∖int 1_________________ (d+ex)∖left(a+cx4∖right)3 ∖,dx

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

Optimal. Leaf size=1352 \[ \text{result too large to display} \]

[Out]

(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)/Sqrt[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 - (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)) + (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*Sq

rt[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] - Sqrt[2]*a^(1/4)*c^(1/4)*x + S

qrt[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[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sq

rt[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)*L

og[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*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)*Log[Sqrt[a] + Sqrt[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[2]*a^(1/4)*c^(1/4)*x

+ Sqrt[c]*x^2])/(16*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)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(12

8*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) - (e^11*Log[a + c*x^4])/(4*(c*d^4 + a*e^4)^3

)

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Rubi [A] time = 3.20965, antiderivative size = 1352, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824 \[ \text{result too large to display} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(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)/Sqrt[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 - (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)) + (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*Sq

rt[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] - Sqrt[2]*a^(1/4)*c^(1/4)*x + S

qrt[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[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sq

rt[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)*L

og[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*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)*Log[Sqrt[a] + Sqrt[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[2]*a^(1/4)*c^(1/4)*x

+ Sqrt[c]*x^2])/(16*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)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(12

8*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)) - (e^11*Log[a + c*x^4])/(4*(c*d^4 + a*e^4)^3

)

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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A] time = 1.52247, size = 835, normalized size = 0.62 \[ \frac{256 \log (d+e x) e^{11}-64 \log \left (c x^4+a\right ) e^{11}+\frac{32 \left (c d^4+a e^4\right )^2 \left (a e^3+c d x \left (d^2-e x d+e^2 x^2\right )\right )}{a \left (c x^4+a\right )^2}+\frac{8 \left (c d^4+a e^4\right ) \left (8 a^2 e^7+a c d x \left (15 d^2-14 e x d+13 e^2 x^2\right ) e^4+c^2 d^5 x \left (7 d^2-6 e x d+5 e^2 x^2\right )\right )}{a^2 \left (c x^4+a\right )}-\frac{2 \sqrt [4]{c} d \left (21 \sqrt{2} c^{5/2} d^{10}-24 \sqrt [4]{a} c^{9/4} e d^9+5 \sqrt{2} \sqrt{a} c^2 e^2 d^8+66 \sqrt{2} a c^{3/2} e^4 d^6-80 a^{5/4} c^{5/4} e^5 d^5+18 \sqrt{2} a^{3/2} c e^6 d^4+77 \sqrt{2} a^2 \sqrt{c} e^8 d^2-120 a^{9/4} \sqrt [4]{c} e^9 d+45 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac{2 \sqrt [4]{c} d \left (21 \sqrt{2} c^{5/2} d^{10}+24 \sqrt [4]{a} c^{9/4} e d^9+5 \sqrt{2} \sqrt{a} c^2 e^2 d^8+66 \sqrt{2} a c^{3/2} e^4 d^6+80 a^{5/4} c^{5/4} e^5 d^5+18 \sqrt{2} a^{3/2} c e^6 d^4+77 \sqrt{2} a^2 \sqrt{c} e^8 d^2+120 a^{9/4} \sqrt [4]{c} e^9 d+45 \sqrt{2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (-21 c^{5/2} d^{11}+5 \sqrt{a} c^2 e^2 d^9-66 a c^{3/2} e^4 d^7+18 a^{3/2} c e^6 d^5-77 a^2 \sqrt{c} e^8 d^3+45 a^{5/2} e^{10} d\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}+\frac{\sqrt{2} \sqrt [4]{c} \left (21 c^{5/2} d^{11}-5 \sqrt{a} c^2 e^2 d^9+66 a c^{3/2} e^4 d^7-18 a^{3/2} c e^6 d^5+77 a^2 \sqrt{c} e^8 d^3-45 a^{5/2} e^{10} d\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}}{256 \left (c d^4+a e^4\right )^3} \]

Antiderivative was successfully veriﬁed.

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

[Out]

((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*(2

1*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*S

qrt[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^11*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 + 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[Sqr

t[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)

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Maple [A] time = 0.035, size = 2106, normalized size = 1.6 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

*d^5*e^6+5/128/(a*e^4+c*d^4)^3*c^2/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c

*a)^(1/4)*x-1)*d^9*e^2+9/128/(a*e^4+c*d^4)^3*c/a/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(

1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)

))*d^5*e^6+5/256/(a*e^4+c*d^4)^3*c^2/a^2/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(

1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^9*e

^2+77/128/(a*e^4+c*d^4)^3*c*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)

*x+1)*d^3*e^8+77/128/(a*e^4+c*d^4)^3*c*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1

/c*a)^(1/4)*x-1)*d^3*e^8+21/256/(a*e^4+c*d^4)^3*c^3*(1/c*a)^(1/4)/a^3*2^(1/2)*ln

((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a

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

^6*e^5+5/32/(a*e^4+c*d^4)^3*c^4/(c*x^4+a)^2*d^9*e^2/a^2*x^7-5/8/(a*e^4+c*d^4)^3*

c^3/(c*x^4+a)^2*d^6*e^5/a*x^6+15/32/(a*e^4+c*d^4)^3*c^2/(c*x^4+a)^2*d^3*x^5*e^8+

7/32/(a*e^4+c*d^4)^3*c^4/(c*x^4+a)^2*d^11/a^2*x^5+1/4/(a*e^4+c*d^4)^3*c^2/(c*x^4

+a)^2*x^4*d^4*e^7+13/16/(a*e^4+c*d^4)^3*c^2/(c*x^4+a)^2*d^5*e^6*x^3+1/8/(a*e^4+c

*d^4)^3*c^2/(c*x^4+a)^2*e^3*d^8-3/16/(a*e^4+c*d^4)^3*c^4/(c*x^4+a)^2*d^10*e/a^2*

x^6+9/16/(a*e^4+c*d^4)^3*c^3/(c*x^4+a)^2*d^5*e^6/a*x^7-15/16/(a*e^4+c*d^4)^3*c/(

a^5*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*a^2*d^2*e^9+11/16/(a*e^4+c*d^4)^3*c^3/(c*x^

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

3*e^11*ln(a^2*(c*x^4+a))+9/64/(a*e^4+c*d^4)^3*c/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2

^(1/2)/(1/c*a)^(1/4)*x+1)*d^5*e^6+33/64/(a*e^4+c*d^4)^3*c^2*(1/c*a)^(1/4)/a^2*2^

(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^7*e^4+9/32/(a*e^4+c*d^4)^3*c^3/(c*x^4+

a)^2*d^9*e^2/a*x^3-5/16/(a*e^4+c*d^4)^3*c^3/(c*x^4+a)^2*d^10*e/a*x^2+21/128/(a*e

^4+c*d^4)^3*c^3*(1/c*a)^(1/4)/a^3*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^11

+19/32/(a*e^4+c*d^4)^3*c/(c*x^4+a)^2*d^3*a*x*e^8+17/32/(a*e^4+c*d^4)^3*c/(c*x^4+

a)^2*d*e^10*a*x^3-9/16/(a*e^4+c*d^4)^3*c/(c*x^4+a)^2*d^2*e^9*a*x^2+21/128/(a*e^4

+c*d^4)^3*c^3*(1/c*a)^(1/4)/a^3*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^11+7

7/256/(a*e^4+c*d^4)^3*c*(1/c*a)^(1/4)/a*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+

(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^3*e^8+e^11*ln(e*x+

d)/(a*e^4+c*d^4)^3+5/128/(a*e^4+c*d^4)^3*c^2/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^

(1/2)/(1/c*a)^(1/4)*x+1)*d^9*e^2+33/128/(a*e^4+c*d^4)^3*c^2*(1/c*a)^(1/4)/a^2*2^

(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2

)+(1/c*a)^(1/2)))*d^7*e^4+33/64/(a*e^4+c*d^4)^3*c^2*(1/c*a)^(1/4)/a^2*2^(1/2)*ar

ctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^7*e^4-7/8/(a*e^4+c*d^4)^3*c^2/(c*x^4+a)^2*d^6*

e^5*x^2+15/16/(a*e^4+c*d^4)^3*c^2/(c*x^4+a)^2*d^7*x*e^4+11/32/(a*e^4+c*d^4)^3*c^

3/(c*x^4+a)^2*d^11/a*x-3/16/(a*e^4+c*d^4)^3*c^3/(a^5*c)^(1/2)*arctan(x^2*(c/a)^(

1/2))*d^10*e+45/128/(a*e^4+c*d^4)^3/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)

^(1/4)*x-1)*d*e^10+45/256/(a*e^4+c*d^4)^3/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^

(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d*e^

10+45/128/(a*e^4+c*d^4)^3/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1

)*d*e^10+1/4/(a*e^4+c*d^4)^3*c/(c*x^4+a)^2*x^4*a*e^11+1/2/(a*e^4+c*d^4)^3*c/(c*x

^4+a)^2*a*d^4*e^7+13/32/(a*e^4+c*d^4)^3*c^2/(c*x^4+a)^2*d*e^10*x^7-7/16/(a*e^4+c

*d^4)^3*c^2/(c*x^4+a)^2*d^2*e^9*x^6

_______________________________________________________________________________________

Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.388769, size = 1, normalized size = 0. \[ \mathit{Done} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Done

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