3.635 \(\int \frac{1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx\)

Optimal. Leaf size=255 \[ -\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e}+\frac{\log (d+e x)}{a^3 e}+\frac{16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 e \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d +
e*x)^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2
)/(4*a^2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^4 - 10*a
*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b
^2 - 4*a*c)^(5/2)*e) + Log[d + e*x]/(a^3*e) - Log[a + b*(d + e*x)^2 + c*(d + e*x
)^4]/(4*a^3*e)

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Rubi [A]  time = 0.939089, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e}+\frac{\log (d+e x)}{a^3 e}+\frac{16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 e \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d +
e*x)^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2
)/(4*a^2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^4 - 10*a
*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b
^2 - 4*a*c)^(5/2)*e) + Log[d + e*x]/(a^3*e) - Log[a + b*(d + e*x)^2 + c*(d + e*x
)^4]/(4*a^3*e)

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Rubi in Sympy [A]  time = 106.21, size = 238, normalized size = 0.93 \[ \frac{- 2 a c + b^{2} + b c \left (d + e x\right )^{2}}{4 a e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )^{2}} + \frac{16 a^{2} c^{2} - 15 a b^{2} c + 2 b^{4} + 2 b c \left (d + e x\right )^{2} \left (- 7 a c + b^{2}\right )}{4 a^{2} e \left (- 4 a c + b^{2}\right )^{2} \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} + \frac{b \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} e \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{\log{\left (\left (d + e x\right )^{2} \right )}}{2 a^{3} e} - \frac{\log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 a^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

(-2*a*c + b**2 + b*c*(d + e*x)**2)/(4*a*e*(-4*a*c + b**2)*(a + b*(d + e*x)**2 +
c*(d + e*x)**4)**2) + (16*a**2*c**2 - 15*a*b**2*c + 2*b**4 + 2*b*c*(d + e*x)**2*
(-7*a*c + b**2))/(4*a**2*e*(-4*a*c + b**2)**2*(a + b*(d + e*x)**2 + c*(d + e*x)*
*4)) + b*(30*a**2*c**2 - 10*a*b**2*c + b**4)*atanh((b + 2*c*(d + e*x)**2)/sqrt(-
4*a*c + b**2))/(2*a**3*e*(-4*a*c + b**2)**(5/2)) + log((d + e*x)**2)/(2*a**3*e)
- log(a + b*(d + e*x)**2 + c*(d + e*x)**4)/(4*a**3*e)

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Mathematica [A]  time = 6.2354, size = 421, normalized size = 1.65 \[ \frac{\log (d+e x)}{a^3 e}+\frac{16 a^2 c^2-15 a b^2 c-14 a b c^2 (d+e x)^2+2 b^4+2 b^3 c (d+e x)^2}{4 a^2 e \left (4 a c-b^2\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\left (-16 a^2 c^2 \sqrt{b^2-4 a c}-30 a^2 b c^2+10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}-b^5\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a^3 e \left (b^2-4 a c\right )^{5/2}}+\frac{\left (-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a^3 e \left (b^2-4 a c\right )^{5/2}}+\frac{2 a c-b^2-b c (d+e x)^2}{4 a e \left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(-b^2 + 2*a*c - b*c*(d + e*x)^2)/(4*a*(-b^2 + 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d
+ e*x)^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + e*x)^2 - 14*a*b*
c^2*(d + e*x)^2)/(4*a^2*(-b^2 + 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4))
+ Log[d + e*x]/(a^3*e) + ((-b^5 + 10*a*b^3*c - 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a
*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b
^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^3*(b^2 - 4*a*c)^(5/2)*e) + ((b^5 - 10*a*b^3
*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2
*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^3*(b^
2 - 4*a*c)^(5/2)*e)

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Maple [C]  time = 0.103, size = 4477, normalized size = 17.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)

[Out]

1/2/a^2/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c^2*d^6-29/4/a/(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/e/(1
6*a^2*c^2-8*a*b^2*c+b^4)*b^2*c^2*d^4+1/a^2/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4
)*b^4*c*d^4-3/a/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c*d^2-7/2/a/(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
*c^3*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2/a^2/(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*c^2*e^5*b^3/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^6-29/4/a/(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*e^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b
^2+1/a^2/(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*e^3*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^4-3/a/(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*e/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^2*b^3*c-21/a/(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*b*c^3*d*e^4/(16*a^2*c^2-8*a*b^2*
c+b^4)*x^5+16/(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*c^3*d*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+24/(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*e/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c^3*d^2-1/2/(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*e/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^2*b*c^2+1/2/a^2/(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*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^5+1/a^2/(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*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^5+1/2/a^2/(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/e/(16*a^2*c^2-8*a*b^2*c+
b^4)*b^5*d^2+ln(e*x+d)/a^3/e-21/a/(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*d^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b*c
^3+3/a^2/(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*d^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^3*c^2-29/a/(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*d^3/(
16*a^2*c^2-8*a*b^2*c+b^4)*x*b^2*c^2+4/a^2/(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*d^3/(16*a^2*c^2-8*a*b^2*c+b^
4)*x*b^4*c-6/a/(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*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^3*c-7/2/a/(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/e/
(16*a^2*c^2-8*a*b^2*c+b^4)*b*c^3*d^6+3/4/a/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4
)*b^4-1/2/a^3/e*sum((c*e^3*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^3+3*c*d*e^2*(16*a^2*c^2
-8*a*b^2*c+b^4)*_R^2+e*(48*a^2*c^3*d^2-24*a*b^2*c^2*d^2+3*b^4*c*d^2+23*a^2*b*c^2
-9*a*b^3*c+b^5)*_R+16*a^2*c^3*d^3-8*a*b^2*c^2*d^3+b^4*c*d^3+23*a^2*b*c^2*d-9*a*b
^3*c*d+b^5*d)/(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))+6*a/(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/e/(16*a^2*c^2-
8*a*b^2*c+b^4)*c^2+16/(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*d^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^3+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
/e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*d^4-21/4/(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/e/(16*a^2*c^2-8*a*b^2*c+b^4
)*c*b^2-1/(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*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b*c^2-1/2/(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/e/(16*a^2
*c^2-8*a*b^2*c+b^4)*b*c^2*d^2+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*e^3*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4
+3/a^2/(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*b^3*c^2*d*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-105/2/a/(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*
e^3*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b*d^2-29/a/(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*c^2*d*e^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^3*b^2+4/a^2/(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*c*d*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*
b^4-105/2/a/(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*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b*c^3*d^4+15/2/a^2/(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*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^3*c^2*d^4-87/2/a/(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*e/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^2*b^2*c^2*d^2+6/a^2/(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*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*
b^4*c*d^2+15/2/a^2/(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*e^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^3*d^2-70/a
/(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*c^3*d^3*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b+10/a^2/(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*c^2*d^3
*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^3

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

[Out]

1/4*(2*(b^3*c^2 - 7*a*b*c^3)*e^6*x^6 + 12*(b^3*c^2 - 7*a*b*c^3)*d*e^5*x^5 + (4*b
^4*c - 29*a*b^2*c^2 + 16*a^2*c^3 + 30*(b^3*c^2 - 7*a*b*c^3)*d^2)*e^4*x^4 + 2*(b^
3*c^2 - 7*a*b*c^3)*d^6 + 4*(10*(b^3*c^2 - 7*a*b*c^3)*d^3 + (4*b^4*c - 29*a*b^2*c
^2 + 16*a^2*c^3)*d)*e^3*x^3 + 3*a*b^4 - 21*a^2*b^2*c + 24*a^3*c^2 + (4*b^4*c - 2
9*a*b^2*c^2 + 16*a^2*c^3)*d^4 + 2*(b^5 - 6*a*b^3*c - a^2*b*c^2 + 15*(b^3*c^2 - 7
*a*b*c^3)*d^4 + 3*(4*b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + 2*(b^5 -
6*a*b^3*c - a^2*b*c^2)*d^2 + 4*(3*(b^3*c^2 - 7*a*b*c^3)*d^5 + (4*b^4*c - 29*a*b^
2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 6*a*b^3*c - a^2*b*c^2)*d)*e*x)/((a^2*b^4*c^2 -
8*a^3*b^2*c^3 + 16*a^4*c^4)*e^9*x^8 + 8*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^
4)*d*e^8*x^7 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3 + 14*(a^2*b^4*c^2 - 8
*a^3*b^2*c^3 + 16*a^4*c^4)*d^2)*e^7*x^6 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 1
6*a^4*c^4)*d^3 + 3*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d)*e^6*x^5 + (a^2*
b^6 - 6*a^3*b^4*c + 32*a^5*c^3 + 70*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d
^4 + 30*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^2)*e^5*x^4 + 4*(14*(a^2*b^4
*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^5 + 10*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*
b*c^3)*d^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d)*e^4*x^3 + 2*(a^3*b^5 - 8*a^
4*b^3*c + 16*a^5*b*c^2 + 14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^6 + 15*
(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^4 + 3*(a^2*b^6 - 6*a^3*b^4*c + 32*a
^5*c^3)*d^2)*e^3*x^2 + 4*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^7 + 3*(
a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^5 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*
c^3)*d^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d)*e^2*x + ((a^2*b^4*c^2 - 8*a
^3*b^2*c^3 + 16*a^4*c^4)*d^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c
- 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^4
+ 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^2)*e) - integrate(((b^4*c - 8*a*b^2
*c^2 + 16*a^2*c^3)*e^3*x^3 + 3*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^2*x^2 + (b
^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 9*a*b^3*c + 23*a^2*b*c^2 + 3*(b^4*
c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e*x + (b^5 - 9*a*b^3*c + 23*a^2*b*c^2)*d)/(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)/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2) + log(e*x + d)/(a^3*e)

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Fricas [A]  time = 1.29001, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^3*(e*x + d)),x, algorithm="fricas")

[Out]

[1/4*(((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^8*x^8 + 8*(b^5*c^2 - 10*a*b^3*c
^3 + 30*a^2*b*c^4)*d*e^7*x^7 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3 + 14*(b^
5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^2)*e^6*x^6 + 4*(14*(b^5*c^2 - 10*a*b^3*c^
3 + 30*a^2*b*c^4)*d^3 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d)*e^5*x^5 + (
b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^8 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 +
60*a^3*b*c^3 + 70*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^4 + 30*(b^6*c - 10*
a*b^4*c^2 + 30*a^2*b^2*c^3)*d^2)*e^4*x^4 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2
+ 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^6 + 4*(14*(b^5*c^2 - 10*a*b^3*c^3
+ 30*a^2*b*c^4)*d^5 + 10*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^3 + (b^7 - 8
*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d)*e^3*x^3 + (b^7 - 8*a*b^5*c + 10*a^2
*b^3*c^2 + 60*a^3*b*c^3)*d^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2 + 14*(b^
5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^6 + 15*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2
*c^3)*d^4 + 3*(b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^2)*e^2*x^2 + 2
*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*d^2 + 4*(2*(b^5*c^2 - 10*a*b^3*c^3 + 30
*a^2*b*c^4)*d^7 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^5 + (b^7 - 8*a*b^5
*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^3 + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2
)*d)*e*x)*log((2*(b^2*c - 4*a*c^2)*e^2*x^2 + 4*(b^2*c - 4*a*c^2)*d*e*x + b^3 - 4
*a*b*c + 2*(b^2*c - 4*a*c^2)*d^2 + (2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4
+ 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*
a*c)*sqrt(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)) + (2*(a*b^3*c^2 - 7*a^2*b*c^3)*e^6*x^6
+ 12*(a*b^3*c^2 - 7*a^2*b*c^3)*d*e^5*x^5 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*
c^3 + 30*(a*b^3*c^2 - 7*a^2*b*c^3)*d^2)*e^4*x^4 + 2*(a*b^3*c^2 - 7*a^2*b*c^3)*d^
6 + 4*(10*(a*b^3*c^2 - 7*a^2*b*c^3)*d^3 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c
^3)*d)*e^3*x^3 + 3*a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + (4*a*b^4*c - 29*a^2*b^2
*c^2 + 16*a^3*c^3)*d^4 + 2*(a*b^5 - 6*a^2*b^3*c - a^3*b*c^2 + 15*(a*b^3*c^2 - 7*
a^2*b*c^3)*d^4 + 3*(4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*d^2)*e^2*x^2 + 2*(a
*b^5 - 6*a^2*b^3*c - a^3*b*c^2)*d^2 + 4*(3*(a*b^3*c^2 - 7*a^2*b*c^3)*d^5 + (4*a*
b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*d^3 + (a*b^5 - 6*a^2*b^3*c - a^3*b*c^2)*d)*
e*x - ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^8*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*d*e^7*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^6*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^5*x^5 + (b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4)*d^8 + (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^4*x^4 + 2*(
b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 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^3*x^3 + 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*(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^2*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^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*x)*log(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) + 4*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^8*
x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^7*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^6*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^5*x^5 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + (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^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 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^3*x^3 + 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*(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 + 1
6*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^2*x^2 + 2*(a*b^5 - 8*
a^2*b^3*c + 16*a^3*b*c^2)*d^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*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/
(((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^9*x^8 + 8*(a^3*b^4*c^2 - 8*a^4*b^
2*c^3 + 16*a^5*c^4)*d*e^8*x^7 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3 + 14
*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^2)*e^7*x^6 + 4*(14*(a^3*b^4*c^2 -
8*a^4*b^2*c^3 + 16*a^5*c^4)*d^3 + 3*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d
)*e^6*x^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3 + 70*(a^3*b^4*c^2 - 8*a^4*b^2*c^
3 + 16*a^5*c^4)*d^4 + 30*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2)*e^5*x^4
+ 4*(14*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5 + 10*(a^3*b^5*c - 8*a^4*
b^3*c^2 + 16*a^5*b*c^3)*d^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d)*e^4*x^3 +
2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2 + 14*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a
^5*c^4)*d^6 + 15*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^4 + 3*(a^3*b^6 - 6
*a^4*b^4*c + 32*a^6*c^3)*d^2)*e^3*x^2 + 4*(2*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a
^5*c^4)*d^7 + 3*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^5 + (a^3*b^6 - 6*a^
4*b^4*c + 32*a^6*c^3)*d^3 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d)*e^2*x + (a
^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d
^8 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^6 + (a^3*b^6 - 6*a^4*b^4*c +
32*a^6*c^3)*d^4 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d^2)*e)*sqrt(b^2 - 4
*a*c)), -1/4*(2*((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^8*x^8 + 8*(b^5*c^2 -
10*a*b^3*c^3 + 30*a^2*b*c^4)*d*e^7*x^7 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^
3 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^2)*e^6*x^6 + 4*(14*(b^5*c^2 - 1
0*a*b^3*c^3 + 30*a^2*b*c^4)*d^3 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d)*e
^5*x^5 + (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^8 + (b^7 - 8*a*b^5*c + 10*a^2
*b^3*c^2 + 60*a^3*b*c^3 + 70*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^4 + 30*(b
^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^2)*e^4*x^4 + a^2*b^5 - 10*a^3*b^3*c + 30
*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^6 + 4*(14*(b^5*c^2 - 10
*a*b^3*c^3 + 30*a^2*b*c^4)*d^5 + 10*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^3
+ (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d)*e^3*x^3 + (b^7 - 8*a*b^5*
c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^
2 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^6 + 15*(b^6*c - 10*a*b^4*c^2 +
30*a^2*b^2*c^3)*d^4 + 3*(b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^2)*e
^2*x^2 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*d^2 + 4*(2*(b^5*c^2 - 10*a*b^
3*c^3 + 30*a^2*b*c^4)*d^7 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^5 + (b^7
- 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^3 + (a*b^6 - 10*a^2*b^4*c + 30*a
^3*b^2*c^2)*d)*e*x)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 +
4*a*c)/(b^2 - 4*a*c)) - (2*(a*b^3*c^2 - 7*a^2*b*c^3)*e^6*x^6 + 12*(a*b^3*c^2 - 7
*a^2*b*c^3)*d*e^5*x^5 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3 + 30*(a*b^3*c^2
- 7*a^2*b*c^3)*d^2)*e^4*x^4 + 2*(a*b^3*c^2 - 7*a^2*b*c^3)*d^6 + 4*(10*(a*b^3*c^
2 - 7*a^2*b*c^3)*d^3 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*d)*e^3*x^3 + 3*
a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + (4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*
d^4 + 2*(a*b^5 - 6*a^2*b^3*c - a^3*b*c^2 + 15*(a*b^3*c^2 - 7*a^2*b*c^3)*d^4 + 3*
(4*a*b^4*c - 29*a^2*b^2*c^2 + 16*a^3*c^3)*d^2)*e^2*x^2 + 2*(a*b^5 - 6*a^2*b^3*c
- a^3*b*c^2)*d^2 + 4*(3*(a*b^3*c^2 - 7*a^2*b*c^3)*d^5 + (4*a*b^4*c - 29*a^2*b^2*
c^2 + 16*a^3*c^3)*d^3 + (a*b^5 - 6*a^2*b^3*c - a^3*b*c^2)*d)*e*x - ((b^4*c^2 - 8
*a*b^2*c^3 + 16*a^2*c^4)*e^8*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^7*
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^6*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^5*x^5 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d
^8 + (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^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2
+ 16*a^2*b*c^3)*d^6 + 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^3*x^3
+ 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*(1
4*(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^2*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^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*x)*log(
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) + 4*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^8*x^8 + 8*(b^4*c^2 -
8*a*b^2*c^3 + 16*a^2*c^4)*d*e^7*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^6*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^5*x^5 + (b^4*c
^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + (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^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 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^3*x^3 + 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*(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^2*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*
b*c^2)*d^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*x)*log(e*x + d))*sqrt(-b^2 + 4*a*c))/(((a^3*b^4*c^2 - 8
*a^4*b^2*c^3 + 16*a^5*c^4)*e^9*x^8 + 8*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4
)*d*e^8*x^7 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3 + 14*(a^3*b^4*c^2 - 8*
a^4*b^2*c^3 + 16*a^5*c^4)*d^2)*e^7*x^6 + 4*(14*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16
*a^5*c^4)*d^3 + 3*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d)*e^6*x^5 + (a^3*b
^6 - 6*a^4*b^4*c + 32*a^6*c^3 + 70*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^
4 + 30*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2)*e^5*x^4 + 4*(14*(a^3*b^4*
c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5 + 10*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b
*c^3)*d^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d)*e^4*x^3 + 2*(a^4*b^5 - 8*a^5
*b^3*c + 16*a^6*b*c^2 + 14*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^6 + 15*(
a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^4 + 3*(a^3*b^6 - 6*a^4*b^4*c + 32*a^
6*c^3)*d^2)*e^3*x^2 + 4*(2*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^7 + 3*(a
^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c
^3)*d^3 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d)*e^2*x + (a^5*b^4 - 8*a^6*b^2
*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^8 + 2*(a^3*b^5*c
- 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^4 +
2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d^2)*e)*sqrt(-b^2 + 4*a*c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.347916, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^3*(e*x + d)),x, algorithm="giac")

[Out]

Done