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

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

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Rubi [A]  time = 0.23, antiderivative size = 145, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.13, size = 141, normalized size = 0.97 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.83, size = 517, normalized size = 3.57 \begin {gather*} \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 \relax (x) - {\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 \relax (x) - {\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 {gather*}

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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giac [A]  time = 0.18, size = 152, normalized size = 1.05 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.07, size = 273, normalized size = 1.88 \begin {gather*} -\frac {2 c e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}+\frac {3 b c d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{3} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{3}}-\frac {b e \ln \relax (x )}{a^{2}}+\frac {b e \ln \left (c \,x^{2}+b x +a \right )}{2 a^{2}}-\frac {c d \ln \relax (x )}{a^{2}}+\frac {c d \ln \left (c \,x^{2}+b x +a \right )}{2 a^{2}}+\frac {b^{2} d \ln \relax (x )}{a^{3}}-\frac {b^{2} d \ln \left (c \,x^{2}+b x +a \right )}{2 a^{3}}-\frac {e}{a x}+\frac {b d}{a^{2} x}-\frac {d}{2 a \,x^{2}} \end {gather*}

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/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)*ar
ctan((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-1/2/a*d/x^2-1/a*e/x+1/a^2/x*b*d-1/a^2*ln(x)*b*e-1/a^2*c*d*ln(x)+1/a^3*ln(x)*b^2*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 2.49, size = 814, normalized size = 5.61 \begin {gather*} \frac {\ln \left (6\,a^3\,c^2\,d-2\,a^2\,b^3\,e+2\,a\,b^4\,d+2\,b^5\,d\,x+7\,a^3\,b\,c\,e-2\,a\,b^4\,e\,x+2\,a\,b^3\,d\,\sqrt {b^2-4\,a\,c}+a^3\,c\,e\,\sqrt {b^2-4\,a\,c}+2\,b^4\,d\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c\,d-2\,a^3\,c^2\,e\,x-2\,a^2\,b^2\,e\,\sqrt {b^2-4\,a\,c}-2\,a\,b^3\,e\,x\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,d\,x+8\,a^2\,b^2\,c\,e\,x+3\,a^2\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^3\,c\,d\,x-3\,a^2\,b\,c\,d\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,d\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,b\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a^2\,\left (2\,c^2\,d+2\,b\,c\,e+c\,e\,\sqrt {b^2-4\,a\,c}\right )+\frac {b^4\,d}{2}-a\,\left (\frac {b^3\,e}{2}+\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {5\,b^2\,c\,d}{2}+\frac {3\,b\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )+\frac {b^3\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\ln \relax (x)\,\left (a\,\left (b\,e+c\,d\right )-b^2\,d\right )}{a^3}-\frac {\frac {d}{2\,a}+\frac {x\,\left (a\,e-b\,d\right )}{a^2}}{x^2}+\frac {\ln \left (2\,a^2\,b^3\,e-6\,a^3\,c^2\,d-2\,a\,b^4\,d-2\,b^5\,d\,x-7\,a^3\,b\,c\,e+2\,a\,b^4\,e\,x+2\,a\,b^3\,d\,\sqrt {b^2-4\,a\,c}+a^3\,c\,e\,\sqrt {b^2-4\,a\,c}+2\,b^4\,d\,x\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b^2\,c\,d+2\,a^3\,c^2\,e\,x-2\,a^2\,b^2\,e\,\sqrt {b^2-4\,a\,c}-2\,a\,b^3\,e\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b\,c^2\,d\,x-8\,a^2\,b^2\,c\,e\,x+3\,a^2\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}+10\,a\,b^3\,c\,d\,x-3\,a^2\,b\,c\,d\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,d\,x\,\sqrt {b^2-4\,a\,c}+4\,a^2\,b\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a^2\,\left (2\,c^2\,d+2\,b\,c\,e-c\,e\,\sqrt {b^2-4\,a\,c}\right )+\frac {b^4\,d}{2}-a\,\left (\frac {b^3\,e}{2}-\frac {b^2\,e\,\sqrt {b^2-4\,a\,c}}{2}+\frac {5\,b^2\,c\,d}{2}-\frac {3\,b\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^3\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{4\,a^4\,c-a^3\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(6*a^3*c^2*d - 2*a^2*b^3*e + 2*a*b^4*d + 2*b^5*d*x + 7*a^3*b*c*e - 2*a*b^4*e*x + 2*a*b^3*d*(b^2 - 4*a*c)^(
1/2) + a^3*c*e*(b^2 - 4*a*c)^(1/2) + 2*b^4*d*x*(b^2 - 4*a*c)^(1/2) - 9*a^2*b^2*c*d - 2*a^3*c^2*e*x - 2*a^2*b^2
*e*(b^2 - 4*a*c)^(1/2) - 2*a*b^3*e*x*(b^2 - 4*a*c)^(1/2) + 9*a^2*b*c^2*d*x + 8*a^2*b^2*c*e*x + 3*a^2*c^2*d*x*(
b^2 - 4*a*c)^(1/2) - 10*a*b^3*c*d*x - 3*a^2*b*c*d*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d*x*(b^2 - 4*a*c)^(1/2) + 4*
a^2*b*c*e*x*(b^2 - 4*a*c)^(1/2))*(a^2*(2*c^2*d + 2*b*c*e + c*e*(b^2 - 4*a*c)^(1/2)) + (b^4*d)/2 - a*((b^3*e)/2
+ (b^2*e*(b^2 - 4*a*c)^(1/2))/2 + (5*b^2*c*d)/2 + (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) + (b^3*d*(b^2 - 4*a*c)^(1/
2))/2))/(4*a^4*c - a^3*b^2) - (log(x)*(a*(b*e + c*d) - b^2*d))/a^3 - (d/(2*a) + (x*(a*e - b*d))/a^2)/x^2 + (lo
g(2*a^2*b^3*e - 6*a^3*c^2*d - 2*a*b^4*d - 2*b^5*d*x - 7*a^3*b*c*e + 2*a*b^4*e*x + 2*a*b^3*d*(b^2 - 4*a*c)^(1/2
) + a^3*c*e*(b^2 - 4*a*c)^(1/2) + 2*b^4*d*x*(b^2 - 4*a*c)^(1/2) + 9*a^2*b^2*c*d + 2*a^3*c^2*e*x - 2*a^2*b^2*e*
(b^2 - 4*a*c)^(1/2) - 2*a*b^3*e*x*(b^2 - 4*a*c)^(1/2) - 9*a^2*b*c^2*d*x - 8*a^2*b^2*c*e*x + 3*a^2*c^2*d*x*(b^2
- 4*a*c)^(1/2) + 10*a*b^3*c*d*x - 3*a^2*b*c*d*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d*x*(b^2 - 4*a*c)^(1/2) + 4*a^2
*b*c*e*x*(b^2 - 4*a*c)^(1/2))*(a^2*(2*c^2*d + 2*b*c*e - c*e*(b^2 - 4*a*c)^(1/2)) + (b^4*d)/2 - a*((b^3*e)/2 -
(b^2*e*(b^2 - 4*a*c)^(1/2))/2 + (5*b^2*c*d)/2 - (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) - (b^3*d*(b^2 - 4*a*c)^(1/2))
/2))/(4*a^4*c - a^3*b^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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