∖int (d+ex)4 ________________________________________________________∖left(a+b(d+ex)2 +c(d+ex)4 ∖right)3∖,dx

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

Optimal. Leaf size=341 \[ \frac{(d+e x) \left (2 a+b (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac{(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{8 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{3 \sqrt{c} \left (-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

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

e*x)^4)^2) - ((d + e*x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/(8*(b^2 - 4*a*c)^

2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[c]*(3*b^2 + 4*a*c - 2*b*Sqrt[

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

4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt[c]*(3*b^2

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

rt[b^2 - 4*a*c]]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi [A] time = 1.89101, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(d+e x) \left (2 a+b (d+e x)^2\right )}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac{(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{8 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{3 \sqrt{c} \left (-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

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

e*x)^4)^2) - ((d + e*x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/(8*(b^2 - 4*a*c)^

2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[c]*(3*b^2 + 4*a*c - 2*b*Sqrt[

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

4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt[c]*(3*b^2

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

rt[b^2 - 4*a*c]]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi in Sympy [A] time = 124.697, size = 314, normalized size = 0.92 \[ - \frac{3 \sqrt{2} \sqrt{c} \left (a c + \frac{3 b^{2}}{4} + \frac{b \sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 e \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{3 \sqrt{2} \sqrt{c} \left (a c + \frac{3 b^{2}}{4} - \frac{b \sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 e \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{\left (2 a + b \left (d + e x\right )^{2}\right ) \left (d + e x\right )}{4 e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )^{2}} - \frac{\left (d + e x\right ) \left (- 4 a c + 7 b^{2} + 12 b c \left (d + e x\right )^{2}\right )}{8 e \left (- 4 a c + b^{2}\right )^{2} \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(2)*sqrt(c)*(a*c + 3*b**2/4 + b*sqrt(-4*a*c + b**2)/2)*atan(sqrt(2)*sqrt(

c)*(d + e*x)/sqrt(b + sqrt(-4*a*c + b**2)))/(2*e*sqrt(b + sqrt(-4*a*c + b**2))*(

-4*a*c + b**2)**(5/2)) + 3*sqrt(2)*sqrt(c)*(a*c + 3*b**2/4 - b*sqrt(-4*a*c + b**

2)/2)*atan(sqrt(2)*sqrt(c)*(d + e*x)/sqrt(b - sqrt(-4*a*c + b**2)))/(2*e*sqrt(b

- sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(5/2)) + (2*a + b*(d + e*x)**2)*(d + e*x

)/(4*e*(-4*a*c + b**2)*(a + b*(d + e*x)**2 + c*(d + e*x)**4)**2) - (d + e*x)*(-4

*a*c + 7*b**2 + 12*b*c*(d + e*x)**2)/(8*e*(-4*a*c + b**2)**2*(a + b*(d + e*x)**2

+ c*(d + e*x)**4))

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Mathematica [A] time = 6.24794, size = 347, normalized size = 1.02 \[ -\frac{-2 a (d+e x)-b (d+e x)^3}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac{4 a c (d+e x)-7 b^2 (d+e x)-12 b c (d+e x)^3}{8 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{3 \sqrt{c} \left (2 b \sqrt{b^2-4 a c}-4 a c-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 \sqrt{2} e \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

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

e*x)^4)^2) + (-7*b^2*(d + e*x) + 4*a*c*(d + e*x) - 12*b*c*(d + e*x)^3)/(8*(b^2

- 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*Sqrt[c]*(-3*b^2 - 4*a*c +

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

*a*c]]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt

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

qrt[b + Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 -

4*a*c]]*e)

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Maple [C] time = 0.094, size = 704, normalized size = 2.1 \[{\frac{1}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) ^{2}} \left ( -{\frac{3\,{c}^{2}{e}^{6}b{x}^{7}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-{\frac{21\,{c}^{2}d{e}^{5}b{x}^{6}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}+{\frac{ \left ( -252\,bc{d}^{2}+4\,ac-19\,{b}^{2} \right ) c{e}^{4}{x}^{5}}{128\,{a}^{2}{c}^{2}-64\,a{b}^{2}c+8\,{b}^{4}}}+{\frac{5\,cd{e}^{3} \left ( -84\,bc{d}^{2}+4\,ac-19\,{b}^{2} \right ){x}^{4}}{128\,{a}^{2}{c}^{2}-64\,a{b}^{2}c+8\,{b}^{4}}}-{\frac{{e}^{2} \left ( 420\,{c}^{2}{d}^{4}b-40\,a{c}^{2}{d}^{2}+190\,{b}^{2}c{d}^{2}+16\,abc+5\,{b}^{3} \right ){x}^{3}}{128\,{a}^{2}{c}^{2}-64\,a{b}^{2}c+8\,{b}^{4}}}-{\frac{de \left ( 252\,{c}^{2}{d}^{4}b-40\,a{c}^{2}{d}^{2}+190\,{b}^{2}c{d}^{2}+48\,abc+15\,{b}^{3} \right ){x}^{2}}{128\,{a}^{2}{c}^{2}-64\,a{b}^{2}c+8\,{b}^{4}}}-{\frac{ \left ( 84\,b{c}^{2}{d}^{6}-20\,a{c}^{2}{d}^{4}+95\,{b}^{2}c{d}^{4}+48\,abc{d}^{2}+15\,{b}^{3}{d}^{2}+12\,{a}^{2}c+3\,a{b}^{2} \right ) x}{128\,{a}^{2}{c}^{2}-64\,a{b}^{2}c+8\,{b}^{4}}}-{\frac{d \left ( 12\,b{c}^{2}{d}^{6}-4\,a{c}^{2}{d}^{4}+19\,{b}^{2}c{d}^{4}+16\,abc{d}^{2}+5\,{b}^{3}{d}^{2}+12\,{a}^{2}c+3\,a{b}^{2} \right ) }{8\, \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) e}} \right ) }+{\frac{3}{16\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -4\,{{\it \_R}}^{2}bc{e}^{2}-8\,{\it \_R}\,bcde-4\,bc{d}^{2}+4\,ac+{b}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \left ( 2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd \right ) }}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

(-3/2*c^2*e^6*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7-21/2*c^2*d*e^5*b/(16*a^2*c^2-8*a*

b^2*c+b^4)*x^6+1/8*(-252*b*c*d^2+4*a*c-19*b^2)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*

x^5+5/8*c*d*e^3*(-84*b*c*d^2+4*a*c-19*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-1/8*e^

2*(420*b*c^2*d^4-40*a*c^2*d^2+190*b^2*c*d^2+16*a*b*c+5*b^3)/(16*a^2*c^2-8*a*b^2*

c+b^4)*x^3-1/8*d*e*(252*b*c^2*d^4-40*a*c^2*d^2+190*b^2*c*d^2+48*a*b*c+15*b^3)/(1

6*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*(84*b*c^2*d^6-20*a*c^2*d^4+95*b^2*c*d^4+48*a*b*

c*d^2+15*b^3*d^2+12*a^2*c+3*a*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/8*d/e*(12*b*c^

2*d^6-4*a*c^2*d^4+19*b^2*c*d^4+16*a*b*c*d^2+5*b^3*d^2+12*a^2*c+3*a*b^2)/(16*a^2*

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

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

+4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e

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

*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+c*d^4+b*d^2+a))

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(12*b*c^2*e^7*x^7 + 84*b*c^2*d*e^6*x^6 + (252*b*c^2*d^2 + 19*b^2*c - 4*a*c^

2)*e^5*x^5 + 12*b*c^2*d^7 + 5*(84*b*c^2*d^3 + (19*b^2*c - 4*a*c^2)*d)*e^4*x^4 +

(420*b*c^2*d^4 + 5*b^3 + 16*a*b*c + 10*(19*b^2*c - 4*a*c^2)*d^2)*e^3*x^3 + (19*b

^2*c - 4*a*c^2)*d^5 + (252*b*c^2*d^5 + 10*(19*b^2*c - 4*a*c^2)*d^3 + 3*(5*b^3 +

16*a*b*c)*d)*e^2*x^2 + (5*b^3 + 16*a*b*c)*d^3 + (84*b*c^2*d^6 + 5*(19*b^2*c - 4*

a*c^2)*d^4 + 3*a*b^2 + 12*a^2*c + 3*(5*b^3 + 16*a*b*c)*d^2)*e*x + 3*(a*b^2 + 4*a

^2*c)*d)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^

3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2

- 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2

*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*c

+ 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^

3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*

d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^

3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b

^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a

*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^

7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*

d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a*b^2*c^3 +

16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b

^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c

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

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

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

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Fricas [A] time = 0.511322, size = 8955, normalized size = 26.26 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(24*b*c^2*e^7*x^7 + 168*b*c^2*d*e^6*x^6 + 2*(252*b*c^2*d^2 + 19*b^2*c - 4*

a*c^2)*e^5*x^5 + 24*b*c^2*d^7 + 10*(84*b*c^2*d^3 + (19*b^2*c - 4*a*c^2)*d)*e^4*x

^4 + 2*(420*b*c^2*d^4 + 5*b^3 + 16*a*b*c + 10*(19*b^2*c - 4*a*c^2)*d^2)*e^3*x^3

+ 2*(19*b^2*c - 4*a*c^2)*d^5 + 2*(252*b*c^2*d^5 + 10*(19*b^2*c - 4*a*c^2)*d^3 +

3*(5*b^3 + 16*a*b*c)*d)*e^2*x^2 + 2*(5*b^3 + 16*a*b*c)*d^3 + 2*(84*b*c^2*d^6 + 5

*(19*b^2*c - 4*a*c^2)*d^4 + 3*a*b^2 + 12*a^2*c + 3*(5*b^3 + 16*a*b*c)*d^2)*e*x -

3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^

2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*

c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16

*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b

^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*

a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c

^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^

3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a

^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 -

6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4

)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c

^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a*b^2*c^

3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a

^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^

3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (a*b^10 - 2

0*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^

5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 12

80*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2

- 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(3*(5*b^4*c + 40*a

*b^2*c^2 + 16*a^2*c^3)*e*x + 3*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d + 3/2*sqr

t(1/2)*((a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^5*b^

5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^3*sqrt(1/((a^2*b^10 - 20*a^3*b^8*

c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)) -

(b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e)*sqrt(-(b^5 + 40*a*b^3*c +

80*a^2*b*c^2 + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280

*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*

c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10 - 20*a^

2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e

^2))) + 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 -

8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 1

4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c

^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6

- 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5

*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 1

6*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c

+ 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^

5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3

*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*

a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 3

2*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a

*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^

4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8

*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (a*b

^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024

*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c

^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10 - 20*a^2*b^8*c + 160*a^3*b

^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(3*(5*b^4*c

+ 40*a*b^2*c^2 + 16*a^2*c^3)*e*x + 3*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d -

3/2*sqrt(1/2)*((a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400

*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^3*sqrt(1/((a^2*b^10 - 20*a

^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*

e^4)) - (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e)*sqrt(-(b^5 + 40*a*b

^3*c + 80*a^2*b*c^2 + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3

+ 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a

^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10

- 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6

*c^5)*e^2))) + 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^

4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*

c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*

a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5

+ (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 +

30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*

c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a

*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6

+ a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*

d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^

3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^

4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c^

2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 +

a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4 + 2*(a*

b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2

- (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4

- 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^

5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10 - 20*a^2*b^8*c + 16

0*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(3*(

5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e*x + 3*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^

3)*d + 3/2*sqrt(1/2)*((a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3

- 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^3*sqrt(1/((a^2*b^10

- 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^

7*c^5)*e^4)) + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e)*sqrt(-(b^5 +

40*a*b^3*c + 80*a^2*b*c^2 - (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*

b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c

+ 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((

a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1

024*a^6*c^5)*e^2))) - 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8

+ 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16

*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c

^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e

^6*x^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)

*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8

*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^

6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c

^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*

b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a

*b^2*c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 -

6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + (

(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3

)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^4

+ 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^

2*b*c^2 - (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*

b^2*c^4 - 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 -

640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4)))/((a*b^10 - 20*a^2*b^8

*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*

log(3*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e*x + 3*(5*b^4*c + 40*a*b^2*c^2 + 16

*a^2*c^3)*d - 3/2*sqrt(1/2)*((a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*

b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^3*sqrt(1/((a

^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 -

1024*a^7*c^5)*e^4)) + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e)*sqrt(

-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 - (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 6

40*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2*sqrt(1/((a^2*b^10 - 20*a^3

*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^

4)))/((a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*

c^4 - 1024*a^6*c^5)*e^2))) + 6*(a*b^2 + 4*a^2*c)*d)/((b^4*c^2 - 8*a*b^2*c^3 + 16

*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c

- 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*

x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 +

16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2

*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 +

4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^

2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*

b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a

*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4

*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b

*c^3)*d^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b

*c^2)*d)*e^2*x + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*

c^2 + 16*a^2*b*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c

+ 32*a^3*c^3)*d^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)

_______________________________________________________________________________________

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

[Out]

Timed out

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{4}}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{3}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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