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

Optimal. Leaf size=81 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 c e \sqrt{b^2-4 a c}}+\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]

[Out]

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

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Rubi [A]  time = 0.26502, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 c e \sqrt{b^2-4 a c}}+\frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.1254, size = 68, normalized size = 0.84 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c e \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 c e} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0693669, size = 77, normalized size = 0.95 \[ \frac{\log \left (a+b (d+e x)^2+c (d+e x)^4\right )-\frac{2 b \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{4 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.007, size = 151, normalized size = 1.9 \[{\frac{1}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ({{\it \_R}}^{3}{e}^{3}+3\,{{\it \_R}}^{2}d{e}^{2}+3\,{\it \_R}\,{d}^{2}e+{d}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{d}^{2}ec{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \]

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),x)

[Out]

1/2/e*sum((_R^3*e^3+3*_R^2*d*e^2+3*_R*d^2*e+d^3)/(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. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.273404, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e x + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} +{\left (2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right ) + \sqrt{b^{2} - 4 \, a c} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, \sqrt{b^{2} - 4 \, a c} c e}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c e}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/4*(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)) + sqrt(b^2 - 4*a*c)*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))/(
sqrt(b^2 - 4*a*c)*c*e), -1/4*(2*b*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)) - sqrt(-b^2 + 4*a*c)*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))/(sq
rt(-b^2 + 4*a*c)*c*e)]

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Sympy [A]  time = 6.71315, size = 280, normalized size = 3.46 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a c e \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 8 a c e \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac{1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} \]

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),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e))*log(2*d*x/e + x**2 +
(-8*a*c*e*(-b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e)) + 2*a + 2
*b**2*e*(-b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e)) + b*d**2)/(b
*e**2)) + (b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e))*log(2*d*x/e
+ x**2 + (-8*a*c*e*(b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e)) +
2*a + 2*b**2*e*(b*sqrt(-4*a*c + b**2)/(4*c*e*(4*a*c - b**2)) + 1/(4*c*e)) + b*d
**2)/(b*e**2))

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

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),x, algorithm="giac")

[Out]

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