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

Optimal. Leaf size=145 \[ -\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (-a b e-a c d+b^2 d\right )}{a^3}+\frac{b d-a e}{a^2 x}+\frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}-\frac{d}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.440851, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (-a b e-a c d+b^2 d\right )}{a^3}+\frac{b d-a e}{a^2 x}+\frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}-\frac{d}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.0405, size = 138, normalized size = 0.95 \[ - \frac{d}{2 a x^{2}} - \frac{a e - b d}{a^{2} x} + \frac{\left (- a b e - a c d + b^{2} d\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (- a b e - a c d + b^{2} d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{3}} + \frac{\left (2 a^{2} c e - a b^{2} e - 3 a b c d + b^{3} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.245929, size = 141, normalized size = 0.97 \[ \frac{\frac{2 \left (-2 a^2 c e+a b^2 e+3 a b c d+b^3 (-d)\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{a^2 d}{x^2}+2 \log (x) \left (-a b e-a c d+b^2 d\right )+\left (a b e+a c d+b^2 (-d)\right ) \log (a+x (b+c x))+\frac{2 a (b d-a e)}{x}}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 273, normalized size = 1.9 \[ -{\frac{d}{2\,a{x}^{2}}}-{\frac{e}{ax}}+{\frac{bd}{{a}^{2}x}}-{\frac{\ln \left ( x \right ) be}{{a}^{2}}}-{\frac{cd\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) be}{2\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) d}{2\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d}{2\,{a}^{3}}}-2\,{\frac{ce}{a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}e}{{a}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{bcd}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{d{b}^{3}}{{a}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d/a/x^2-e/a/x+1/a^2/x*b*d-1/a^2*ln(x)*b*e-c*d*ln(x)/a^2+1/a^3*ln(x)*b^2*d+1
/2/a^2*ln(c*x^2+b*x+a)*b*e+1/2/a^2*c*ln(c*x^2+b*x+a)*d-1/2/a^3*ln(c*x^2+b*x+a)*b
^2*d-2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*e+1/a^2/(4*a*c-
b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*e*b^2+3/a^2/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*b*d-1/a^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*d*b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.683915, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left ({\left (a b e -{\left (b^{2} - a c\right )} d\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (a b e -{\left (b^{2} - a c\right )} d\right )} x^{2} \log \left (x\right ) - a^{2} d + 2 \,{\left (a b d - a^{2} e\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} a^{3} x^{2}}, -\frac{2 \,{\left ({\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left ({\left (a b e -{\left (b^{2} - a c\right )} d\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (a b e -{\left (b^{2} - a c\right )} d\right )} x^{2} \log \left (x\right ) - a^{2} d + 2 \,{\left (a b d - a^{2} e\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e)*x^2*log((b^3 - 4*a*b*c + 2*(b^2*
c - 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 +
b*x + a)) + ((a*b*e - (b^2 - a*c)*d)*x^2*log(c*x^2 + b*x + a) - 2*(a*b*e - (b^2
- a*c)*d)*x^2*log(x) - a^2*d + 2*(a*b*d - a^2*e)*x)*sqrt(b^2 - 4*a*c))/(sqrt(b^
2 - 4*a*c)*a^3*x^2), -1/2*(2*((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e)*x^2*arcta
n(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - ((a*b*e - (b^2 - a*c)*d)*x^2*
log(c*x^2 + b*x + a) - 2*(a*b*e - (b^2 - a*c)*d)*x^2*log(x) - a^2*d + 2*(a*b*d -
a^2*e)*x)*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3*x^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269292, size = 205, normalized size = 1.41 \[ -\frac{{\left (b^{2} d - a c d - a b e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac{{\left (b^{2} d - a c d - a b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} d - 3 \, a b c d - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{a^{2} d - 2 \,{\left (a b d - a^{2} e\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^2*d - a*c*d - a*b*e)*ln(c*x^2 + b*x + a)/a^3 + (b^2*d - a*c*d - a*b*e)*l
n(abs(x))/a^3 - (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*arctan((2*c*x + b)/sqr
t(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) - 1/2*(a^2*d - 2*(a*b*d - a^2*e)*x)/(a
^3*x^2)