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

Optimal. Leaf size=121 \[ -\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{x (c d-b e)}{c^2}+\frac{e x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.148198, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {800, 634, 618, 206, 628} \[ -\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{x (c d-b e)}{c^2}+\frac{e x^2}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0787626, size = 119, normalized size = 0.98 \[ \frac{\frac{2 \left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\left (-a c e+b^2 e-b c d\right ) \log (a+x (b+c x))+2 c x (c d-b e)+c^2 e x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 241, normalized size = 2. \begin{align*}{\frac{e{x}^{2}}{2\,c}}-{\frac{bex}{{c}^{2}}}+{\frac{dx}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ae}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}e}{2\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bd}{2\,{c}^{2}}}+3\,{\frac{abe}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{ad}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}e}{{c}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{{c}^{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(x^2*(e*x+d)/(c*x^2+b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56432, size = 880, normalized size = 7.27 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{2} + \sqrt{b^{2} - 4 \, a c}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b 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 ) + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d -{\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{2} - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b 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 ) + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d -{\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} x -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((b^2*c^2 - 4*a*c^3)*e*x^2 + sqrt(b^2 - 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*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)) + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c
- 4*a*b*c^2)*e)*x - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^2 + b*x + a))/(b^2*c^3
- 4*a*c^4), 1/2*((b^2*c^2 - 4*a*c^3)*e*x^2 - 2*sqrt(-b^2 + 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*ar
ctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*((b^2*c^2 - 4*a*c^3)*d - (b^3*c - 4*a*b*c^2)*e)*x - ((
b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^2 + b*x + a))/(b^2*c^3 - 4*a*c^4)]

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Sympy [B]  time = 2.28383, size = 609, normalized size = 5.03 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right ) \log{\left (x + \frac{2 a^{2} c e - a b^{2} e + a b c d + 4 a c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right ) - b^{2} c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right ) \log{\left (x + \frac{2 a^{2} c e - a b^{2} e + a b c d + 4 a c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right ) - b^{2} c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{2 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{2 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \frac{e x^{2}}{2 c} - \frac{x \left (b e - c d\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32579, size = 165, normalized size = 1.36 \begin{align*} \frac{c x^{2} e + 2 \, c d x - 2 \, b x e}{2 \, c^{2}} - \frac{{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, 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^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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