\(\int \frac {1}{(d+e x) (a+c x^4)^3} \, dx\) [189]

Optimal result

Integrand size = 17, antiderivative size = 1133 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Output:

1/8*e^3/(a*e^4+c*d^4)/(c*x^4+a)^2+1/8*c*x*(d*e^2*x^2-d^2*e*x+d^3)/a/(a*e^4 
+c*d^4)/(c*x^4+a)^2+1/4*e^7/(a*e^4+c*d^4)^2/(c*x^4+a)+1/4*c*e^4*x*(d*e^2*x 
^2-d^2*e*x+d^3)/a/(a*e^4+c*d^4)^2/(c*x^4+a)+1/32*c*x*(5*d*e^2*x^2-6*d^2*e* 
x+7*d^3)/a^2/(a*e^4+c*d^4)/(c*x^4+a)-1/2*c^(1/2)*d^2*e^9*arctan(c^(1/2)*x^ 
2/a^(1/2))/a^(1/2)/(a*e^4+c*d^4)^3-1/4*c^(1/2)*d^2*e^5*arctan(c^(1/2)*x^2/ 
a^(1/2))/a^(3/2)/(a*e^4+c*d^4)^2-3/16*c^(1/2)*d^2*e*arctan(c^(1/2)*x^2/a^( 
1/2))/a^(5/2)/(a*e^4+c*d^4)+1/4*c^(1/4)*d*e^8*(c^(1/2)*d^2+a^(1/2)*e^2)*ar 
ctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4)/(a*e^4+c*d^4)^3+1/16*c^ 
(1/4)*d*e^4*(3*c^(1/2)*d^2+a^(1/2)*e^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4 
))*2^(1/2)/a^(7/4)/(a*e^4+c*d^4)^2+1/128*c^(1/4)*d*(21*c^(1/2)*d^2+5*a^(1/ 
2)*e^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(11/4)/(a*e^4+c*d^4 
)+1/4*c^(1/4)*d*e^8*(c^(1/2)*d^2+a^(1/2)*e^2)*arctan(1+2^(1/2)*c^(1/4)*x/a 
^(1/4))*2^(1/2)/a^(3/4)/(a*e^4+c*d^4)^3+1/16*c^(1/4)*d*e^4*(3*c^(1/2)*d^2+ 
a^(1/2)*e^2)*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(7/4)/(a*e^4+c* 
d^4)^2+1/128*c^(1/4)*d*(21*c^(1/2)*d^2+5*a^(1/2)*e^2)*arctan(1+2^(1/2)*c^( 
1/4)*x/a^(1/4))*2^(1/2)/a^(11/4)/(a*e^4+c*d^4)+1/4*c^(1/4)*d*e^8*(c^(1/2)* 
d^2-a^(1/2)*e^2)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))* 
2^(1/2)/a^(3/4)/(a*e^4+c*d^4)^3+1/16*c^(1/4)*d*e^4*(3*c^(1/2)*d^2-a^(1/2)* 
e^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)/(a*e^4+c*d^4)^2+1/128*c^(1/4)*d*(21*c^(1/2)*d^2-5*a^(1/2)*e^2)*arct...
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 835, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\frac {\frac {32 \left (c d^4+a e^4\right )^2 \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )^2}+\frac {8 \left (c d^4+a e^4\right ) \left (8 a^2 e^7+c^2 d^5 x \left (7 d^2-6 d e x+5 e^2 x^2\right )+a c d e^4 x \left (15 d^2-14 d e x+13 e^2 x^2\right )\right )}{a^2 \left (a+c x^4\right )}-\frac {2 \sqrt [4]{c} d \left (21 \sqrt {2} c^{5/2} d^{10}-24 \sqrt [4]{a} c^{9/4} d^9 e+5 \sqrt {2} \sqrt {a} c^2 d^8 e^2+66 \sqrt {2} a c^{3/2} d^6 e^4-80 a^{5/4} c^{5/4} d^5 e^5+18 \sqrt {2} a^{3/2} c d^4 e^6+77 \sqrt {2} a^2 \sqrt {c} d^2 e^8-120 a^{9/4} \sqrt [4]{c} d e^9+45 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {2 \sqrt [4]{c} d \left (21 \sqrt {2} c^{5/2} d^{10}+24 \sqrt [4]{a} c^{9/4} d^9 e+5 \sqrt {2} \sqrt {a} c^2 d^8 e^2+66 \sqrt {2} a c^{3/2} d^6 e^4+80 a^{5/4} c^{5/4} d^5 e^5+18 \sqrt {2} a^{3/2} c d^4 e^6+77 \sqrt {2} a^2 \sqrt {c} d^2 e^8+120 a^{9/4} \sqrt [4]{c} d e^9+45 \sqrt {2} a^{5/2} e^{10}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+256 e^{11} \log (d+e x)+\frac {\sqrt {2} \sqrt [4]{c} \left (-21 c^{5/2} d^{11}+5 \sqrt {a} c^2 d^9 e^2-66 a c^{3/2} d^7 e^4+18 a^{3/2} c d^5 e^6-77 a^2 \sqrt {c} d^3 e^8+45 a^{5/2} d e^{10}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (21 c^{5/2} d^{11}-5 \sqrt {a} c^2 d^9 e^2+66 a c^{3/2} d^7 e^4-18 a^{3/2} c d^5 e^6+77 a^2 \sqrt {c} d^3 e^8-45 a^{5/2} d e^{10}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4}}-64 e^{11} \log \left (a+c x^4\right )}{256 \left (c d^4+a e^4\right )^3} \] Input:

Integrate[1/((d + e*x)*(a + c*x^4)^3),x]
 

Output:

((32*(c*d^4 + a*e^4)^2*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)))/(a*(a + c* 
x^4)^2) + (8*(c*d^4 + a*e^4)*(8*a^2*e^7 + c^2*d^5*x*(7*d^2 - 6*d*e*x + 5*e 
^2*x^2) + a*c*d*e^4*x*(15*d^2 - 14*d*e*x + 13*e^2*x^2)))/(a^2*(a + c*x^4)) 
 - (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 - 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sq 
rt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqrt[2]*a*c^(3/2)*d^6*e^4 - 80*a^(5/4)*c^(5 
/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 + 77*Sqrt[2]*a^2*Sqrt[c]*d^2*e^ 
8 - 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2]*a^(5/2)*e^10)*ArcTan[1 - (Sqrt[ 
2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + (2*c^(1/4)*d*(21*Sqrt[2]*c^(5/2)*d^10 + 
 24*a^(1/4)*c^(9/4)*d^9*e + 5*Sqrt[2]*Sqrt[a]*c^2*d^8*e^2 + 66*Sqrt[2]*a*c 
^(3/2)*d^6*e^4 + 80*a^(5/4)*c^(5/4)*d^5*e^5 + 18*Sqrt[2]*a^(3/2)*c*d^4*e^6 
 + 77*Sqrt[2]*a^2*Sqrt[c]*d^2*e^8 + 120*a^(9/4)*c^(1/4)*d*e^9 + 45*Sqrt[2] 
*a^(5/2)*e^10)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + 256*e^1 
1*Log[d + e*x] + (Sqrt[2]*c^(1/4)*(-21*c^(5/2)*d^11 + 5*Sqrt[a]*c^2*d^9*e^ 
2 - 66*a*c^(3/2)*d^7*e^4 + 18*a^(3/2)*c*d^5*e^6 - 77*a^2*Sqrt[c]*d^3*e^8 + 
 45*a^(5/2)*d*e^10)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] 
)/a^(11/4) + (Sqrt[2]*c^(1/4)*(21*c^(5/2)*d^11 - 5*Sqrt[a]*c^2*d^9*e^2 + 6 
6*a*c^(3/2)*d^7*e^4 - 18*a^(3/2)*c*d^5*e^6 + 77*a^2*Sqrt[c]*d^3*e^8 - 45*a 
^(5/2)*d*e^10)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^( 
11/4) - 64*e^11*Log[a + c*x^4])/(256*(c*d^4 + a*e^4)^3)
 

Rubi [A] (verified)

Time = 2.95 (sec) , antiderivative size = 1352, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 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.

\(\displaystyle \int \frac {1}{\left (a+c x^4\right )^3 (d+e x)} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{12}}{(d+e x) \left (a e^4+c d^4\right )^3}-\frac {c e^4 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (a+c x^4\right )^2 \left (a e^4+c d^4\right )^2}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (a+c x^4\right )^3 \left (a e^4+c d^4\right )}-\frac {c e^8 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (d+e x) e^{11}}{\left (c d^4+a e^4\right )^3}-\frac {\log \left (c x^4+a\right ) e^{11}}{4 \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^9}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^8}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^8}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{4 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {\left (a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )\right ) e^4}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^2}-\frac {3 \sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{16 a^{5/2} \left (c d^4+a e^4\right )}+\frac {a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{8 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )^2}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{128 \sqrt {2} a^{11/4} \left (c d^4+a e^4\right )}+\frac {c x \left (7 d^3-6 e x d^2+5 e^2 x^2 d\right )}{32 a^2 \left (c d^4+a e^4\right ) \left (c x^4+a\right )}\)

Input:

Int[1/((d + e*x)*(a + c*x^4)^3),x]
 

Output:

(c*x*(7*d^3 - 6*d^2*e*x + 5*d*e^2*x^2))/(32*a^2*(c*d^4 + a*e^4)*(a + c*x^4 
)) + (a*e^3 + c*x*(d^3 - d^2*e*x + d*e^2*x^2))/(8*a*(c*d^4 + a*e^4)*(a + c 
*x^4)^2) + (e^4*(a*e^3 + c*x*(d^3 - d^2*e*x + d*e^2*x^2)))/(4*a*(c*d^4 + a 
*e^4)^2*(a + c*x^4)) - (Sqrt[c]*d^2*e^9*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2* 
Sqrt[a]*(c*d^4 + a*e^4)^3) - (Sqrt[c]*d^2*e^5*ArcTan[(Sqrt[c]*x^2)/Sqrt[a] 
])/(4*a^(3/2)*(c*d^4 + a*e^4)^2) - (3*Sqrt[c]*d^2*e*ArcTan[(Sqrt[c]*x^2)/S 
qrt[a]])/(16*a^(5/2)*(c*d^4 + a*e^4)) - (c^(1/4)*d*e^8*(Sqrt[c]*d^2 + Sqrt 
[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^ 
4 + a*e^4)^3) - (c^(1/4)*d*e^4*(3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (S 
qrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2) - (c^(1/ 
4)*d*(21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/ 
4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) + (c^(1/4)*d*e^8*(Sqrt[c]*d^2 + 
 Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)* 
(c*d^4 + a*e^4)^3) + (c^(1/4)*d*e^4*(3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 
 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^2) + ( 
c^(1/4)*d*(21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/ 
a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) + (e^11*Log[d + e*x])/(c*d 
^4 + a*e^4)^3 - (c^(1/4)*d*e^8*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - S 
qrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4 
)^3) - (c^(1/4)*d*e^4*(3*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 677, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {c \left (\frac {\frac {c d \,e^{2} \left (13 a^{2} e^{8}+18 a c \,d^{4} e^{4}+5 d^{8} c^{2}\right ) x^{7}}{32 a^{2}}-\frac {d^{2} e c \left (7 a^{2} e^{8}+10 a c \,d^{4} e^{4}+3 d^{8} c^{2}\right ) x^{6}}{16 a^{2}}+\frac {c \,d^{3} \left (15 a^{2} e^{8}+22 a c \,d^{4} e^{4}+7 d^{8} c^{2}\right ) x^{5}}{32 a^{2}}+\left (\frac {1}{4} a \,e^{11}+\frac {1}{4} d^{4} e^{7} c \right ) x^{4}+\frac {d \,e^{2} \left (17 a^{2} e^{8}+26 a c \,d^{4} e^{4}+9 d^{8} c^{2}\right ) x^{3}}{32 a}-\frac {d^{2} e \left (9 a^{2} e^{8}+14 a c \,d^{4} e^{4}+5 d^{8} c^{2}\right ) x^{2}}{16 a}+\frac {d^{3} \left (19 a^{2} e^{8}+30 a c \,d^{4} e^{4}+11 d^{8} c^{2}\right ) x}{32 a}+\frac {e^{3} \left (3 a^{2} e^{8}+4 a c \,d^{4} e^{4}+d^{8} c^{2}\right )}{8 c}}{\left (c \,x^{4}+a \right )^{2}}+\frac {\frac {\left (77 a^{2} d^{3} e^{8}+66 a c \,d^{7} e^{4}+21 c^{2} d^{11}\right ) \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 {\left (-60 a^{2} d^{2} e^{9}-40 a c \,d^{6} e^{5}-12 c^{2} d^{10} e \right ) \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{2 \sqrt {a c}}+\frac {\left (45 a^{2} d \,e^{10}+18 a c \,d^{5} e^{6}+5 c^{2} d^{9} e^{2}\right ) \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}}}-\frac {8 a^{2} e^{11} \ln \left (c \,x^{4}+a \right )}{c}}{32 a^{2}}\right )}{\left (e^{4} a +c \,d^{4}\right )^{3}}+\frac {e^{11} \ln \left (e x +d \right )}{\left (e^{4} a +c \,d^{4}\right )^{3}}\)	\(677\)
risch	\(\text {Expression too large to display}\)	\(1312\)

Input:

int(1/(e*x+d)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

c/(a*e^4+c*d^4)^3*((1/32*c*d*e^2*(13*a^2*e^8+18*a*c*d^4*e^4+5*c^2*d^8)/a^2 
*x^7-1/16*d^2*e*c*(7*a^2*e^8+10*a*c*d^4*e^4+3*c^2*d^8)/a^2*x^6+1/32*c*d^3* 
(15*a^2*e^8+22*a*c*d^4*e^4+7*c^2*d^8)/a^2*x^5+(1/4*a*e^11+1/4*d^4*e^7*c)*x 
^4+1/32*d*e^2*(17*a^2*e^8+26*a*c*d^4*e^4+9*c^2*d^8)/a*x^3-1/16*d^2*e*(9*a^ 
2*e^8+14*a*c*d^4*e^4+5*c^2*d^8)/a*x^2+1/32*d^3*(19*a^2*e^8+30*a*c*d^4*e^4+ 
11*c^2*d^8)/a*x+1/8*e^3*(3*a^2*e^8+4*a*c*d^4*e^4+c^2*d^8)/c)/(c*x^4+a)^2+1 
/32/a^2*(1/8*(77*a^2*d^3*e^8+66*a*c*d^7*e^4+21*c^2*d^11)*(a/c)^(1/4)/a*2^( 
1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2 
)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^( 
1/4)*x-1))+1/2*(-60*a^2*d^2*e^9-40*a*c*d^6*e^5-12*c^2*d^10*e)/(a*c)^(1/2)* 
arctan((c/a)^(1/2)*x^2)+1/8*(45*a^2*d*e^10+18*a*c*d^5*e^6+5*c^2*d^9*e^2)/c 
/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c 
)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan 
(2^(1/2)/(a/c)^(1/4)*x-1))-8*a^2*e^11/c*ln(c*x^4+a)))+e^11*ln(e*x+d)/(a*e^ 
4+c*d^4)^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1015, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="maxima")
 

Output:

e^11*log(e*x + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12 
) - 1/256*c*(sqrt(2)*(32*sqrt(2)*a^(11/4)*c^(1/4)*e^11 - 21*c^3*d^11 + 5*s 
qrt(a)*c^(5/2)*d^9*e^2 - 66*a*c^2*d^7*e^4 + 18*a^(3/2)*c^(3/2)*d^5*e^6 - 7 
7*a^2*c*d^3*e^8 + 45*a^(5/2)*sqrt(c)*d*e^10)*log(sqrt(c)*x^2 + sqrt(2)*a^( 
1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(32*sqrt(2)*a^(11/4) 
*c^(1/4)*e^11 + 21*c^3*d^11 - 5*sqrt(a)*c^(5/2)*d^9*e^2 + 66*a*c^2*d^7*e^4 
 - 18*a^(3/2)*c^(3/2)*d^5*e^6 + 77*a^2*c*d^3*e^8 - 45*a^(5/2)*sqrt(c)*d*e^ 
10)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4 
)) - 2*(21*sqrt(2)*a^(1/4)*c^(13/4)*d^11 + 5*sqrt(2)*a^(3/4)*c^(11/4)*d^9* 
e^2 + 66*sqrt(2)*a^(5/4)*c^(9/4)*d^7*e^4 + 18*sqrt(2)*a^(7/4)*c^(7/4)*d^5* 
e^6 + 77*sqrt(2)*a^(9/4)*c^(5/4)*d^3*e^8 + 45*sqrt(2)*a^(11/4)*c^(3/4)*d*e 
^10 + 24*sqrt(a)*c^3*d^10*e + 80*a^(3/2)*c^2*d^6*e^5 + 120*a^(5/2)*c*d^2*e 
^9)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a 
)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)) - 2*(21*sqrt(2)*a^(1/4 
)*c^(13/4)*d^11 + 5*sqrt(2)*a^(3/4)*c^(11/4)*d^9*e^2 + 66*sqrt(2)*a^(5/4)* 
c^(9/4)*d^7*e^4 + 18*sqrt(2)*a^(7/4)*c^(7/4)*d^5*e^6 + 77*sqrt(2)*a^(9/4)* 
c^(5/4)*d^3*e^8 + 45*sqrt(2)*a^(11/4)*c^(3/4)*d*e^10 - 24*sqrt(a)*c^3*d^10 
*e - 80*a^(3/2)*c^2*d^6*e^5 - 120*a^(5/2)*c*d^2*e^9)*arctan(1/2*sqrt(2)*(2 
*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt 
(sqrt(a)*sqrt(c))*c^(5/4)))/(a^2*c^3*d^12 + 3*a^3*c^2*d^8*e^4 + 3*a^4*c...
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1311, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(c*x^4+a)^3,x, algorithm="giac")
 

Output:

e^12*log(abs(e*x + d))/(c^3*d^12*e + 3*a*c^2*d^8*e^5 + 3*a^2*c*d^4*e^9 + a 
^3*e^13) - 1/4*e^11*log(abs(c*x^4 + a))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^ 
2*c*d^4*e^8 + a^3*e^12) - 1/64*(75*sqrt(2)*a*c^2*d^2*e^3 - 51*sqrt(2)*sqrt 
(a*c)*c^2*d^4*e - 21*(a*c^3)^(1/4)*c^2*d^5 - 45*(a*c^3)^(1/4)*a*c*d*e^4 - 
122*(a*c^3)^(3/4)*d^3*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/ 
(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^6 + 9*sqrt(2)*a^4*c^2*d^2*e^4 + 9*sqrt(2)* 
sqrt(a*c)*a^3*c^2*d^4*e^2 + sqrt(2)*sqrt(a*c)*a^4*c*e^6 - 6*(a*c^3)^(1/4)* 
a^3*c^2*d^5*e - 6*(a*c^3)^(1/4)*a^4*c*d*e^5 - 16*(a*c^3)^(3/4)*a^3*d^3*e^3 
) + 1/64*(75*sqrt(2)*a*c^2*d^2*e^3 + 51*sqrt(2)*sqrt(a*c)*c^2*d^4*e + 21*( 
a*c^3)^(1/4)*c^2*d^5 + 45*(a*c^3)^(1/4)*a*c*d*e^4 + 122*(a*c^3)^(3/4)*d^3* 
e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)* 
a^3*c^3*d^6 + 9*sqrt(2)*a^4*c^2*d^2*e^4 + 9*sqrt(2)*sqrt(a*c)*a^3*c^2*d^4* 
e^2 + sqrt(2)*sqrt(a*c)*a^4*c*e^6 + 6*(a*c^3)^(1/4)*a^3*c^2*d^5*e + 6*(a*c 
^3)^(1/4)*a^4*c*d*e^5 + 16*(a*c^3)^(3/4)*a^3*d^3*e^3) + 1/256*(21*sqrt(2)* 
(a*c^3)^(1/4)*c^4*d^11 + 66*sqrt(2)*(a*c^3)^(1/4)*a*c^3*d^7*e^4 + 77*sqrt( 
2)*(a*c^3)^(1/4)*a^2*c^2*d^3*e^8 - 5*sqrt(2)*(a*c^3)^(3/4)*c^2*d^9*e^2 - 1 
8*sqrt(2)*(a*c^3)^(3/4)*a*c*d^5*e^6 - 45*sqrt(2)*(a*c^3)^(3/4)*a^2*d*e^10) 
*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^5*d^12 + 3*a^4*c^4*d^ 
8*e^4 + 3*a^5*c^3*d^4*e^8 + a^6*c^2*e^12) - 1/256*(21*sqrt(2)*(a*c^3)^(1/4 
)*c^4*d^11 + 66*sqrt(2)*(a*c^3)^(1/4)*a*c^3*d^7*e^4 + 77*sqrt(2)*(a*c^3...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 23.61 (sec) , antiderivative size = 2720, normalized size of antiderivative = 2.40 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + c*x^4)^3*(d + e*x)),x)
 

Output:

symsum(log((194481*c^7*d^13*e^6 + 871362*a*c^6*d^9*e^10 + 425984*a^3*c^4*d 
*e^18 + 1148881*a^2*c^5*d^5*e^14)/(1048576*(a^12*e^16 + a^8*c^4*d^16 + 4*a 
^11*c*d^4*e^12 + 4*a^9*c^3*d^12*e^4 + 6*a^10*c^2*d^8*e^8)) + root(80530636 
8*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^3 
*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152*a 
^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z^2 + 
 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 
676881*a*c*d^4*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)*(root(8053063 
68*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c^ 
3*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152* 
a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z^2 
+ 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z + 
 676881*a*c*d^4*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)*(root(805306 
368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11*c 
^3*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 43057152 
*a^7*c*d^4*e^6*z^2 + 11599872*a^6*c^2*d^8*e^2*z^2 + 100663296*a^8*e^10*z^2 
 + 9652224*a^4*c*d^4*e^5*z + 2709504*a^3*c^2*d^8*e*z + 16777216*a^5*e^9*z 
+ 676881*a*c*d^4*e^4 + 194481*c^2*d^8 + 1048576*a^2*e^8, z, k)*(root(80530 
6368*a^12*c^2*d^8*e^4*z^4 + 805306368*a^13*c*d^4*e^8*z^4 + 268435456*a^11* 
c^3*d^12*z^4 + 268435456*a^14*e^12*z^4 + 268435456*a^11*e^11*z^3 + 4305...
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 4499, normalized size of antiderivative = 3.97 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)/(c*x^4+a)^3,x)
 

Output:

( - 90*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**4*d*e**10 - 36*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)))*a**3*c*d**5*e**6 - 180*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**3*c*d*e**10*x 
**4 - 10*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqr 
t(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*c**2*d**9*e**2 - 72*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*c**2*d**5*e**6*x**4 - 90*c**(1/4)*a**(3/4)*sqrt(2)*at 
an((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))* 
a**2*c**2*d*e**10*x**8 - 20*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**3*d**9*e**2*x 
**4 - 36*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqr 
t(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c**3*d**5*e**6*x**8 - 10*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**4*d**9*e**2*x**8 - 154*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* 
*4*d**3*e**8 - 132*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**3*c*d**7*e**4 - 308*c**( 
3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c...