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

Optimal. Leaf size=211 \[ \frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}-\frac {\log (x) (2 b d-a e)}{a^3}-\frac {-a b e-6 a c d+2 b^2 d}{a^2 x \left (b^2-4 a c\right )}-\frac {\left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

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Rubi [A]  time = 0.36, antiderivative size = 211, 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 (6 a^2 b c e+12 a^2 c^2 d-12 a b^2 c d-a b^3 e+2 b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {-a b e-6 a c d+2 b^2 d}{a^2 x \left (b^2-4 a c\right )}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}-\frac {\log (x) (2 b d-a e)}{a^3}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \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^2*(a + b*x + c*x^2)^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.34, size = 192, normalized size = 0.91 \begin {gather*} \frac {-\frac {2 \left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 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}}-\frac {2 a \left (b^2 (c d x-a e)-a b c (3 d+e x)+2 a c (a e-c d x)+b^3 d\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+(2 b d-a e) \log (a+x (b+c x))+2 \log (x) (a e-2 b d)-\frac {2 a d}{x}}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*d)/x - (2*a*(b^3*d + 2*a*c*(a*e - c*d*x) + b^2*(-(a*e) + c*d*x) - a*b*c*(3*d + e*x)))/((b^2 - 4*a*c)*(a
+ x*(b + c*x))) - (2*(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTan[(b + 2*c*x)/Sqrt[
-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(-2*b*d + a*e)*Log[x] + (2*b*d - a*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^2 \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^2*(a + b*x + c*x^2)^2),x]

[Out]

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

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fricas [B]  time = 2.03, size = 1615, normalized size = 7.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3*b*c^2)*e)*x^2 + ((2*(b^4*c - 6*a*b^2
*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2*b*c^2)*e)*x^3 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b
^2*c)*e)*x^2 + (2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d - (a^2*b^3 - 6*a^3*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^2*b^4 - 8*a^3*b^2*c
+ 16*a^4*c^2)*d + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*e)*x - ((
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c
+ 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d
- (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x)*log(c*x^2 + b*x + a) + 2*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3
)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2
*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^
2)*e)*x)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2 +
(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x), -1/2*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*
a^3*b*c^2)*e)*x^2 + 2*((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2*b*c^2)*e)*x^3 + (2*(b^5 - 6*a
*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b^2*c)*e)*x^2 + (2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d - (a^2*b^3 - 6
*a^3*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^2*b^4 - 8*a^3*
b^2*c + 16*a^4*c^2)*d + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 6*a^3*b^2*c + 8*a^4*c^2)*e)*
x - ((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a
*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*
c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x)*log(c*x^2 + b*x + a) + 2*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*d - (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d - (a*b^5 -
8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*
a^4*c^2)*e)*x)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*
x^2 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x)]

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giac [A]  time = 0.16, size = 245, normalized size = 1.16 \begin {gather*} \frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{2} - 6 \, a c^{2} d x^{2} - a b c x^{2} e + 2 \, b^{3} d x - 7 \, a b c d x - a b^{2} x e + 2 \, a^{2} c x e + a b^{2} d - 4 \, a^{2} c d}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c x^{3} + b x^{2} + a x\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b
^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)) - (2*b^2*c*d*x^2 - 6*a*c^2*d*x^2 - a*b*c*x^2*e + 2*b^3*d*x - 7*a*b*c*d*x - a
*b^2*x*e + 2*a^2*c*x*e + a*b^2*d - 4*a^2*c*d)/((a^2*b^2 - 4*a^3*c)*(c*x^3 + b*x^2 + a*x)) + 1/2*(2*b*d - a*e)*
log(c*x^2 + b*x + a)/a^3 - (2*b*d - a*e)*log(abs(x))/a^3

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maple [B]  time = 0.06, size = 582, normalized size = 2.76 \begin {gather*} -\frac {b c e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {6 b 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}-\frac {2 c^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {12 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}+\frac {b^{3} 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 {b^{2} c d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {12 b^{2} 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^{2}}-\frac {2 b^{4} 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 {b^{2} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {3 b c d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {2 c e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a}+\frac {b^{3} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) a^{2}}+\frac {4 b c d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{3}}+\frac {2 c e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right )}+\frac {e \ln \relax (x )}{a^{2}}-\frac {2 b d \ln \relax (x )}{a^{3}}-\frac {d}{a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b*e-2/a/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*d+1/a^2/(c*x^2+b*x+a)*c/(4*a*c-b^2)
*x*b^2*d+2/(c*x^2+b*x+a)/(4*a*c-b^2)*c*e-1/a/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*e-3/a/(c*x^2+b*x+a)/(4*a*c-b^2)*b*c
*d+1/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*b^3*d-2/a/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*e+1/2/a^2/(4*a*c-b^2)*ln(c*x^2+b*x+
a)*b^2*e+4/a^2/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b*d-1/a^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3*d-6/a/(4*a*c-b^2)^(3/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*e-12/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d+1/
a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e+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*d-2/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d-1/a^2*d/x+1/a^2*e*ln(
x)-2/a^3*ln(x)*b*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^2/(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 = 3.09, size = 1366, normalized size = 6.47 \begin {gather*} \ln \left (96\,a^5\,c^3\,e-2\,a^2\,b^6\,e+4\,a\,b^7\,d+4\,b^8\,d\,x+174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-84\,a^4\,b^2\,c^2\,e-2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-46\,a^2\,b^5\,c\,d-216\,a^4\,b\,c^3\,d+23\,a^3\,b^4\,c\,e+48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^5\,c\,e\,x+120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+194\,a^2\,b^4\,c^2\,d\,x-276\,a^3\,b^2\,c^3\,d\,x-94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}-\frac {e}{2\,a^2}+\frac {b\,d}{a^3}\right )-\ln \left (2\,a^2\,b^6\,e-96\,a^5\,c^3\,e-4\,a\,b^7\,d-4\,b^8\,d\,x-174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+84\,a^4\,b^2\,c^2\,e+2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+46\,a^2\,b^5\,c\,d+216\,a^4\,b\,c^3\,d-23\,a^3\,b^4\,c\,e-48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a^2\,b^5\,c\,e\,x-120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-194\,a^2\,b^4\,c^2\,d\,x+276\,a^3\,b^2\,c^3\,d\,x+94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}+\frac {e}{2\,a^2}-\frac {b\,d}{a^3}\right )-\frac {\frac {d}{a}-\frac {x\,\left (2\,c\,e\,a^2-e\,a\,b^2-7\,c\,d\,a\,b+2\,d\,b^3\right )}{a^2\,\left (4\,a\,c-b^2\right )}+\frac {c\,x^2\,\left (-2\,d\,b^2+a\,e\,b+6\,a\,c\,d\right )}{a^2\,\left (4\,a\,c-b^2\right )}}{c\,x^3+b\,x^2+a\,x}+\frac {\ln \relax (x)\,\left (a\,e-2\,b\,d\right )}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(96*a^5*c^3*e - 2*a^2*b^6*e + 4*a*b^7*d + 4*b^8*d*x + 174*a^3*b^3*c^2*d - 2*a^2*b^3*e*(-(4*a*c - b^2)^3)^(1
/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 84*a^4*b^2*c^2*e - 2*a*b^7*e*x + 4*a*b^4*d*(-(4*a*c - b^2)^3)^(1/
2) - 46*a^2*b^5*c*d - 216*a^4*b*c^3*d + 23*a^3*b^4*c*e + 48*a^4*c^4*d*x + 4*b^5*d*x*(-(4*a*c - b^2)^3)^(1/2) +
9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^4*e*x*(-(4*a*c - b^2)^3)^(1/2) + 24*a^2*b^5*c*e*x + 120*a^4*b*c^
3*e*x - 18*a^2*b^2*c*d*(-(4*a*c - b^2)^3)^(1/2) + 194*a^2*b^4*c^2*d*x - 276*a^3*b^2*c^3*d*x - 94*a^3*b^3*c^2*e
*x - 12*a^3*c^2*e*x*(-(4*a*c - b^2)^3)^(1/2) - 48*a*b^6*c*d*x - 24*a*b^3*c*d*x*(-(4*a*c - b^2)^3)^(1/2) + 30*a
^2*b*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b^2*c*e*x*(-(4*a*c - b^2)^3)^(1/2))*((b^4*d*(-(4*a*c - b^2)^3)^
(1/2) + 6*a^2*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - (a*b^3*e*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a*b^2*c*d*(-(4*a*c - b
^2)^3)^(1/2) + 3*a^2*b*c*e*(-(4*a*c - b^2)^3)^(1/2))/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) -
e/(2*a^2) + (b*d)/a^3) - log(2*a^2*b^6*e - 96*a^5*c^3*e - 4*a*b^7*d - 4*b^8*d*x - 174*a^3*b^3*c^2*d - 2*a^2*b^
3*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 84*a^4*b^2*c^2*e + 2*a*b^7*e*x + 4*a*b^4
*d*(-(4*a*c - b^2)^3)^(1/2) + 46*a^2*b^5*c*d + 216*a^4*b*c^3*d - 23*a^3*b^4*c*e - 48*a^4*c^4*d*x + 4*b^5*d*x*(
-(4*a*c - b^2)^3)^(1/2) + 9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^4*e*x*(-(4*a*c - b^2)^3)^(1/2) - 24*a^2
*b^5*c*e*x - 120*a^4*b*c^3*e*x - 18*a^2*b^2*c*d*(-(4*a*c - b^2)^3)^(1/2) - 194*a^2*b^4*c^2*d*x + 276*a^3*b^2*c
^3*d*x + 94*a^3*b^3*c^2*e*x - 12*a^3*c^2*e*x*(-(4*a*c - b^2)^3)^(1/2) + 48*a*b^6*c*d*x - 24*a*b^3*c*d*x*(-(4*a
*c - b^2)^3)^(1/2) + 30*a^2*b*c^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b^2*c*e*x*(-(4*a*c - b^2)^3)^(1/2))*((
b^4*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - (a*b^3*e*(-(4*a*c - b^2)^3)^(1/2))/2 -
6*a*b^2*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c*e*(-(4*a*c - b^2)^3)^(1/2))/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b
^4*c + 48*a^5*b^2*c^2) + e/(2*a^2) - (b*d)/a^3) - (d/a - (x*(2*b^3*d - a*b^2*e + 2*a^2*c*e - 7*a*b*c*d))/(a^2*
(4*a*c - b^2)) + (c*x^2*(a*b*e - 2*b^2*d + 6*a*c*d))/(a^2*(4*a*c - b^2)))/(a*x + b*x^2 + c*x^3) + (log(x)*(a*e
- 2*b*d))/a^3

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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**2/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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