3.149 \(\int \frac{B+A x}{(c+2 b x+a x^2)^2} \, dx\)

Optimal. Leaf size=90 \[ -\frac{-x (A b-a B)-A c+b B}{2 \left (b^2-a c\right ) \left (a x^2+2 b x+c\right )}-\frac{(A b-a B) \tanh ^{-1}\left (\frac{a x+b}{\sqrt{b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \]

[Out]

-(b*B - A*c - (A*b - a*B)*x)/(2*(b^2 - a*c)*(c + 2*b*x + a*x^2)) - ((A*b - a*B)*ArcTanh[(b + a*x)/Sqrt[b^2 - a
*c]])/(2*(b^2 - a*c)^(3/2))

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Rubi [A]  time = 0.0709111, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {638, 618, 206} \[ -\frac{-x (A b-a B)-A c+b B}{2 \left (b^2-a c\right ) \left (a x^2+2 b x+c\right )}-\frac{(A b-a B) \tanh ^{-1}\left (\frac{a x+b}{\sqrt{b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(B + A*x)/(c + 2*b*x + a*x^2)^2,x]

[Out]

-(b*B - A*c - (A*b - a*B)*x)/(2*(b^2 - a*c)*(c + 2*b*x + a*x^2)) - ((A*b - a*B)*ArcTanh[(b + a*x)/Sqrt[b^2 - a
*c]])/(2*(b^2 - 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{B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx &=-\frac{b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}+\frac{(A b-a B) \int \frac{1}{c+2 b x+a x^2} \, dx}{2 \left (b^2-a c\right )}\\ &=-\frac{b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}-\frac{(A b-a B) \operatorname{Subst}\left (\int \frac{1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 a x\right )}{b^2-a c}\\ &=-\frac{b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}-\frac{(A b-a B) \tanh ^{-1}\left (\frac{b+a x}{\sqrt{b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0867668, size = 88, normalized size = 0.98 \[ \frac{\frac{(A b-a B) \tan ^{-1}\left (\frac{a x+b}{\sqrt{a c-b^2}}\right )}{\sqrt{a c-b^2}}+\frac{-a B x+A b x+A c-b B}{x (a x+2 b)+c}}{2 \left (b^2-a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(B + A*x)/(c + 2*b*x + a*x^2)^2,x]

[Out]

((-(b*B) + A*c + A*b*x - a*B*x)/(c + x*(2*b + a*x)) + ((A*b - a*B)*ArcTan[(b + a*x)/Sqrt[-b^2 + a*c]])/Sqrt[-b
^2 + a*c])/(2*(b^2 - a*c))

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Maple [A]  time = 0.007, size = 146, normalized size = 1.6 \begin{align*}{\frac{ \left ( -2\,Ab+2\,Ba \right ) x+2\,bB-2\,Ac}{ \left ( 4\,ac-4\,{b}^{2} \right ) \left ( a{x}^{2}+2\,bx+c \right ) }}-2\,{\frac{Ab}{ \left ( 4\,ac-4\,{b}^{2} \right ) \sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,ax+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) }+2\,{\frac{Ba}{ \left ( 4\,ac-4\,{b}^{2} \right ) \sqrt{ac-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,ax+2\,b}{\sqrt{ac-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A*x+B)/(a*x^2+2*b*x+c)^2,x)

[Out]

((-2*A*b+2*B*a)*x+2*b*B-2*A*c)/(4*a*c-4*b^2)/(a*x^2+2*b*x+c)-2/(4*a*c-4*b^2)/(a*c-b^2)^(1/2)*arctan(1/2*(2*a*x
+2*b)/(a*c-b^2)^(1/2))*A*b+2/(4*a*c-4*b^2)/(a*c-b^2)^(1/2)*arctan(1/2*(2*a*x+2*b)/(a*c-b^2)^(1/2))*B*a

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.98165, size = 932, normalized size = 10.36 \begin{align*} \left [-\frac{2 \, B b^{3} + 2 \, A a c^{2} -{\left ({\left (B a^{2} - A a b\right )} x^{2} +{\left (B a - A b\right )} c + 2 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{b^{2} - a c} \log \left (\frac{a^{2} x^{2} + 2 \, a b x + 2 \, b^{2} - a c + 2 \, \sqrt{b^{2} - a c}{\left (a x + b\right )}}{a x^{2} + 2 \, b x + c}\right ) - 2 \,{\left (B a b + A b^{2}\right )} c + 2 \,{\left (B a b^{2} - A b^{3} -{\left (B a^{2} - A a b\right )} c\right )} x}{4 \,{\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3} +{\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2}\right )} x^{2} + 2 \,{\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}, -\frac{B b^{3} + A a c^{2} -{\left ({\left (B a^{2} - A a b\right )} x^{2} +{\left (B a - A b\right )} c + 2 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{-b^{2} + a c} \arctan \left (-\frac{\sqrt{-b^{2} + a c}{\left (a x + b\right )}}{b^{2} - a c}\right ) -{\left (B a b + A b^{2}\right )} c +{\left (B a b^{2} - A b^{3} -{\left (B a^{2} - A a b\right )} c\right )} x}{2 \,{\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3} +{\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2}\right )} x^{2} + 2 \,{\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*B*b^3 + 2*A*a*c^2 - ((B*a^2 - A*a*b)*x^2 + (B*a - A*b)*c + 2*(B*a*b - A*b^2)*x)*sqrt(b^2 - a*c)*log((
a^2*x^2 + 2*a*b*x + 2*b^2 - a*c + 2*sqrt(b^2 - a*c)*(a*x + b))/(a*x^2 + 2*b*x + c)) - 2*(B*a*b + A*b^2)*c + 2*
(B*a*b^2 - A*b^3 - (B*a^2 - A*a*b)*c)*x)/(b^4*c - 2*a*b^2*c^2 + a^2*c^3 + (a*b^4 - 2*a^2*b^2*c + a^3*c^2)*x^2
+ 2*(b^5 - 2*a*b^3*c + a^2*b*c^2)*x), -1/2*(B*b^3 + A*a*c^2 - ((B*a^2 - A*a*b)*x^2 + (B*a - A*b)*c + 2*(B*a*b
- A*b^2)*x)*sqrt(-b^2 + a*c)*arctan(-sqrt(-b^2 + a*c)*(a*x + b)/(b^2 - a*c)) - (B*a*b + A*b^2)*c + (B*a*b^2 -
A*b^3 - (B*a^2 - A*a*b)*c)*x)/(b^4*c - 2*a*b^2*c^2 + a^2*c^3 + (a*b^4 - 2*a^2*b^2*c + a^3*c^2)*x^2 + 2*(b^5 -
2*a*b^3*c + a^2*b*c^2)*x)]

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Sympy [B]  time = 1.11121, size = 323, normalized size = 3.59 \begin{align*} - \frac{\sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) \log{\left (x + \frac{- A b^{2} + B a b - a^{2} c^{2} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) + 2 a b^{2} c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) - b^{4} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right )}{- A a b + B a^{2}} \right )}}{4} + \frac{\sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) \log{\left (x + \frac{- A b^{2} + B a b + a^{2} c^{2} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) - 2 a b^{2} c \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) + b^{4} \sqrt{- \frac{1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right )}{- A a b + B a^{2}} \right )}}{4} + \frac{- A c + B b + x \left (- A b + B a\right )}{2 a c^{2} - 2 b^{2} c + x^{2} \left (2 a^{2} c - 2 a b^{2}\right ) + x \left (4 a b c - 4 b^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A*x+B)/(a*x**2+2*b*x+c)**2,x)

[Out]

-sqrt(-1/(a*c - b**2)**3)*(-A*b + B*a)*log(x + (-A*b**2 + B*a*b - a**2*c**2*sqrt(-1/(a*c - b**2)**3)*(-A*b + B
*a) + 2*a*b**2*c*sqrt(-1/(a*c - b**2)**3)*(-A*b + B*a) - b**4*sqrt(-1/(a*c - b**2)**3)*(-A*b + B*a))/(-A*a*b +
B*a**2))/4 + sqrt(-1/(a*c - b**2)**3)*(-A*b + B*a)*log(x + (-A*b**2 + B*a*b + a**2*c**2*sqrt(-1/(a*c - b**2)*
*3)*(-A*b + B*a) - 2*a*b**2*c*sqrt(-1/(a*c - b**2)**3)*(-A*b + B*a) + b**4*sqrt(-1/(a*c - b**2)**3)*(-A*b + B*
a))/(-A*a*b + B*a**2))/4 + (-A*c + B*b + x*(-A*b + B*a))/(2*a*c**2 - 2*b**2*c + x**2*(2*a**2*c - 2*a*b**2) + x
*(4*a*b*c - 4*b**3))

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Giac [A]  time = 1.06969, size = 124, normalized size = 1.38 \begin{align*} -\frac{{\left (B a - A b\right )} \arctan \left (\frac{a x + b}{\sqrt{-b^{2} + a c}}\right )}{2 \,{\left (b^{2} - a c\right )} \sqrt{-b^{2} + a c}} - \frac{B a x - A b x + B b - A c}{2 \,{\left (a x^{2} + 2 \, b x + c\right )}{\left (b^{2} - a c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*a - A*b)*arctan((a*x + b)/sqrt(-b^2 + a*c))/((b^2 - a*c)*sqrt(-b^2 + a*c)) - 1/2*(B*a*x - A*b*x + B*b
- A*c)/((a*x^2 + 2*b*x + c)*(b^2 - a*c))