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

Optimal. Leaf size=99 \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0470867, antiderivative size = 99, 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 = {777, 618, 206} \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 777

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

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{x (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=\frac{a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(b d-2 a e) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=\frac{a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 (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 )}{b^2-4 a c}\\ &=\frac{a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a 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 = 0.0850892, size = 99, normalized size = 1. \[ \frac{a b e-2 a c (d+e x)+b x (b e-c d)}{c \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*e + b*(-(c*d) + b*e)*x - 2*a*c*(d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) - (2*(b*d - 2*a*e)*ArcTan[
(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)

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Maple [A]  time = 0.007, size = 147, normalized size = 1.5 \begin{align*}{\frac{1}{c{x}^{2}+bx+a} \left ( -{\frac{ \left ( 2\,ace-{b}^{2}e+bcd \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( be-2\,cd \right ) }{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+4\,{\frac{ae}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{bd}{ \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(x*(e*x+d)/(c*x^2+b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.39855, size = 1108, normalized size = 11.19 \begin{align*} \left [\frac{{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c 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 ) + 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e +{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac{2 \,{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c 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 ) - 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d +{\left (a b^{3} - 4 \, a^{2} b c\right )} e -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((a*b*c*d - 2*a^2*c*e + (b*c^2*d - 2*a*c^2*e)*x^2 + (b^2*c*d - 2*a*b*c*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)) + 2*(a*b^2*c - 4*a^2*c^2)*d - (a*
b^3 - 4*a^2*b*c)*e + ((b^3*c - 4*a*b*c^2)*d - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*e)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 1
6*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x), -(2*(a*b*c*d -
2*a^2*c*e + (b*c^2*d - 2*a*c^2*e)*x^2 + (b^2*c*d - 2*a*b*c*e)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c
)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(a*b^2*c - 4*a^2*c^2)*d + (a*b^3 - 4*a^2*b*c)*e - ((b^3*c - 4*a*b*c^2)*d - (b
^4 - 6*a*b^2*c + 8*a^2*c^2)*e)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)
*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)]

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Sympy [B]  time = 1.61985, size = 379, normalized size = 3.83 \begin{align*} - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} + \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} - \frac{- a b e + 2 a c d + x \left (2 a c e - b^{2} e + b c d\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27882, size = 153, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (b d - 2 \, a 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{b c d x - b^{2} x e + 2 \, a c x e + 2 \, a c d - a b e}{{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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