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

Optimal. Leaf size=283 \[ -\frac {\left (-2 a b e-2 a c d+3 b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac {\log (x) \left (-2 a b e-2 a c d+3 b^2 d\right )}{a^4}-\frac {-2 a b e-8 a c d+3 b^2 d}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac {6 a^2 c e-2 a b^2 e-11 a b c d+3 b^3 d}{a^3 x \left (b^2-4 a c\right )}+\frac {\left (-12 a^3 c^2 e+12 a^2 b^2 c e+30 a^2 b c^2 d-2 a b^4 e-20 a b^3 c d+3 b^5 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

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Rubi [A]  time = 0.61, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {822, 800, 634, 618, 206, 628} \begin {gather*} \frac {\left (12 a^2 b^2 c e+30 a^2 b c^2 d-12 a^3 c^2 e-20 a b^3 c d-2 a b^4 e+3 b^5 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}-\frac {-2 a b e-8 a c d+3 b^2 d}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac {\left (-2 a b e-2 a c d+3 b^2 d\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac {6 a^2 c e-2 a b^2 e-11 a b c d+3 b^3 d}{a^3 x \left (b^2-4 a c\right )}+\frac {\log (x) \left (-2 a b e-2 a c d+3 b^2 d\right )}{a^4}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*b^2*d - 8*a*c*d - 2*a*b*e)/(2*a^2*(b^2 - 4*a*c)*x^2) + (3*b^3*d - 11*a*b*c*d - 2*a*b^2*e + 6*a^2*c*e)/(a^3
*(b^2 - 4*a*c)*x) + (b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + ((
3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e)*ArcTanh[(b + 2*c*x)/Sqrt[
b^2 - 4*a*c]])/(a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2*d - 2*a*c*d - 2*a*b*e)*Log[x])/a^4 - ((3*b^2*d - 2*a*c*d -
2*a*b*e)*Log[a + b*x + c*x^2])/(2*a^4)

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]

Rule 822

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

Rubi steps

\begin {align*} \int \frac {d+e x}{x^3 \left (a+b x+c x^2\right )^2} \, dx &=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {-3 b^2 d+8 a c d+2 a b e-3 c (b d-2 a e) x}{x^3 \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {-3 b^2 d+8 a c d+2 a b e}{a x^3}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^2 x^2}-\frac {\left (-b^2+4 a c\right ) \left (-3 b^2 d+2 a c d+2 a b e\right )}{a^3 x}+\frac {3 b^5 d-17 a b^3 c d+19 a^2 b c^2 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a c d-2 a b e\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2 d-8 a c d-2 a b e}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^3 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)}{a^4}-\frac {\int \frac {3 b^5 d-17 a b^3 c d+19 a^2 b c^2 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a c d-2 a b e\right ) x}{a+b x+c x^2} \, dx}{a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2 d-8 a c d-2 a b e}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^3 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^4}-\frac {\left (3 b^5 d-20 a b^3 c d+30 a^2 b c^2 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2 d-8 a c d-2 a b e}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^3 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac {\left (3 b^5 d-20 a b^3 c d+30 a^2 b c^2 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2 d-8 a c d-2 a b e}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b^3 d-11 a b c d-2 a b^2 e+6 a^2 c e}{a^3 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )}+\frac {\left (3 b^5 d-20 a b^3 c d+30 a^2 b c^2 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a c d-2 a b e\right ) \log \left (a+b x+c x^2\right )}{2 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 253, normalized size = 0.89 \begin {gather*} \frac {\frac {2 a \left (2 a^2 c^2 (d+e x)+b^3 (c d x-a e)-a b^2 c (4 d+e x)+3 a b c (a e-c d x)+b^4 d\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {a^2 d}{x^2}+\frac {2 \left (-12 a^3 c^2 e+12 a^2 b^2 c e+30 a^2 b c^2 d-2 a b^4 e-20 a b^3 c d+3 b^5 d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+2 \log (x) \left (-2 a b e-2 a c d+3 b^2 d\right )+\left (2 a b e+2 a c d-3 b^2 d\right ) \log (a+x (b+c x))-\frac {2 a (a e-2 b d)}{x}}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a^2*d)/x^2) - (2*a*(-2*b*d + a*e))/x + (2*a*(b^4*d + 3*a*b*c*(a*e - c*d*x) + b^3*(-(a*e) + c*d*x) + 2*a^2*
c^2*(d + e*x) - a*b^2*c*(4*d + e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(3*b^5*d - 20*a*b^3*c*d + 30*a^2*
b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3
/2) + 2*(3*b^2*d - 2*a*c*d - 2*a*b*e)*Log[x] + (-3*b^2*d + 2*a*c*d + 2*a*b*e)*Log[a + x*(b + c*x)])/(2*a^4)

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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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 4.08, size = 2003, normalized size = 7.08

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b^2*c^2 + 12*a^4*c^3)*e)*x^3 + (
(6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a^3*b^3*c + 28*a^4*b*c^2)*e)*x^2
+ (((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e)*x^4 + ((3*b^6 - 20
*a*b^4*c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e)*x^3 + ((3*a*b^5 - 20*a^2*b^3*c + 30*a^
3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e)*x^2)*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)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*d + (3*(a^2*b
^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x - (((3*b^6*c - 26*a*b^4*c^2 +
64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e)*x^4 + ((3*b^7 - 26*a*b^5*c + 6
4*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + ((3*a*b^6 - 26*a^2*b^4*c +
64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*log(c*x^2 + b*x + a) + 2*((
(3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e)*x^4 +
((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + (
(3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*lo
g(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^4 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^3 + (a^5*b^4 -
8*a^6*b^2*c + 16*a^7*c^2)*x^2), 1/2*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b
^2*c^2 + 12*a^4*c^3)*e)*x^3 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a
^3*b^3*c + 28*a^4*b*c^2)*e)*x^2 + 2*(((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 +
6*a^3*c^3)*e)*x^4 + ((3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e)*x^3 +
((3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e)*x^2)*sqrt(-b^2 + 4*a*c)
*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*d + (3*(a^2*b^5
- 8*a^3*b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x - (((3*b^6*c - 26*a*b^4*c^2 + 64
*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e)*x^4 + ((3*b^7 - 26*a*b^5*c + 64*a
^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + ((3*a*b^6 - 26*a^2*b^4*c + 64
*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*log(c*x^2 + b*x + a) + 2*(((3*
b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e)*x^4 + ((
3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e)*x^3 + ((3*
a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e)*x^2)*log(x
))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^4 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^3 + (a^5*b^4 - 8*a
^6*b^2*c + 16*a^7*c^2)*x^2)]

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giac [A]  time = 0.17, size = 345, normalized size = 1.22 \begin {gather*} -\frac {{\left (3 \, b^{5} d - 20 \, a b^{3} c d + 30 \, a^{2} b c^{2} d - 2 \, a b^{4} e + 12 \, a^{2} b^{2} c e - 12 \, a^{3} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} + \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {a^{3} b^{2} d - 4 \, a^{4} c d - 2 \, {\left (3 \, a b^{3} c d - 11 \, a^{2} b c^{2} d - 2 \, a^{2} b^{2} c e + 6 \, a^{3} c^{2} e\right )} x^{3} - {\left (6 \, a b^{4} d - 25 \, a^{2} b^{2} c d + 8 \, a^{3} c^{2} d - 4 \, a^{2} b^{3} e + 14 \, a^{3} b c e\right )} x^{2} - {\left (3 \, a^{2} b^{3} d - 12 \, a^{3} b c d - 2 \, a^{3} b^{2} e + 8 \, a^{4} c e\right )} x}{2 \, {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{4} 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)^2,x, algorithm="giac")

[Out]

-(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e)*arctan((2*c*x + b)/sqrt
(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(3*b^2*d - 2*a*c*d - 2*a*b*e)*log(c*x^2 + b*x +
a)/a^4 + (3*b^2*d - 2*a*c*d - 2*a*b*e)*log(abs(x))/a^4 - 1/2*(a^3*b^2*d - 4*a^4*c*d - 2*(3*a*b^3*c*d - 11*a^2
*b*c^2*d - 2*a^2*b^2*c*e + 6*a^3*c^2*e)*x^3 - (6*a*b^4*d - 25*a^2*b^2*c*d + 8*a^3*c^2*d - 4*a^2*b^3*e + 14*a^3
*b*c*e)*x^2 - (3*a^2*b^3*d - 12*a^3*b*c*d - 2*a^3*b^2*e + 8*a^4*c*e)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*a^4*x
^2)

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maple [B]  time = 0.07, size = 770, normalized size = 2.72 \begin {gather*} -\frac {2 c^{2} e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {12 c^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a}+\frac {b^{2} c e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {12 b^{2} c e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}+\frac {3 b \,c^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {30 b \,c^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {2 b^{4} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {b^{3} c d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {20 b^{3} c d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {3 b^{5} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {3 b c e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {2 c^{2} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}+\frac {b^{3} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {4 b^{2} c d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {4 b c e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{2}}+\frac {4 c^{2} d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{4} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {b^{3} e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{3}}-\frac {7 b^{2} c d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{3}}+\frac {3 b^{4} d \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) a^{4}}-\frac {2 b e \ln \relax (x )}{a^{3}}-\frac {2 c d \ln \relax (x )}{a^{3}}+\frac {3 b^{2} d \ln \relax (x )}{a^{4}}-\frac {e}{a^{2} x}+\frac {2 b d}{a^{3} x}-\frac {d}{2 a^{2} 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)^2,x)

[Out]

-1/2/a^2*d/x^2-1/a^2*e/x-20/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d+4/a^2/(4*a*c-b^2
)*c*ln(c*x^2+b*x+a)*b*e+4/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c*d-2/a/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*e-3/a/(c*x
^2+b*x+a)/(4*a*c-b^2)*b*c*e-7/a^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d+12/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b
)/(4*a*c-b^2)^(1/2))*b^2*c*e+30/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d-1/a^3/(c*x^2
+b*x+a)*c/(4*a*c-b^2)*x*b^3*d+1/a^2/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^2*e+3/a^2/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*
b*d+2/a^3/x*b*d-2/a^3*ln(x)*b*e+3/a^4*ln(x)*b^2*d-2/a/(c*x^2+b*x+a)/(4*a*c-b^2)*c^2*d+1/a^2/(c*x^2+b*x+a)/(4*a
*c-b^2)*b^3*e-1/a^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*d+4/a^2/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*d-1/a^3/(4*a*c-b^2)*
ln(c*x^2+b*x+a)*b^3*e+3/2/a^4/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^4*d-12/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))*c^2*e-2/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*e+3/a^4/(4*a*c-b^2)^(3/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*d-2/a^3*c*d*ln(x)

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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)^2,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 = 3.06, size = 1661, normalized size = 5.87

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(192*a^5*c^4*d - 4*a^2*b^7*e + 6*a*b^8*d + 6*b^9*d*x + 307*a^3*b^4*c^2*d - 492*a^4*b^2*c^3*d + 4*a^2*b^4*e*
(-(4*a*c - b^2)^3)^(1/2) - 174*a^4*b^3*c^2*e + 6*a^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^8*e*x - 6*a*b^5*d*
(-(4*a*c - b^2)^3)^(1/2) - 73*a^2*b^6*c*d + 46*a^3*b^5*c*e + 216*a^5*b*c^3*e - 6*b^6*d*x*(-(4*a*c - b^2)^3)^(1
/2) - 48*a^5*c^4*e*x + 312*a^4*b*c^4*d*x + 4*a*b^5*e*x*(-(4*a*c - b^2)^3)^(1/2) + 48*a^2*b^6*c*e*x + 31*a^2*b^
3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 27*a^3*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 18*a^3*b^2*c*e*(-(4*a*c - b^2)^3)^(
1/2) + 339*a^2*b^5*c^2*d*x - 602*a^3*b^3*c^3*d*x + 24*a^3*c^3*d*x*(-(4*a*c - b^2)^3)^(1/2) - 194*a^3*b^4*c^2*e
*x + 276*a^4*b^2*c^3*e*x - 76*a*b^7*c*d*x - 69*a^2*b^2*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 40*a*b^4*c*d*x*(-(4*
a*c - b^2)^3)^(1/2) - 24*a^2*b^3*c*e*x*(-(4*a*c - b^2)^3)^(1/2) + 30*a^3*b*c^2*e*x*(-(4*a*c - b^2)^3)^(1/2))*(
((3*b^5*d*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2
) - 10*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*b^2*c*e*(-(4*a*c -
b^2)^3)^(1/2))/(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2) - (3*b^2*d)/(2*a^4) + (b*e)/a^3 + (c*d)
/a^3) - log(4*a^2*b^7*e - 192*a^5*c^4*d - 6*a*b^8*d - 6*b^9*d*x - 307*a^3*b^4*c^2*d + 492*a^4*b^2*c^3*d + 4*a^
2*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 174*a^4*b^3*c^2*e + 6*a^4*c^2*e*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^8*e*x - 6*
a*b^5*d*(-(4*a*c - b^2)^3)^(1/2) + 73*a^2*b^6*c*d - 46*a^3*b^5*c*e - 216*a^5*b*c^3*e - 6*b^6*d*x*(-(4*a*c - b^
2)^3)^(1/2) + 48*a^5*c^4*e*x - 312*a^4*b*c^4*d*x + 4*a*b^5*e*x*(-(4*a*c - b^2)^3)^(1/2) - 48*a^2*b^6*c*e*x + 3
1*a^2*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 27*a^3*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 18*a^3*b^2*c*e*(-(4*a*c - b
^2)^3)^(1/2) - 339*a^2*b^5*c^2*d*x + 602*a^3*b^3*c^3*d*x + 24*a^3*c^3*d*x*(-(4*a*c - b^2)^3)^(1/2) + 194*a^3*b
^4*c^2*e*x - 276*a^4*b^2*c^3*e*x + 76*a*b^7*c*d*x - 69*a^2*b^2*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 40*a*b^4*c*d
*x*(-(4*a*c - b^2)^3)^(1/2) - 24*a^2*b^3*c*e*x*(-(4*a*c - b^2)^3)^(1/2) + 30*a^3*b*c^2*e*x*(-(4*a*c - b^2)^3)^
(1/2))*(((3*b^5*d*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - a*b^4*e*(-(4*a*c - b^2)
^3)^(1/2) - 10*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 15*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*b^2*c*e*(-
(4*a*c - b^2)^3)^(1/2))/(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2) + (3*b^2*d)/(2*a^4) - (b*e)/a^3
- (c*d)/a^3) - (d/(2*a) + (x*(2*a*e - 3*b*d))/(2*a^2) + (x^2*(6*b^4*d + 8*a^2*c^2*d - 4*a*b^3*e - 25*a*b^2*c*
d + 14*a^2*b*c*e))/(2*a^3*(4*a*c - b^2)) + (c*x^3*(3*b^3*d - 2*a*b^2*e + 6*a^2*c*e - 11*a*b*c*d))/(a^3*(4*a*c
- b^2)))/(a*x^2 + b*x^3 + c*x^4) - (log(x)*(a*(2*b*e + 2*c*d) - 3*b^2*d))/a^4

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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)**2,x)

[Out]

Timed out

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