3.1529 \(\int \frac{b+2 c x}{(d+e x) (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=130 \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]

[Out]

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

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Rubi [A]  time = 0.175605, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {800, 634, 618, 206, 628} \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{\int \frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{a+b x+c x^2} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}-\frac{\left (\left (b^2-4 a c\right ) e\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}+\frac{(2 c d-b e) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (\left (b^2-4 a c\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c d^2-b d e+a e^2}\\ &=\frac{\sqrt{b^2-4 a c} e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c d^2-b d e+a e^2}-\frac{(2 c d-b e) \log (d+e x)}{c d^2-b d e+a e^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.1241, size = 116, normalized size = 0.89 \[ \frac{\sqrt{4 a c-b^2} (2 c d-b e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 e \left (b^2-4 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 233, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( ex+d \right ) be}{a{e}^{2}-bde+c{d}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) cd}{a{e}^{2}-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) eb}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) d}{a{e}^{2}-bde+c{d}^{2}}}+4\,{\frac{ace}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{2}e}{a{e}^{2}-bde+c{d}^{2}}\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((2*c*x+b)/(e*x+d)/(c*x^2+b*x+a),x)

[Out]

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

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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((2*c*x+b)/(e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.27525, size = 527, normalized size = 4.05 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} e \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 ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} e \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 167.156, size = 3312, normalized size = 25.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e - 2*c*d)*log(x + (-2*a**2*b*e**5*(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 + 4*a**2*c*d*e**4*(b*e - 2
*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 + a**2*c*e**4*(b*e - 2*c*d)/(a*e**2 - b*d*e + c*d**2) + 4*a*b**2*d*e**4*
(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 - 12*a*b*c*d**2*e**3*(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)*
*2 - 2*a*b*c*d*e**3*(b*e - 2*c*d)/(a*e**2 - b*d*e + c*d**2) - a*b*c*e**3 + 8*a*c**2*d**3*e**2*(b*e - 2*c*d)**2
/(a*e**2 - b*d*e + c*d**2)**2 + 2*a*c**2*d**2*e**2*(b*e - 2*c*d)/(a*e**2 - b*d*e + c*d**2) + 3*a*c**2*d*e**2 -
2*b**3*d**2*e**3*(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 + 2*b**3*e**3 + 8*b**2*c*d**3*e**2*(b*e - 2*c*
d)**2/(a*e**2 - b*d*e + c*d**2)**2 + b**2*c*d**2*e**2*(b*e - 2*c*d)/(a*e**2 - b*d*e + c*d**2) - 9*b**2*c*d*e**
2 - 10*b*c**2*d**4*e*(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 - 2*b*c**2*d**3*e*(b*e - 2*c*d)/(a*e**2 - b
*d*e + c*d**2) + 12*b*c**2*d**2*e + 4*c**3*d**5*(b*e - 2*c*d)**2/(a*e**2 - b*d*e + c*d**2)**2 + c**3*d**4*(b*e
- 2*c*d)/(a*e**2 - b*d*e + c*d**2) - 5*c**3*d**3)/(a*c**2*e**3 + 2*b**2*c*e**3 - 9*b*c**2*d*e**2 + 9*c**3*d**
2*e))/(a*e**2 - b*d*e + c*d**2) + (-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*
e**2 - b*d*e + c*d**2)))*log(x + (-2*a**2*b*e**5*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e
- 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + 4*a**2*c*d*e**4*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d*
*2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + a**2*c*e**4*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*
e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) + 4*a*b**2*d*e**4*(-e*sqrt(-4*a*c + b**2)/(2*(a*e*
*2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 - 12*a*b*c*d**2*e**3*(-e*sqrt(-4*a*c +
b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 - 2*a*b*c*d*e**3*(-e*sq
rt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) - a*b*c*e**3 +
8*a*c**2*d**3*e**2*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e +
c*d**2)))**2 + 2*a*c**2*d**2*e**2*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*
e**2 - b*d*e + c*d**2))) + 3*a*c**2*d*e**2 - 2*b**3*d**2*e**3*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d
**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + 2*b**3*e**3 + 8*b**2*c*d**3*e**2*(-e*sqrt(-4*a*c + b
**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + b**2*c*d**2*e**2*(-e*sq
rt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) - 9*b**2*c*d*e*
*2 - 10*b*c**2*d**4*e*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e
+ c*d**2)))**2 - 2*b*c**2*d**3*e*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*
e**2 - b*d*e + c*d**2))) + 12*b*c**2*d**2*e + 4*c**3*d**5*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)
) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + c**3*d**4*(-e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c
*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) - 5*c**3*d**3)/(a*c**2*e**3 + 2*b**2*c*e**3 - 9*b*c**2*
d*e**2 + 9*c**3*d**2*e)) + (e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b
*d*e + c*d**2)))*log(x + (-2*a**2*b*e**5*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/
(2*(a*e**2 - b*d*e + c*d**2)))**2 + 4*a**2*c*d*e**4*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*
e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + a**2*c*e**4*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)
) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) + 4*a*b**2*d*e**4*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e +
c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 - 12*a*b*c*d**2*e**3*(e*sqrt(-4*a*c + b**2)/(2*(a*
e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 - 2*a*b*c*d*e**3*(e*sqrt(-4*a*c + b*
*2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) - a*b*c*e**3 + 8*a*c**2*d**3*
e**2*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 +
2*a*c**2*d**2*e**2*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c
*d**2))) + 3*a*c**2*d*e**2 - 2*b**3*d**2*e**3*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*
c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + 2*b**3*e**3 + 8*b**2*c*d**3*e**2*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 -
b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 + b**2*c*d**2*e**2*(e*sqrt(-4*a*c + b**2)/(
2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2))) - 9*b**2*c*d*e**2 - 10*b*c**2*d**4
*e*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2)))**2 - 2*
b*c**2*d**3*e*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(a*e**2 - b*d*e + c*d**2
))) + 12*b*c**2*d**2*e + 4*c**3*d**5*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)/(2*(
a*e**2 - b*d*e + c*d**2)))**2 + c**3*d**4*(e*sqrt(-4*a*c + b**2)/(2*(a*e**2 - b*d*e + c*d**2)) - (b*e - 2*c*d)
/(2*(a*e**2 - b*d*e + c*d**2))) - 5*c**3*d**3)/(a*c**2*e**3 + 2*b**2*c*e**3 - 9*b*c**2*d*e**2 + 9*c**3*d**2*e)
)

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Giac [A]  time = 1.13831, size = 201, normalized size = 1.55 \begin{align*} \frac{{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (2 \, c d e - b e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (b^{2} e - 4 \, a c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*c*d - b*e)*log(c*x^2 + b*x + a)/(c*d^2 - b*d*e + a*e^2) - (2*c*d*e - b*e^2)*log(abs(x*e + d))/(c*d^2*e
- b*d*e^2 + a*e^3) - (b^2*e - 4*a*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c*d^2 - b*d*e + a*e^2)*sqrt(-b
^2 + 4*a*c))