3.888 \(\int \frac{d+e x}{x^3 (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=145 \[ -\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (-a b e-a c d+b^2 d\right )}{a^3}+\frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b d-a e}{a^2 x}-\frac{d}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.229522, antiderivative size = 145, 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} \[ -\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (-a b e-a c d+b^2 d\right )}{a^3}+\frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b d-a e}{a^2 x}-\frac{d}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(a + b*x + c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.134211, size = 141, normalized size = 0.97 \[ \frac{\frac{2 \left (-2 a^2 c e+a b^2 e+3 a b c d+b^3 (-d)\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{a^2 d}{x^2}+2 \log (x) \left (-a b e-a c d+b^2 d\right )+\left (a b e+a c d+b^2 (-d)\right ) \log (a+x (b+c x))+\frac{2 a (b d-a e)}{x}}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 273, normalized size = 1.9 \begin{align*} -{\frac{d}{2\,a{x}^{2}}}-{\frac{e}{ax}}+{\frac{bd}{{a}^{2}x}}-{\frac{\ln \left ( x \right ) be}{{a}^{2}}}-{\frac{cd\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) be}{2\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) d}{2\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d}{2\,{a}^{3}}}-2\,{\frac{ce}{a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}e}{{a}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{bcd}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}d}{{a}^{3}}\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((e*x+d)/x^3/(c*x^2+b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.9901, size = 1104, normalized size = 7.61 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c}{\left ({\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \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^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (x\right ) -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d + 2 \,{\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} x^{2} \log \left (x\right ) -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d + 2 \,{\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [A]  time = 1.31907, size = 205, normalized size = 1.41 \begin{align*} -\frac{{\left (b^{2} d - a c d - a b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac{{\left (b^{2} d - a c d - a b e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} d - 3 \, a b c d - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{a^{2} d - 2 \,{\left (a b d - a^{2} e\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

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