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

Optimal. Leaf size=97 \[ \frac{2 a+b (d+e x)^2}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.269304, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 a+b (d+e x)^2}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.0064, size = 82, normalized size = 0.85 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{2 a + b \left (d + e x\right )^{2}}{2 e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.228051, size = 100, normalized size = 1.03 \[ \frac{\frac{2 a+b (d+e x)^2}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac{2 b \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.031, size = 276, normalized size = 2.9 \[{\frac{1}{c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a} \left ( -{\frac{{x}^{2}be}{8\,ac-2\,{b}^{2}}}-{\frac{bdx}{4\,ac-{b}^{2}}}-{\frac{b{d}^{2}+2\,a}{2\,e \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{b}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -e{\it \_R}-d \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1/2*b*e/(4*a*c-b^2)*x^2-b*d/(4*a*c-b^2)*x-1/2/e*(b*d^2+2*a)/(4*a*c-b^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)+1/2*b/e*sum((-_R*e-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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -b \int -\frac{e x + d}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{4} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} x^{3} +{\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{2} x^{2} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e x}\,{d x} + \frac{b e^{2} x^{2} + 2 \, b d e x + b d^{2} + 2 \, a}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{5} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{4} x^{3} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{3} x^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2} x +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.318688, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((b*c*e^4*x^4 + 4*b*c*d*e^3*x^3 + b*c*d^4 + (6*b*c*d^2 + b^2)*e^2*x^2 + b^
2*d^2 + 2*(2*b*c*d^3 + b^2*d)*e*x + a*b)*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)) - (b*e^2
*x^2 + 2*b*d*e*x + b*d^2 + 2*a)*sqrt(b^2 - 4*a*c))/(((b^2*c - 4*a*c^2)*e^5*x^4 +
4*(b^2*c - 4*a*c^2)*d*e^4*x^3 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^3*x
^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e^2*x + ((b^2*c - 4*a*c^2)*
d^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*d^2)*e)*sqrt(b^2 - 4*a*c)), 1/2*(2*(b*c*
e^4*x^4 + 4*b*c*d*e^3*x^3 + b*c*d^4 + (6*b*c*d^2 + b^2)*e^2*x^2 + b^2*d^2 + 2*(2
*b*c*d^3 + b^2*d)*e*x + a*b)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqr
t(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (b*e^2*x^2 + 2*b*d*e*x + b*d^2 + 2*a)*sqrt(-b^2
+ 4*a*c))/(((b^2*c - 4*a*c^2)*e^5*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^4*x^3 + (b^3 -
4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^3*x^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 -
4*a*b*c)*d)*e^2*x + ((b^2*c - 4*a*c^2)*d^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*
d^2)*e)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 164.745, size = 493, normalized size = 5.08 \[ \frac{b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2} + 2 b c d^{2}}{2 b c e^{2}} \right )}}{2 e} - \frac{b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2} + 2 b c d^{2}}{2 b c e^{2}} \right )}}{2 e} - \frac{2 a + b d^{2} + 2 b d e x + b e^{2} x^{2}}{8 a^{2} c e - 2 a b^{2} e + 8 a b c d^{2} e + 8 a c^{2} d^{4} e - 2 b^{3} d^{2} e - 2 b^{2} c d^{4} e + x^{4} \left (8 a c^{2} e^{5} - 2 b^{2} c e^{5}\right ) + x^{3} \left (32 a c^{2} d e^{4} - 8 b^{2} c d e^{4}\right ) + x^{2} \left (8 a b c e^{3} + 48 a c^{2} d^{2} e^{3} - 2 b^{3} e^{3} - 12 b^{2} c d^{2} e^{3}\right ) + x \left (16 a b c d e^{2} + 32 a c^{2} d^{3} e^{2} - 4 b^{3} d e^{2} - 8 b^{2} c d^{3} e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*sqrt(-1/(4*a*c - b**2)**3)*log(2*d*x/e + x**2 + (-16*a**2*b*c**2*sqrt(-1/(4*a*
c - b**2)**3) + 8*a*b**3*c*sqrt(-1/(4*a*c - b**2)**3) - b**5*sqrt(-1/(4*a*c - b*
*2)**3) + b**2 + 2*b*c*d**2)/(2*b*c*e**2))/(2*e) - b*sqrt(-1/(4*a*c - b**2)**3)*
log(2*d*x/e + x**2 + (16*a**2*b*c**2*sqrt(-1/(4*a*c - b**2)**3) - 8*a*b**3*c*sqr
t(-1/(4*a*c - b**2)**3) + b**5*sqrt(-1/(4*a*c - b**2)**3) + b**2 + 2*b*c*d**2)/(
2*b*c*e**2))/(2*e) - (2*a + b*d**2 + 2*b*d*e*x + b*e**2*x**2)/(8*a**2*c*e - 2*a*
b**2*e + 8*a*b*c*d**2*e + 8*a*c**2*d**4*e - 2*b**3*d**2*e - 2*b**2*c*d**4*e + x*
*4*(8*a*c**2*e**5 - 2*b**2*c*e**5) + x**3*(32*a*c**2*d*e**4 - 8*b**2*c*d*e**4) +
x**2*(8*a*b*c*e**3 + 48*a*c**2*d**2*e**3 - 2*b**3*e**3 - 12*b**2*c*d**2*e**3) +
x*(16*a*b*c*d*e**2 + 32*a*c**2*d**3*e**2 - 4*b**3*d*e**2 - 8*b**2*c*d**3*e**2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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