3.2366 \(\int \frac{(A+B x) (d+e x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]

[Out]

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

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Rubi [A]  time = 0.128048, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {773, 634, 618, 206, 628} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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

Mathematica [A]  time = 0.0900425, size = 108, normalized size = 1. \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x)) (A c e-b B e+B c d)+2 B c e x}{2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 261, normalized size = 2.4 \begin{align*}{\frac{Bex}{c}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bBe}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,c}}+2\,{\frac{Ad}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{aBe}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{Abe}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}Be}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{Bbd}{c}\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((B*x+A)*(e*x+d)/(c*x^2+b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40404, size = 805, normalized size = 7.45 \begin{align*} \left [\frac{2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} e x + \sqrt{b^{2} - 4 \, a c}{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} 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 ) +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} d -{\left (B b^{3} + 4 \, A a c^{2} -{\left (4 \, B a b + A b^{2}\right )} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac{2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} e x + 2 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} d -{\left (B b^{3} + 4 \, A a c^{2} -{\left (4 \, B a b + A b^{2}\right )} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.53529, size = 677, normalized size = 6.27 \begin{align*} \frac{B e x}{c} + \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15821, size = 151, normalized size = 1.4 \begin{align*} \frac{B x e}{c} + \frac{{\left (B c d - B b e + A c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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