### 3.2194 $$\int \frac{d+e x}{(a+b x+c x^2)^2} \, dx$$

Optimal. Leaf size=87 $\frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0383568, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {638, 618, 206} $\frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(a + b*x + c*x^2)^2,x]

[Out]

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

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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])

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 c d-b e) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(2 (2 c d-b e)) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0823219, size = 88, normalized size = 1.01 $\frac{\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 a e-b d+b e x-2 c d x}{a+x (b+c x)}}{b^2-4 a c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.154, size = 118, normalized size = 1.4 \begin{align*}{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}-2\,{\frac{be}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{cd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d-2*a*e+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)-2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b
*e+4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*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)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90407, size = 988, normalized size = 11.36 \begin{align*} \left [\frac{{\left (2 \, a c d - a b e +{\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt{b^{2} - 4 \, a c} \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 (b^{3} - 4 \, a b c\right )} d + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e -{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, \frac{2 \,{\left (2 \, a c d - a b e +{\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (b^{3} - 4 \, a b c\right )} d + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e -{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((2*a*c*d - a*b*e + (2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b^2*e)*x)*sqrt(b^2 - 4*a*c)*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)) - (b^3 - 4*a*b*c)*d + 2*(a*b^2 - 4*a^2*c)*e -
(2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*x)/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*
a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x), (2*(2*a*c*d - a*b*e + (2*c^2*d - b*c*e)*x^2 + (2*b*c*d - b
^2*e)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*d + 2*(a*b
^2 - 4*a^2*c)*e - (2*(b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*x)/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c -
8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x)]

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Sympy [B]  time = 1.10025, size = 359, normalized size = 4.13 \begin{align*} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \frac{2 a e - b d + x \left (b e - 2 c d\right )}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x + (-16*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + 8*a
*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) - b**4*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**2*e - 2*
b*c*d)/(2*b*c*e - 4*c**2*d)) - sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x + (16*a**2*c**2*sqrt(-1/(4*a*c -
b**2)**3)*(b*e - 2*c*d) - 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**4*sqrt(-1/(4*a*c - b**2)**
3)*(b*e - 2*c*d) + b**2*e - 2*b*c*d)/(2*b*c*e - 4*c**2*d)) - (2*a*e - b*d + x*(b*e - 2*c*d))/(4*a**2*c - a*b**
2 + x**2*(4*a*c**2 - b**2*c) + x*(4*a*b*c - b**3))

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Giac [A]  time = 1.14903, size = 134, normalized size = 1.54 \begin{align*} -\frac{2 \,{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c d x - b x e + b d - 2 \, a e}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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