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

Optimal. Leaf size=228 \[ \frac{b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac{2 b \log (d+e x)}{a^3 e f^3}-\frac{b^2-3 a c}{a^2 e f^3 \left (b^2-4 a c\right ) (d+e x)^2}-\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{a^3 e f^3 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[Out]

-((b^2 - 3*a*c)/(a^2*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d +
 e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4
)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a
*c]])/(a^3*(b^2 - 4*a*c)^(3/2)*e*f^3) - (2*b*Log[d + e*x])/(a^3*e*f^3) + (b*Log[
a + b*(d + e*x)^2 + c*(d + e*x)^4])/(2*a^3*e*f^3)

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Rubi [A]  time = 0.761414, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242 \[ \frac{b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac{2 b \log (d+e x)}{a^3 e f^3}-\frac{b^2-3 a c}{a^2 e f^3 \left (b^2-4 a c\right ) (d+e x)^2}-\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{a^3 e f^3 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 3*a*c)/(a^2*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2)) + (b^2 - 2*a*c + b*c*(d +
 e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4
)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a
*c]])/(a^3*(b^2 - 4*a*c)^(3/2)*e*f^3) - (2*b*Log[d + e*x])/(a^3*e*f^3) + (b*Log[
a + b*(d + e*x)^2 + c*(d + e*x)^4])/(2*a^3*e*f^3)

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Rubi in Sympy [A]  time = 104.26, size = 209, normalized size = 0.92 \[ \frac{- 2 a c + b^{2} + b c \left (d + e x\right )^{2}}{2 a e f^{3} \left (d + e x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} - \frac{- 3 a c + b^{2}}{a^{2} e f^{3} \left (d + e x\right )^{2} \left (- 4 a c + b^{2}\right )} - \frac{b \log{\left (\left (d + e x\right )^{2} \right )}}{a^{3} e f^{3}} + \frac{b \log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{2 a^{3} e f^{3}} - \frac{\left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} e f^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*a*c + b**2 + b*c*(d + e*x)**2)/(2*a*e*f**3*(d + e*x)**2*(-4*a*c + b**2)*(a +
 b*(d + e*x)**2 + c*(d + e*x)**4)) - (-3*a*c + b**2)/(a**2*e*f**3*(d + e*x)**2*(
-4*a*c + b**2)) - b*log((d + e*x)**2)/(a**3*e*f**3) + b*log(a + b*(d + e*x)**2 +
 c*(d + e*x)**4)/(2*a**3*e*f**3) - (6*a**2*c**2 - 6*a*b**2*c + b**4)*atanh((b +
2*c*(d + e*x)**2)/sqrt(-4*a*c + b**2))/(a**3*e*f**3*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.857838, size = 287, normalized size = 1.26 \[ \frac{\frac{\left (6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{a \left (-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right )}{\left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{a}{(d+e x)^2}-4 b \log (d+e x)}{2 a^3 e f^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(a/(d + e*x)^2) + (a*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2)
)/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - 4*b*Log[d + e*x] + ((b^
4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*L
og[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2) + ((-b^4 + 6*a*
b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b + S
qrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(2*a^3*e*f^3)

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Maple [C]  time = 0.015, size = 1047, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/f^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)*c^2*e/(4*a*c-b^2)*x^2+1/2/f^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)*c*e/(4*a*c-b^2)*x^2*b
^2-2/f^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)*c^2*d/(4*a*c-b^2)*x+1/f^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)*c*d/(4*a*c-b^2)*x*b^2-1
/f^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)/e/(4*a*c-b^2)*c^2*d^2+1/2/f^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)/e/(4*a*c-b^2)*b^2*c*d^2
-3/2/f^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)/e/(4*a*c-b^2)*b*c+1/2/f^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)/e/(4*a*c-b^2)*b^3+1/f^3
/a^3/e*sum((b*e^3*c*(4*a*c-b^2)*_R^3+3*b*d*e^2*c*(4*a*c-b^2)*_R^2+e*(12*a*b*c^2*
d^2-3*b^3*c*d^2-3*a^2*c^2+5*a*b^2*c-b^4)*_R+4*a*b*c^2*d^3-b^3*c*d^3-3*a^2*c^2*d+
5*a*b^2*c*d-b^4*d)/(4*a*c-b^2)/(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))-1/2/f^3/a^2/e/(e*x+d)^2-2*b*ln(e*x+d)/
a^3/e/f^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)^2*(e*f*x + d*f)^3),x, algorithm="maxima")

[Out]

-1/2*(2*(b^2*c - 3*a*c^2)*e^4*x^4 + 8*(b^2*c - 3*a*c^2)*d*e^3*x^3 + 2*(b^2*c - 3
*a*c^2)*d^4 + (2*b^3 - 7*a*b*c + 12*(b^2*c - 3*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a
^2*c + (2*b^3 - 7*a*b*c)*d^2 + 2*(4*(b^2*c - 3*a*c^2)*d^3 + (2*b^3 - 7*a*b*c)*d)
*e*x)/((a^2*b^2*c - 4*a^3*c^2)*e^7*f^3*x^6 + 6*(a^2*b^2*c - 4*a^3*c^2)*d*e^6*f^3
*x^5 + (a^2*b^3 - 4*a^3*b*c + 15*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^5*f^3*x^4 + 4*(5
*(a^2*b^2*c - 4*a^3*c^2)*d^3 + (a^2*b^3 - 4*a^3*b*c)*d)*e^4*f^3*x^3 + (a^3*b^2 -
 4*a^4*c + 15*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 6*(a^2*b^3 - 4*a^3*b*c)*d^2)*e^3*f^3
*x^2 + 2*(3*(a^2*b^2*c - 4*a^3*c^2)*d^5 + 2*(a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2
 - 4*a^4*c)*d)*e^2*f^3*x + ((a^2*b^2*c - 4*a^3*c^2)*d^6 + (a^2*b^3 - 4*a^3*b*c)*
d^4 + (a^3*b^2 - 4*a^4*c)*d^2)*e*f^3) + 2*integrate(((b^3*c - 4*a*b*c^2)*e^3*x^3
 + 3*(b^3*c - 4*a*b*c^2)*d*e^2*x^2 + (b^3*c - 4*a*b*c^2)*d^3 + (b^4 - 5*a*b^2*c
+ 3*a^2*c^2 + 3*(b^3*c - 4*a*b*c^2)*d^2)*e*x + (b^4 - 5*a*b^2*c + 3*a^2*c^2)*d)/
((b^2*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d
^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3
- 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x), x)/(a^3*f
^3) - 2*b*log(e*x + d)/(a^3*e*f^3)

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Fricas [A]  time = 1.59886, 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)^2*(e*f*x + d*f)^3),x, algorithm="fricas")

[Out]

[-1/2*(((b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^6*x^6 + 6*(b^4*c - 6*a*b^2*c^2 + 6*a
^2*c^3)*d*e^5*x^5 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2 + 15*(b^4*c - 6*a*b^2*c^2 + 6
*a^2*c^3)*d^2)*e^4*x^4 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^6 + 4*(5*(b^4*c - 6
*a*b^2*c^2 + 6*a^2*c^3)*d^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d)*e^3*x^3 + (b^5
- 6*a*b^3*c + 6*a^2*b*c^2)*d^4 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + 15*(b^4*c -
6*a*b^2*c^2 + 6*a^2*c^3)*d^4 + 6*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^2)*e^2*x^2 +
(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d^2 + 2*(3*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d
^5 + 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^3 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*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^2*c - 3*a^2*c^2)*e^4*x^4 + 8*(a*
b^2*c - 3*a^2*c^2)*d*e^3*x^3 + 2*(a*b^2*c - 3*a^2*c^2)*d^4 + (2*a*b^3 - 7*a^2*b*
c + 12*(a*b^2*c - 3*a^2*c^2)*d^2)*e^2*x^2 + a^2*b^2 - 4*a^3*c + (2*a*b^3 - 7*a^2
*b*c)*d^2 + 2*(4*(a*b^2*c - 3*a^2*c^2)*d^3 + (2*a*b^3 - 7*a^2*b*c)*d)*e*x - ((b^
3*c - 4*a*b*c^2)*e^6*x^6 + 6*(b^3*c - 4*a*b*c^2)*d*e^5*x^5 + (b^4 - 4*a*b^2*c +
15*(b^3*c - 4*a*b*c^2)*d^2)*e^4*x^4 + (b^3*c - 4*a*b*c^2)*d^6 + 4*(5*(b^3*c - 4*
a*b*c^2)*d^3 + (b^4 - 4*a*b^2*c)*d)*e^3*x^3 + (b^4 - 4*a*b^2*c)*d^4 + (15*(b^3*c
 - 4*a*b*c^2)*d^4 + a*b^3 - 4*a^2*b*c + 6*(b^4 - 4*a*b^2*c)*d^2)*e^2*x^2 + (a*b^
3 - 4*a^2*b*c)*d^2 + 2*(3*(b^3*c - 4*a*b*c^2)*d^5 + 2*(b^4 - 4*a*b^2*c)*d^3 + (a
*b^3 - 4*a^2*b*c)*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^3*c - 4*a*b*c^2)*e^6*x^6 +
6*(b^3*c - 4*a*b*c^2)*d*e^5*x^5 + (b^4 - 4*a*b^2*c + 15*(b^3*c - 4*a*b*c^2)*d^2)
*e^4*x^4 + (b^3*c - 4*a*b*c^2)*d^6 + 4*(5*(b^3*c - 4*a*b*c^2)*d^3 + (b^4 - 4*a*b
^2*c)*d)*e^3*x^3 + (b^4 - 4*a*b^2*c)*d^4 + (15*(b^3*c - 4*a*b*c^2)*d^4 + a*b^3 -
 4*a^2*b*c + 6*(b^4 - 4*a*b^2*c)*d^2)*e^2*x^2 + (a*b^3 - 4*a^2*b*c)*d^2 + 2*(3*(
b^3*c - 4*a*b*c^2)*d^5 + 2*(b^4 - 4*a*b^2*c)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e*x)*l
og(e*x + d))*sqrt(b^2 - 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2)*e^7*f^3*x^6 + 6*(a^3*b
^2*c - 4*a^4*c^2)*d*e^6*f^3*x^5 + (a^3*b^3 - 4*a^4*b*c + 15*(a^3*b^2*c - 4*a^4*c
^2)*d^2)*e^5*f^3*x^4 + 4*(5*(a^3*b^2*c - 4*a^4*c^2)*d^3 + (a^3*b^3 - 4*a^4*b*c)*
d)*e^4*f^3*x^3 + (a^4*b^2 - 4*a^5*c + 15*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 6*(a^3*b^
3 - 4*a^4*b*c)*d^2)*e^3*f^3*x^2 + 2*(3*(a^3*b^2*c - 4*a^4*c^2)*d^5 + 2*(a^3*b^3
- 4*a^4*b*c)*d^3 + (a^4*b^2 - 4*a^5*c)*d)*e^2*f^3*x + ((a^3*b^2*c - 4*a^4*c^2)*d
^6 + (a^3*b^3 - 4*a^4*b*c)*d^4 + (a^4*b^2 - 4*a^5*c)*d^2)*e*f^3)*sqrt(b^2 - 4*a*
c)), 1/2*(2*((b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^6*x^6 + 6*(b^4*c - 6*a*b^2*c^2
+ 6*a^2*c^3)*d*e^5*x^5 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2 + 15*(b^4*c - 6*a*b^2*c^
2 + 6*a^2*c^3)*d^2)*e^4*x^4 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^6 + 4*(5*(b^4*
c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d)*e^3*x^3 +
(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^4 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + 15*(b^4
*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^4 + 6*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^2)*e^2*x
^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d^2 + 2*(3*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c
^3)*d^5 + 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^3 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*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^2*c - 3*a^2*c^2)*e^4*x^4 + 8*(a*b^2*c - 3*a^2*c^2)*d*e^3
*x^3 + 2*(a*b^2*c - 3*a^2*c^2)*d^4 + (2*a*b^3 - 7*a^2*b*c + 12*(a*b^2*c - 3*a^2*
c^2)*d^2)*e^2*x^2 + a^2*b^2 - 4*a^3*c + (2*a*b^3 - 7*a^2*b*c)*d^2 + 2*(4*(a*b^2*
c - 3*a^2*c^2)*d^3 + (2*a*b^3 - 7*a^2*b*c)*d)*e*x - ((b^3*c - 4*a*b*c^2)*e^6*x^6
 + 6*(b^3*c - 4*a*b*c^2)*d*e^5*x^5 + (b^4 - 4*a*b^2*c + 15*(b^3*c - 4*a*b*c^2)*d
^2)*e^4*x^4 + (b^3*c - 4*a*b*c^2)*d^6 + 4*(5*(b^3*c - 4*a*b*c^2)*d^3 + (b^4 - 4*
a*b^2*c)*d)*e^3*x^3 + (b^4 - 4*a*b^2*c)*d^4 + (15*(b^3*c - 4*a*b*c^2)*d^4 + a*b^
3 - 4*a^2*b*c + 6*(b^4 - 4*a*b^2*c)*d^2)*e^2*x^2 + (a*b^3 - 4*a^2*b*c)*d^2 + 2*(
3*(b^3*c - 4*a*b*c^2)*d^5 + 2*(b^4 - 4*a*b^2*c)*d^3 + (a*b^3 - 4*a^2*b*c)*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^3*c - 4*a*b*c^2)*e^6*x^6 + 6*(b^3*c - 4*a*b*c^2)*d*
e^5*x^5 + (b^4 - 4*a*b^2*c + 15*(b^3*c - 4*a*b*c^2)*d^2)*e^4*x^4 + (b^3*c - 4*a*
b*c^2)*d^6 + 4*(5*(b^3*c - 4*a*b*c^2)*d^3 + (b^4 - 4*a*b^2*c)*d)*e^3*x^3 + (b^4
- 4*a*b^2*c)*d^4 + (15*(b^3*c - 4*a*b*c^2)*d^4 + a*b^3 - 4*a^2*b*c + 6*(b^4 - 4*
a*b^2*c)*d^2)*e^2*x^2 + (a*b^3 - 4*a^2*b*c)*d^2 + 2*(3*(b^3*c - 4*a*b*c^2)*d^5 +
 2*(b^4 - 4*a*b^2*c)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e*x)*log(e*x + d))*sqrt(-b^2 +
 4*a*c))/(((a^3*b^2*c - 4*a^4*c^2)*e^7*f^3*x^6 + 6*(a^3*b^2*c - 4*a^4*c^2)*d*e^6
*f^3*x^5 + (a^3*b^3 - 4*a^4*b*c + 15*(a^3*b^2*c - 4*a^4*c^2)*d^2)*e^5*f^3*x^4 +
4*(5*(a^3*b^2*c - 4*a^4*c^2)*d^3 + (a^3*b^3 - 4*a^4*b*c)*d)*e^4*f^3*x^3 + (a^4*b
^2 - 4*a^5*c + 15*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 6*(a^3*b^3 - 4*a^4*b*c)*d^2)*e^3
*f^3*x^2 + 2*(3*(a^3*b^2*c - 4*a^4*c^2)*d^5 + 2*(a^3*b^3 - 4*a^4*b*c)*d^3 + (a^4
*b^2 - 4*a^5*c)*d)*e^2*f^3*x + ((a^3*b^2*c - 4*a^4*c^2)*d^6 + (a^3*b^3 - 4*a^4*b
*c)*d^4 + (a^4*b^2 - 4*a^5*c)*d^2)*e*f^3)*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*f*x+d*f)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{2}{\left (e f x + d f\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^2*(e*f*x + d*f)^3), x)

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