3.886 \(\int \frac{d+e x}{x (a+b x+c x^2)} \, dx\)
Optimal. Leaf size=71 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b x+c x^2\right )}{2 a}+\frac{d \log (x)}{a} \]
[Out]
((b*d - 2*a*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + (d*Log[x])/a - (d*Log[a + b*x +
c*x^2])/(2*a)
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Rubi [A] time = 0.0942628, antiderivative size = 71, 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, Rules used =
{800, 634, 618, 206, 628} \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{d \log \left (a+b x+c x^2\right )}{2 a}+\frac{d \log (x)}{a} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)/(x*(a + b*x + c*x^2)),x]
[Out]
((b*d - 2*a*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + (d*Log[x])/a - (d*Log[a + b*x +
c*x^2])/(2*a)
Rule 800
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rubi steps
\begin{align*} \int \frac{d+e x}{x \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{d}{a x}+\frac{-b d+a e-c d x}{a \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{d \log (x)}{a}+\frac{\int \frac{-b d+a e-c d x}{a+b x+c x^2} \, dx}{a}\\ &=\frac{d \log (x)}{a}-\frac{d \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a}+\frac{(-b d+2 a e) \int \frac{1}{a+b x+c x^2} \, dx}{2 a}\\ &=\frac{d \log (x)}{a}-\frac{d \log \left (a+b x+c x^2\right )}{2 a}-\frac{(-b d+2 a e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a}\\ &=\frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}+\frac{d \log (x)}{a}-\frac{d \log \left (a+b x+c x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.108707, size = 71, normalized size = 1. \[ -\frac{\frac{2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+d (\log (a+x (b+c x))-2 \log (x))}{2 a} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)/(x*(a + b*x + c*x^2)),x]
[Out]
-((2*(b*d - 2*a*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + d*(-2*Log[x] + Log[a + x*(b +
c*x)]))/(2*a)
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Maple [A] time = 0.006, size = 100, normalized size = 1.4 \begin{align*}{\frac{d\ln \left ( x \right ) }{a}}-{\frac{d\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a}}+2\,{\frac{e}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{bd}{a}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)/x/(c*x^2+b*x+a),x)
[Out]
d*ln(x)/a-1/2*d*ln(c*x^2+b*x+a)/a+2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*e-1/a/(4*a*c-b^2)^(1
/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x/(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.79476, size = 540, normalized size = 7.61 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (x\right ) + \sqrt{b^{2} - 4 \, a c}{\left (b d - 2 \, a e\right )} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}, -\frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (x\right ) - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (b d - 2 \, a e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x/(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
[-1/2*((b^2 - 4*a*c)*d*log(c*x^2 + b*x + a) - 2*(b^2 - 4*a*c)*d*log(x) + sqrt(b^2 - 4*a*c)*(b*d - 2*a*e)*log((
2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)))/(a*b^2 - 4*a^2*c), -1/2
*((b^2 - 4*a*c)*d*log(c*x^2 + b*x + a) - 2*(b^2 - 4*a*c)*d*log(x) - 2*sqrt(-b^2 + 4*a*c)*(b*d - 2*a*e)*arctan(
-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(a*b^2 - 4*a^2*c)]
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Sympy [B] time = 58.8523, size = 1498, normalized size = 21.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x/(c*x**2+b*x+a),x)
[Out]
(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))*log(x + (-4*a**4*b*c*e*(-d/(2*a) - sqrt(-4
*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 - 24*a**4*c**2*d*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e -
b*d)/(2*a*(4*a*c - b**2)))**2 + 4*a**4*c*e**2*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2
))) + a**3*b**3*e*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 + 14*a**3*b**2*c*d*(-
d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 - a**3*b**2*e**2*(-d/(2*a) - sqrt(-4*a*c
+ b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) - 8*a**3*b*c*d*e*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*
a*(4*a*c - b**2))) + 12*a**3*c**2*d**2*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) - 4
*a**3*c*d*e**2 - 2*a**2*b**4*d*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 + 2*a**2
*b**3*d*e*(-d/(2*a) - sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) - 3*a**2*b**2*c*d**2*(-d/(2*a) -
sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) + a**2*b**2*d*e**2 + 12*a**2*b*c*d**2*e + 12*a**2*c**
2*d**3 - 3*a*b**3*d**2*e - 11*a*b**2*c*d**3 + 2*b**4*d**3)/(2*a**3*c*e**3 - 3*a**2*b*c*d*e**2 + 18*a**2*c**2*d
**2*e - 3*a*b**2*c*d**2*e - 9*a*b*c**2*d**3 + 2*b**3*c*d**3)) + (-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/
(2*a*(4*a*c - b**2)))*log(x + (-4*a**4*b*c*e*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)
))**2 - 24*a**4*c**2*d*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 + 4*a**4*c*e**2*
(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) + a**3*b**3*e*(-d/(2*a) + sqrt(-4*a*c + b*
*2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 + 14*a**3*b**2*c*d*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2
*a*(4*a*c - b**2)))**2 - a**3*b**2*e**2*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) -
8*a**3*b*c*d*e*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) + 12*a**3*c**2*d**2*(-d/(2*
a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2))) - 4*a**3*c*d*e**2 - 2*a**2*b**4*d*(-d/(2*a) + sqr
t(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c - b**2)))**2 + 2*a**2*b**3*d*e*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a
*e - b*d)/(2*a*(4*a*c - b**2))) - 3*a**2*b**2*c*d**2*(-d/(2*a) + sqrt(-4*a*c + b**2)*(2*a*e - b*d)/(2*a*(4*a*c
- b**2))) + a**2*b**2*d*e**2 + 12*a**2*b*c*d**2*e + 12*a**2*c**2*d**3 - 3*a*b**3*d**2*e - 11*a*b**2*c*d**3 +
2*b**4*d**3)/(2*a**3*c*e**3 - 3*a**2*b*c*d*e**2 + 18*a**2*c**2*d**2*e - 3*a*b**2*c*d**2*e - 9*a*b*c**2*d**3 +
2*b**3*c*d**3)) + d*log(x)/a
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Giac [A] time = 1.31705, size = 97, normalized size = 1.37 \begin{align*} -\frac{d \log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac{d \log \left ({\left | x \right |}\right )}{a} - \frac{{\left (b d - 2 \, a e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x/(c*x^2+b*x+a),x, algorithm="giac")
[Out]
-1/2*d*log(c*x^2 + b*x + a)/a + d*log(abs(x))/a - (b*d - 2*a*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-
b^2 + 4*a*c)*a)