (3 d+e x^2\right )-c d^2 x \left (d+3 e x^2\right )\right )}{a+c x^4}-6 \sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+a^{3/2} e^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+a^{3/2} e^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 \sqrt {2} \left (-c^{3/2} d^3+\sqrt {a} c d^2 e-a \sqrt {c} d e^2+a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+3 \sqrt {2} \left (c^{3/2} d^3-\sqrt {a} c d^2 e+a \sqrt {c} d e^2-a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 a^{7/4} c^{7/4}} \] Input:

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

Output:

((-8*a^(3/4)*c^(3/4)*(a*e^2*x*(3*d + e*x^2) - c*d^2*x*(d + 3*e*x^2)))/(a + 
 c*x^4) - 6*Sqrt[2]*(c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^( 
3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 6*Sqrt[2]*(c^(3/2)*d^3 
 + Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^ 
(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*(-(c^(3/2)*d^3) + Sqrt[a]*c*d^2*e - a*Sqrt[c 
]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x 
^2] + 3*Sqrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 - a^(3/2) 
*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(32*a^(7/4)* 
c^(7/4))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {1519, 25, 2397, 27, 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.

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

\(\Big \downarrow \) 1519

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2397

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 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]
 

rule 1519

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim 
p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c 
*(4*p + 2*q + 1))   Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* 
x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x 
], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
 

rule 2397

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, 
x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, 
 x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S 
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] 
 + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))   Int[(a + b*x^n)^(p + 1)* 
ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, 
n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
 

Maple [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {-\frac {e \left (a \,e^{2}-3 c \,d^{2}\right ) x^{3}}{4 a c}-\frac {d \left (3 a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (e \left (a \,e^{2}+c \,d^{2}\right ) \textit {\_R}^{2}+d \left (a \,e^{2}+c \,d^{2}\right )\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 a \,c^{2}}\)	\(119\)
default	\(\frac {-\frac {e \left (a \,e^{2}-3 c \,d^{2}\right ) x^{3}}{4 a c}-\frac {d \left (3 a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 a c}\)	\(284\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (207) = 414\).

Time = 0.73 (sec) , antiderivative size = 2116, normalized size of antiderivative = 7.90 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*(4*(3*c*d^2*e - a*e^3)*x^3 - 3*(a*c^2*x^4 + a^2*c)*sqrt(-(2*c^2*d^5*e 
 + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^ 
2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^ 
10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))*log(-27*(c^5*d^10 + 3*a*c^4*d^8*e^2 
+ 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 - a^5*e^10)*x + 
27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5*c^2*d*e^6 + a^6* 
c^5*e*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6 
*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))*sqrt(-(2 
*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c 
^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^ 
5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))) + 3*(a*c^2*x^4 + a^2*c)*s 
qrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c^6*d^12 
+ 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 
 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))*log(-27*(c^5*d^10 + 
 3*a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 
 - a^5*e^10)*x - 27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5 
*c^2*d*e^6 + a^6*c^5*e*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^ 
4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^ 
7*c^7)))*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(- 
(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^...
 

Sympy [A] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{7} + t^{2} \cdot \left (9216 a^{6} c^{4} d e^{5} + 18432 a^{5} c^{5} d^{3} e^{3} + 9216 a^{4} c^{6} d^{5} e\right ) + 81 a^{6} e^{12} + 486 a^{5} c d^{2} e^{10} + 1215 a^{4} c^{2} d^{4} e^{8} + 1620 a^{3} c^{3} d^{6} e^{6} + 1215 a^{2} c^{4} d^{8} e^{4} + 486 a c^{5} d^{10} e^{2} + 81 c^{6} d^{12}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{6} c^{5} e + 432 t a^{5} c^{2} d e^{6} + 720 t a^{4} c^{3} d^{3} e^{4} + 144 t a^{3} c^{4} d^{5} e^{2} - 144 t a^{2} c^{5} d^{7}}{27 a^{5} e^{10} + 81 a^{4} c d^{2} e^{8} + 54 a^{3} c^{2} d^{4} e^{6} - 54 a^{2} c^{3} d^{6} e^{4} - 81 a c^{4} d^{8} e^{2} - 27 c^{5} d^{10}} \right )} \right )\right )} + \frac {x^{3} \left (- a e^{3} + 3 c d^{2} e\right ) + x \left (- 3 a d e^{2} + c d^{3}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \] Input:

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

Output:

RootSum(65536*_t**4*a**7*c**7 + _t**2*(9216*a**6*c**4*d*e**5 + 18432*a**5* 
c**5*d**3*e**3 + 9216*a**4*c**6*d**5*e) + 81*a**6*e**12 + 486*a**5*c*d**2* 
e**10 + 1215*a**4*c**2*d**4*e**8 + 1620*a**3*c**3*d**6*e**6 + 1215*a**2*c* 
*4*d**8*e**4 + 486*a*c**5*d**10*e**2 + 81*c**6*d**12, Lambda(_t, _t*log(x 
+ (4096*_t**3*a**6*c**5*e + 432*_t*a**5*c**2*d*e**6 + 720*_t*a**4*c**3*d** 
3*e**4 + 144*_t*a**3*c**4*d**5*e**2 - 144*_t*a**2*c**5*d**7)/(27*a**5*e**1 
0 + 81*a**4*c*d**2*e**8 + 54*a**3*c**2*d**4*e**6 - 54*a**2*c**3*d**6*e**4 
- 81*a*c**4*d**8*e**2 - 27*c**5*d**10)))) + (x**3*(-a*e**3 + 3*c*d**2*e) + 
 x*(-3*a*d*e**2 + c*d**3))/(4*a**2*c + 4*a*c**2*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\frac {{\left (3 \, c d^{2} e - a e^{3}\right )} x^{3} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {3 \, {\left (c d^{2} + a e^{2}\right )} {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, a c} \] Input:

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

Output:

1/4*((3*c*d^2*e - a*e^3)*x^3 + (c*d^3 - 3*a*d*e^2)*x)/(a*c^2*x^4 + a^2*c) 
+ 3/32*(c*d^2 + a*e^2)*(2*sqrt(2)*(sqrt(c)*d + sqrt(a)*e)*arctan(1/2*sqrt( 
2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a) 
*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(sqrt(c)*d + sqrt(a)*e)*arctan 
(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)) 
)/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(sqrt(c)*d - sqrt(a)*e 
)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) 
 - sqrt(2)*(sqrt(c)*d - sqrt(a)*e)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/ 
4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (207) = 414\).

Time = 0.14 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.60 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\frac {3 \, c d^{2} e x^{3} - a e^{3} x^{3} + c d^{3} x - 3 \, a d e^{2} x}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{4}} \] Input:

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

Output:

1/4*(3*c*d^2*e*x^3 - a*e^3*x^3 + c*d^3*x - 3*a*d*e^2*x)/((c*x^4 + a)*a*c) 
+ 3/16*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^3 
)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*( 
a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^4) + 3/16*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + 
 (a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^3)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)* 
arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^4) + 3/ 
32*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 - (a*c^3)^(3 
/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt( 
a/c))/(a^2*c^4) - 3/32*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^ 
2*d*e^2 - (a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*log(x^2 - sqrt(2)*x 
*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4)
 

Mupad [B] (verification not implemented)

Time = 17.64 (sec) , antiderivative size = 2560, normalized size of antiderivative = 9.55 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((d*x*(3*a*e^2 - c*d^2))/(4*a*c) + (e*x^3*(a*e^2 - 3*c*d^2))/(4*a*c))/(a 
 + c*x^4) - 2*atanh((9*c^3*d^6*x*((9*e^6*(-a^7*c^7)^(1/2))/(256*a^4*c^7) - 
 (9*d^5*e)/(128*a^3*c) - (9*d^3*e^3)/(64*a^2*c^2) - (9*d^6*(-a^7*c^7)^(1/2 
))/(256*a^7*c^4) - (9*d*e^5)/(128*a*c^3) + (9*d^2*e^4*(-a^7*c^7)^(1/2))/(2 
56*a^5*c^6) - (9*d^4*e^2*(-a^7*c^7)^(1/2))/(256*a^6*c^5))^(1/2))/(2*((27*c 
*d^6*e^3)/16 - (27*a^3*e^9)/(32*c^2) + (27*c^2*d^8*e)/(32*a) - (27*a^2*d^2 
*e^7)/(16*c) + (27*d^9*(-a^7*c^7)^(1/2))/(32*a^5*c) - (27*d*e^8*(-a^7*c^7) 
^(1/2))/(32*a*c^5) - (27*d^3*e^6*(-a^7*c^7)^(1/2))/(16*a^2*c^4) + (27*d^7* 
e^2*(-a^7*c^7)^(1/2))/(16*a^4*c^2))) + (9*a*e^6*x*((9*e^6*(-a^7*c^7)^(1/2) 
)/(256*a^4*c^7) - (9*d^5*e)/(128*a^3*c) - (9*d^3*e^3)/(64*a^2*c^2) - (9*d^ 
6*(-a^7*c^7)^(1/2))/(256*a^7*c^4) - (9*d*e^5)/(128*a*c^3) + (9*d^2*e^4*(-a 
^7*c^7)^(1/2))/(256*a^5*c^6) - (9*d^4*e^2*(-a^7*c^7)^(1/2))/(256*a^6*c^5)) 
^(1/2))/(2*((27*a*e^9)/(32*c^2) + (27*d^2*e^7)/(16*c) - (27*c*d^6*e^3)/(16 
*a^2) - (27*c^2*d^8*e)/(32*a^3) - (27*d^9*(-a^7*c^7)^(1/2))/(32*a^7*c) + ( 
27*d*e^8*(-a^7*c^7)^(1/2))/(32*a^3*c^5) + (27*d^3*e^6*(-a^7*c^7)^(1/2))/(1 
6*a^4*c^4) - (27*d^7*e^2*(-a^7*c^7)^(1/2))/(16*a^6*c^2))) + (9*c*d^2*e^4*x 
*((9*e^6*(-a^7*c^7)^(1/2))/(256*a^4*c^7) - (9*d^5*e)/(128*a^3*c) - (9*d^3* 
e^3)/(64*a^2*c^2) - (9*d^6*(-a^7*c^7)^(1/2))/(256*a^7*c^4) - (9*d*e^5)/(12 
8*a*c^3) + (9*d^2*e^4*(-a^7*c^7)^(1/2))/(256*a^5*c^6) - (9*d^4*e^2*(-a^7*c 
^7)^(1/2))/(256*a^6*c^5))^(1/2))/(2*((27*a*e^9)/(32*c^2) + (27*d^2*e^7)...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1304, normalized size of antiderivative = 4.87 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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