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

Optimal. Leaf size=327 \[ \frac {e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac {2 e^2 (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

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Rubi [A]  time = 0.53, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {822, 800, 634, 618, 206, 628} \begin {gather*} \frac {e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac {2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac {2 e^2 (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.60, size = 275, normalized size = 0.84 \begin {gather*} \frac {-\frac {\left (e (a e-b d)+c d^2\right ) \left (c e (-a e-2 b d+b e x)+b^2 e^2+c^2 d (d-2 e x)\right )}{a+x (b+c x)}-2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac {2 e (b e-2 c d) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-\frac {e^2 (b e-2 c d) \left (e (a e-b d)+c d^2\right )}{d+e x}}{\left (e (a e-b d)+c d^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 43.72, size = 4494, normalized size = 13.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((b^2*c^3 - 4*a*c^4)*d^5 - 3*(b^3*c^2 - 4*a*b*c^3)*d^4*e + (3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^2 - (b
^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^2*e^3 - 3*(a^2*b^2*c - 4*a^3*c^2)*d*e^4 + (a^2*b^3 - 4*a^3*b*c)*e^5 - 2*(2*(b^
2*c^3 - 4*a*c^4)*d^3*e^2 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^4 - (a*b^3*
c - 4*a^2*b*c^2)*e^5)*x^2 - (2*a*c^3*d^4*e - 3*a*b*c^2*d^3*e^2 + 3*(a*b^2*c - 2*a^2*c^2)*d^2*e^3 - (a*b^3 - 3*
a^2*b*c)*d*e^4 + (2*c^4*d^3*e^2 - 3*b*c^3*d^2*e^3 + 3*(b^2*c^2 - 2*a*c^3)*d*e^4 - (b^3*c - 3*a*b*c^2)*e^5)*x^3
+ (2*c^4*d^4*e - b*c^3*d^3*e^2 - 6*a*c^3*d^2*e^3 + (2*b^3*c - 3*a*b*c^2)*d*e^4 - (b^4 - 3*a*b^2*c)*e^5)*x^2 +
(2*b*c^3*d^4*e - (3*b^2*c^2 - 2*a*c^3)*d^3*e^2 + 3*(b^3*c - 3*a*b*c^2)*d^2*e^3 - (b^4 - 6*a*b^2*c + 6*a^2*c^2
)*d*e^4 - (a*b^3 - 3*a^2*b*c)*e^5)*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)) - ((b^2*c^3 - 4*a*c^4)*d^4*e + 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 - (5*b^4*c
- 22*a*b^2*c^2 + 8*a^2*c^3)*d^2*e^3 + 2*(b^5 - 3*a*b^3*c - 4*a^2*b*c^2)*d*e^4 - (2*a*b^4 - 9*a^2*b^2*c + 4*a^
3*c^2)*e^5)*x - (3*(a*b^2*c^2 - 4*a^2*c^3)*d^3*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^3 + (a*b^4 - 5*a^2*b^2*c
+ 4*a^3*c^2)*d*e^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (b^4*c - 5*a*b^2*c^2 + 4
*a^2*c^3)*e^5)*x^3 + (3*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - (2*b^4*c - 7*a*b^2*c^2 - 4*a^2*c^3)*d*e^4 + (b^5 - 5*a*b
^3*c + 4*a^2*b*c^2)*e^5)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 - 3*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e^3
+ (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^4 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^5)*x)*log(c*x^2 + b*x + a) + 2*
(3*(a*b^2*c^2 - 4*a^2*c^3)*d^3*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^3 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d*e
^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*e^5)*x
^3 + (3*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - (2*b^4*c - 7*a*b^2*c^2 - 4*a^2*c^3)*d*e^4 + (b^5 - 5*a*b^3*c + 4*a^2*b*c
^2)*e^5)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 - 3*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e^3 + (b^5 - 8*a*b^3
*c + 16*a^2*b*c^2)*d*e^4 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^5)*x)*log(e*x + d))/((a*b^2*c^3 - 4*a^2*c^4)*d^
7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 - (a*b^5 + 2*a^2*b^3*c
- 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d^2*e^5 + (
a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^2 + 3*(b^4*c^2 - 3*a*b^2
*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*
c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3 + ((b^2*c^4 - 4*a*c^5)*d^7
- 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3*c^2 + 12*a^2*b*c^3)
*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^3*c)*d^2*e^5 - (3*a^2*b^
4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a*b*c^4)*d^7 - (3*b^4*c^2
- 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^
3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^4*c^2)*d^2*e^5 - 2*(a^3*
b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x), -((b^2*c^3 - 4*a*c^4)*d^5 - 3*(b^3*c^2 - 4*a*b*c^3)*d^4*
e + (3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^2 - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^2*e^3 - 3*(a^2*b^2*c - 4*
a^3*c^2)*d*e^4 + (a^2*b^3 - 4*a^3*b*c)*e^5 - 2*(2*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2*e^
3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d*e^4 - (a*b^3*c - 4*a^2*b*c^2)*e^5)*x^2 - 2*(2*a*c^3*d^4*e - 3*a*b*c^2*
d^3*e^2 + 3*(a*b^2*c - 2*a^2*c^2)*d^2*e^3 - (a*b^3 - 3*a^2*b*c)*d*e^4 + (2*c^4*d^3*e^2 - 3*b*c^3*d^2*e^3 + 3*(
b^2*c^2 - 2*a*c^3)*d*e^4 - (b^3*c - 3*a*b*c^2)*e^5)*x^3 + (2*c^4*d^4*e - b*c^3*d^3*e^2 - 6*a*c^3*d^2*e^3 + (2*
b^3*c - 3*a*b*c^2)*d*e^4 - (b^4 - 3*a*b^2*c)*e^5)*x^2 + (2*b*c^3*d^4*e - (3*b^2*c^2 - 2*a*c^3)*d^3*e^2 + 3*(b^
3*c - 3*a*b*c^2)*d^2*e^3 - (b^4 - 6*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 - 3*a^2*b*c)*e^5)*x)*sqrt(-b^2 + 4*a*c
)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - ((b^2*c^3 - 4*a*c^4)*d^4*e + 2*(b^3*c^2 - 4*a*b*c^3)
*d^3*e^2 - (5*b^4*c - 22*a*b^2*c^2 + 8*a^2*c^3)*d^2*e^3 + 2*(b^5 - 3*a*b^3*c - 4*a^2*b*c^2)*d*e^4 - (2*a*b^4 -
9*a^2*b^2*c + 4*a^3*c^2)*e^5)*x - (3*(a*b^2*c^2 - 4*a^2*c^3)*d^3*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^3 + (a
*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d*e^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (b^4*
c - 5*a*b^2*c^2 + 4*a^2*c^3)*e^5)*x^3 + (3*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - (2*b^4*c - 7*a*b^2*c^2 - 4*a^2*c^3)*d
*e^4 + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^5)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 - 3*(b^4*c - 5*a*b^2*c^2 +
4*a^2*c^3)*d^2*e^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^4 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^5)*x)*log(c*
x^2 + b*x + a) + 2*(3*(a*b^2*c^2 - 4*a^2*c^3)*d^3*e^2 - 3*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^3 + (a*b^4 - 5*a^2*b^2
*c + 4*a^3*c^2)*d*e^4 + (3*(b^2*c^3 - 4*a*c^4)*d^2*e^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d*e^4 + (b^4*c - 5*a*b^2*c^2
+ 4*a^2*c^3)*e^5)*x^3 + (3*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - (2*b^4*c - 7*a*b^2*c^2 - 4*a^2*c^3)*d*e^4 + (b^5 - 5*
a*b^3*c + 4*a^2*b*c^2)*e^5)*x^2 + (3*(b^3*c^2 - 4*a*b*c^3)*d^3*e^2 - 3*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e
^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^4 + (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e^5)*x)*log(e*x + d))/((a*b^2*
c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 - (
a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3 - 4*a
^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^2 + 3
*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c - 3*a
^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3 + ((b^2
*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3*
c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^3*c)*
d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a*b*c^
4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15*
a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^4*c^2
)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x)]

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giac [B]  time = 0.24, size = 741, normalized size = 2.27 \begin {gather*} -\frac {2 \, {\left (2 \, c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} - 6 \, a c^{2} d e^{5} - b^{3} e^{6} + 3 \, a b c e^{6}\right )} \arctan \left (\frac {{\left (2 \, c d - \frac {2 \, c d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}} + \frac {\frac {2 \, c d e^{6}}{x e + d} - \frac {b e^{7}}{x e + d}}{c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6} + 2 \, a c d^{2} e^{6} - 2 \, a b d e^{7} + a^{2} e^{8}} + \frac {\frac {3 \, c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + b^{2} c e^{4} - a c^{2} e^{4}}{c d^{2} - b d e + a e^{2}} - \frac {{\left (4 \, c^{3} d^{3} e^{3} - 6 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - 4 \, a c^{2} d e^{5} - b^{3} e^{6} + 2 \, a b c e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e + a e^{2}\right )} {\left (x e + d\right )}}}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - 6*a*c^2*d*e^5 - b^3*e^6 + 3*a*b*c*e^6)*arctan((2*c*d - 2
*c*d^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*a*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/((c^3*d^6 -
3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*
d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) + (3*c^2*d^2*e^2 - 3*b*c*d*e^3 + b^2*e^4 - a*c*e^4)*log
(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^3*d^6 - 3
*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d
^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) + (2*c*d*e^6/(x*e + d) - b*e^7/(x*e + d))/(c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2
*d^2*e^6 + 2*a*c*d^2*e^6 - 2*a*b*d*e^7 + a^2*e^8) + ((3*c^3*d^2*e^2 - 3*b*c^2*d*e^3 + b^2*c*e^4 - a*c^2*e^4)/(
c*d^2 - b*d*e + a*e^2) - (4*c^3*d^3*e^3 - 6*b*c^2*d^2*e^4 + 4*b^2*c*d*e^5 - 4*a*c^2*d*e^5 - b^3*e^6 + 2*a*b*c*
e^6)*e^(-1)/((c*d^2 - b*d*e + a*e^2)*(x*e + d)))/((c*d^2 - b*d*e + a*e^2)^2*(c - 2*c*d/(x*e + d) + c*d^2/(x*e
+ d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2))

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maple [B]  time = 0.08, size = 1177, normalized size = 3.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((2*c*x+b)/(e*x+d)^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 = 8.68, size = 3631, normalized size = 11.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(c^2*d^2*e - 2*b^2*e^3 + a*c*e^3 + 3*b*c*d*e^2))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^
3*e + 2*a*c*d^2*e^2) - (c^2*d^3 + b^2*d*e^2 + a*b*e^3 - 3*a*c*d*e^2 - 2*b*c*d^2*e)/(a^2*e^4 + c^2*d^4 + b^2*d^
2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) + (2*x^2*(2*c^2*d*e^2 - b*c*e^3))/(a^2*e^4 + c^2*d^4 + b^2*
d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2))/(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3) + (log
(d + e*x)*(e^4*(2*a*c - 2*b^2) - 6*c^2*d^2*e^2 + 6*b*c*d*e^3))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*
e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3) +
(log((27*d*e^6*(b^2 - 4*a*c)^(7/2))/16 + (9*e^7*x*(b^2 - 4*a*c)^(7/2))/16 + 8*a*b^6*e^7 - 4*b*c^6*d^7 + 8*b^7
*e^7*x - 8*c^7*d^7*x + 4*c^6*d^7*(b^2 - 4*a*c)^(1/2) - 72*a^4*c^3*e^7 + (57*b^2*e^7*x*(b^2 - 4*a*c)^(5/2))/16
+ (51*b^4*e^7*x*(b^2 - 4*a*c)^(3/2))/16 + (11*b^6*e^7*x*(b^2 - 4*a*c)^(1/2))/16 - 60*a^2*b^4*c*e^7 - 8*b^2*c^5
*d^6*e - 4*b^6*c*d^2*e^5 + (75*c^2*d^3*e^4*(b^2 - 4*a*c)^(5/2))/4 + 25*c^4*d^5*e^2*(b^2 - 4*a*c)^(3/2) + 132*a
^3*b^2*c^2*e^7 - 408*a^2*c^5*d^4*e^3 + 456*a^3*c^4*d^2*e^5 + 20*b^3*c^4*d^5*e^2 - 28*b^4*c^3*d^4*e^3 + 16*b^5*
c^2*d^3*e^4 + (9*a*b*e^7*(b^2 - 4*a*c)^(5/2))/4 + 88*a*c^6*d^6*e + (9*a*b^3*e^7*(b^2 - 4*a*c)^(3/2))/2 + (5*a*
b^5*e^7*(b^2 - 4*a*c)^(1/2))/4 + (111*b^2*d*e^6*(b^2 - 4*a*c)^(5/2))/16 - (79*b^4*d*e^6*(b^2 - 4*a*c)^(3/2))/1
6 - (59*b^6*d*e^6*(b^2 - 4*a*c)^(1/2))/16 - 40*a*b^5*c*d*e^6 + (23*b^2*c^2*d^3*e^4*(b^2 - 4*a*c)^(3/2))/2 - 45
*b^2*c^4*d^5*e^2*(b^2 - 4*a*c)^(1/2) + 65*b^3*c^3*d^4*e^3*(b^2 - 4*a*c)^(1/2) - (185*b^4*c^2*d^3*e^4*(b^2 - 4*
a*c)^(1/2))/4 - 64*a*b^5*c*e^7*x + 28*b*c^6*d^6*e*x - 48*b^6*c*d*e^6*x - 504*a^2*b^2*c^3*d^2*e^5 - 21*b*c*d^2*
e^5*(b^2 - 4*a*c)^(5/2) + 8*b*c^5*d^6*e*(b^2 - 4*a*c)^(1/2) + 44*c^6*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 164*a*b*c^5
*d^5*e^2 - 348*a^3*b*c^3*d*e^6 - 108*a^3*b*c^3*e^7*x + 200*a*c^6*d^5*e^2*x + 216*a^3*c^4*d*e^6*x - 37*b*c^3*d^
4*e^3*(b^2 - 4*a*c)^(3/2) + 7*b^3*c*d^2*e^5*(b^2 - 4*a*c)^(3/2) + 18*b^5*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) + (57*c
^2*d^2*e^5*x*(b^2 - 4*a*c)^(5/2))/4 + 51*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(3/2) + 284*a*b^2*c^4*d^4*e^3 - 228*a*b^3
*c^3*d^3*e^4 + 124*a*b^4*c^2*d^2*e^5 + 516*a^2*b*c^4*d^3*e^4 + 240*a^2*b^3*c^2*d*e^6 + 156*a^2*b^3*c^2*e^7*x -
600*a^2*c^5*d^3*e^4*x - 92*b^2*c^5*d^5*e^2*x + 160*b^3*c^4*d^4*e^3*x - 180*b^4*c^3*d^3*e^4*x + 124*b^5*c^2*d^
2*e^5*x - 102*b*c^3*d^3*e^4*x*(b^2 - 4*a*c)^(3/2) - 132*b*c^5*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 800*a*b^2*c^4*d^
3*e^4*x - 700*a*b^3*c^3*d^2*e^5*x + 900*a^2*b*c^4*d^2*e^5*x - 612*a^2*b^2*c^3*d*e^6*x - (57*b*c*d*e^6*x*(b^2 -
4*a*c)^(5/2))/4 + (153*b^2*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(3/2))/2 + 165*b^2*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(1/2) -
110*b^3*c^3*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) + (165*b^4*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2))/4 - (51*b^3*c*d*e^6*x
*(b^2 - 4*a*c)^(3/2))/2 - (33*b^5*c*d*e^6*x*(b^2 - 4*a*c)^(1/2))/4 - 500*a*b*c^5*d^4*e^3*x + 328*a*b^4*c^2*d*e
^6*x)*(e^3*((3*c*d*(b^2 - 4*a*c)^(3/2))/2 - 3*b*c*d*(4*a*c - b^2) + (3*b^2*c*d*(b^2 - 4*a*c)^(1/2))/2) - e^4*(
(4*a*c - b^2)^2/4 + (3*b*(b^2 - 4*a*c)^(3/2))/4 - (3*b^2*(4*a*c - b^2))/4 + (b^3*(b^2 - 4*a*c)^(1/2))/4) + e^2
*(3*c^2*d^2*(4*a*c - b^2) - 3*b*c^2*d^2*(b^2 - 4*a*c)^(1/2)) + 2*c^3*d^3*e*(b^2 - 4*a*c)^(1/2)))/((4*a*c - b^2
)*((4*a*c - b^2)*((3*a*d^2*e^4)/4 - (3*b*d^3*e^3)/2 + (3*c*d^4*e^2)/4) + a^3*e^6 + c^3*d^6 - (5*b^3*d^3*e^3)/2
+ (15*a*b^2*d^2*e^4)/4 + (15*b^2*c*d^4*e^2)/4 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e)) - (log((27*d*e^6*(b^2 - 4*a*c
)^(7/2))/16 + (9*e^7*x*(b^2 - 4*a*c)^(7/2))/16 - 8*a*b^6*e^7 + 4*b*c^6*d^7 - 8*b^7*e^7*x + 8*c^7*d^7*x + 4*c^6
*d^7*(b^2 - 4*a*c)^(1/2) + 72*a^4*c^3*e^7 + (57*b^2*e^7*x*(b^2 - 4*a*c)^(5/2))/16 + (51*b^4*e^7*x*(b^2 - 4*a*c
)^(3/2))/16 + (11*b^6*e^7*x*(b^2 - 4*a*c)^(1/2))/16 + 60*a^2*b^4*c*e^7 + 8*b^2*c^5*d^6*e + 4*b^6*c*d^2*e^5 + (
75*c^2*d^3*e^4*(b^2 - 4*a*c)^(5/2))/4 + 25*c^4*d^5*e^2*(b^2 - 4*a*c)^(3/2) - 132*a^3*b^2*c^2*e^7 + 408*a^2*c^5
*d^4*e^3 - 456*a^3*c^4*d^2*e^5 - 20*b^3*c^4*d^5*e^2 + 28*b^4*c^3*d^4*e^3 - 16*b^5*c^2*d^3*e^4 + (9*a*b*e^7*(b^
2 - 4*a*c)^(5/2))/4 - 88*a*c^6*d^6*e + (9*a*b^3*e^7*(b^2 - 4*a*c)^(3/2))/2 + (5*a*b^5*e^7*(b^2 - 4*a*c)^(1/2))
/4 + (111*b^2*d*e^6*(b^2 - 4*a*c)^(5/2))/16 - (79*b^4*d*e^6*(b^2 - 4*a*c)^(3/2))/16 - (59*b^6*d*e^6*(b^2 - 4*a
*c)^(1/2))/16 + 40*a*b^5*c*d*e^6 + (23*b^2*c^2*d^3*e^4*(b^2 - 4*a*c)^(3/2))/2 - 45*b^2*c^4*d^5*e^2*(b^2 - 4*a*
c)^(1/2) + 65*b^3*c^3*d^4*e^3*(b^2 - 4*a*c)^(1/2) - (185*b^4*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2))/4 + 64*a*b^5*c*e
^7*x - 28*b*c^6*d^6*e*x + 48*b^6*c*d*e^6*x + 504*a^2*b^2*c^3*d^2*e^5 - 21*b*c*d^2*e^5*(b^2 - 4*a*c)^(5/2) + 8*
b*c^5*d^6*e*(b^2 - 4*a*c)^(1/2) + 44*c^6*d^6*e*x*(b^2 - 4*a*c)^(1/2) + 164*a*b*c^5*d^5*e^2 + 348*a^3*b*c^3*d*e
^6 + 108*a^3*b*c^3*e^7*x - 200*a*c^6*d^5*e^2*x - 216*a^3*c^4*d*e^6*x - 37*b*c^3*d^4*e^3*(b^2 - 4*a*c)^(3/2) +
7*b^3*c*d^2*e^5*(b^2 - 4*a*c)^(3/2) + 18*b^5*c*d^2*e^5*(b^2 - 4*a*c)^(1/2) + (57*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(
5/2))/4 + 51*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(3/2) - 284*a*b^2*c^4*d^4*e^3 + 228*a*b^3*c^3*d^3*e^4 - 124*a*b^4*c^2
*d^2*e^5 - 516*a^2*b*c^4*d^3*e^4 - 240*a^2*b^3*c^2*d*e^6 - 156*a^2*b^3*c^2*e^7*x + 600*a^2*c^5*d^3*e^4*x + 92*
b^2*c^5*d^5*e^2*x - 160*b^3*c^4*d^4*e^3*x + 180*b^4*c^3*d^3*e^4*x - 124*b^5*c^2*d^2*e^5*x - 102*b*c^3*d^3*e^4*
x*(b^2 - 4*a*c)^(3/2) - 132*b*c^5*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) - 800*a*b^2*c^4*d^3*e^4*x + 700*a*b^3*c^3*d^2*
e^5*x - 900*a^2*b*c^4*d^2*e^5*x + 612*a^2*b^2*c^3*d*e^6*x - (57*b*c*d*e^6*x*(b^2 - 4*a*c)^(5/2))/4 + (153*b^2*
c^2*d^2*e^5*x*(b^2 - 4*a*c)^(3/2))/2 + 165*b^2*c^4*d^4*e^3*x*(b^2 - 4*a*c)^(1/2) - 110*b^3*c^3*d^3*e^4*x*(b^2
- 4*a*c)^(1/2) + (165*b^4*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2))/4 - (51*b^3*c*d*e^6*x*(b^2 - 4*a*c)^(3/2))/2 - (3
3*b^5*c*d*e^6*x*(b^2 - 4*a*c)^(1/2))/4 + 500*a*b*c^5*d^4*e^3*x - 328*a*b^4*c^2*d*e^6*x)*(e^3*((3*c*d*(b^2 - 4*
a*c)^(3/2))/2 + 3*b*c*d*(4*a*c - b^2) + (3*b^2*c*d*(b^2 - 4*a*c)^(1/2))/2) - e^4*((3*b*(b^2 - 4*a*c)^(3/2))/4
- (4*a*c - b^2)^2/4 + (3*b^2*(4*a*c - b^2))/4 + (b^3*(b^2 - 4*a*c)^(1/2))/4) - e^2*(3*c^2*d^2*(4*a*c - b^2) +
3*b*c^2*d^2*(b^2 - 4*a*c)^(1/2)) + 2*c^3*d^3*e*(b^2 - 4*a*c)^(1/2)))/((4*a*c - b^2)*((4*a*c - b^2)*((3*a*d^2*e
^4)/4 - (3*b*d^3*e^3)/2 + (3*c*d^4*e^2)/4) + a^3*e^6 + c^3*d^6 - (5*b^3*d^3*e^3)/2 + (15*a*b^2*d^2*e^4)/4 + (1
5*b^2*c*d^4*e^2)/4 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e))

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

[Out]

Timed out

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