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

Optimal. Leaf size=397 \[ \frac {c e x \left (b^2-4 a c\right )-\left (b^2-4 a c\right ) (c d-b e)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

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Rubi [A]  time = 0.82, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 8, 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 \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e+3 b^2 c d e-2 b^3 e^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \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 )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^
2) - (e*(3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3
*a*e))*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (e*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2
*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3) - (e^4*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)
^3 + (e^4*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(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) \left (a+b x+c x^2\right )^3} \, dx &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\left (b^2-4 a c\right ) e (c d-2 b e)-3 c \left (b^2-4 a c\right ) e^2 x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \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}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {2 \left (b^2-4 a c\right ) e \left (c^3 d^3+b^3 e^3-c^2 d e (b d-5 a e)-b c e^2 (b d+4 a e)\right )+2 c \left (b^2-4 a c\right ) e^2 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (b^2-4 a c\right )^2 e^5 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {2 \left (b^2-4 a c\right ) e \left (c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e \int \frac {c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\left (e^4 (2 c d-b e)\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (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 (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end {align*}

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Mathematica [A]  time = 1.11, size = 356, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (-\frac {2 e \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\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} \left (e (b d-a e)-c d^2\right )^3}+\frac {e \left (b c \left (c d (d-2 e x)-7 a e^2\right )+2 c^2 \left (a e (4 d-3 e x)+c d^2 x\right )+2 b^3 e^2+b^2 c e (2 e x-3 d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}+\frac {2 e^4 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac {e^4 (2 c d-b e) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}+\frac {b e-c d+c e x}{(a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(c*d) + b*e + c*e*x)/((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^2) + (e*(2*b^3*e^2 + b^2*c*e*(-3*d + 2*e
*x) + 2*c^2*(c*d^2*x + a*e*(4*d - 3*e*x)) + b*c*(-7*a*e^2 + c*d*(d - 2*e*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*
d) + a*e))^2*(a + x*(b + c*x))) - (2*e*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d
+ a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(-(c*d^2) +
e*(b*d - a*e))^3) + (2*e^4*(-2*c*d + b*e)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e^4*(2*c*d - b*e)*Log[
a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2

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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) \left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 83.85, size = 6667, normalized size = 16.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^5 - (3*b^5*c^2 - 23*a*b^3*c^3 + 44*a^2*b*c^4)*d^4*e + (3*b^6*c -
18*a*b^4*c^2 + 8*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*e^2 - (b^7 + a*b^5*c - 50*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^2*e^3
+ (4*a*b^6 - 27*a^2*b^4*c + 32*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^4 - (3*a^2*b^5 - 23*a^3*b^3*c + 44*a^4*b*c^2)*e^
5 - 2*((b^2*c^5 - 4*a*c^6)*d^4*e - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e^2 + 2*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^2
*e^3 - (b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)*d*e^4 + (a*b^4*c^2 - 7*a^2*b^2*c^3 + 12*a^3*c^4)*e^5)*x^3 - (3*(b
^3*c^4 - 4*a*b*c^5)*d^4*e - 8*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^3*e^2 + 9*(b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b
*c^4)*d^2*e^3 - 4*(b^6*c - 6*a*b^4*c^2 + 6*a^2*b^2*c^3 + 8*a^3*c^4)*d*e^4 + (4*a*b^5*c - 29*a^2*b^3*c^2 + 52*a
^3*b*c^3)*e^5)*x^2 - (2*a^2*c^4*d^4*e - 4*a^2*b*c^3*d^3*e^2 + 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2)
*d*e^4 - (a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*
c^3 - 6*a*b*c^4)*d*e^4 - (b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 +
12*a*b*c^4*d^2*e^3 + 2*(b^4*c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2
*c^4 + 2*a*c^5)*d^4*e - 4*(b^3*c^3 + 2*a*b*c^4)*d^3*e^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*
b^3*c^2 - 12*a^2*b*c^3)*d*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*d^4*e -
4*a*b^2*c^3*d^3*e^2 + 12*a^2*b*c^3*d^2*e^3 + 2*(a*b^4*c - 6*a^2*b^2*c^2)*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3
*b*c^2)*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)) - 2*((b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 3*(b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)*d^3*e^
2 + 3*(b^6*c - 6*a*b^4*c^2 + 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^3 - (b^7 - 4*a*b^5*c - 10*a^2*b^3*c^2 + 40*a^3*b
*c^3)*d*e^4 + (a*b^6 - 6*a^2*b^4*c + 3*a^3*b^2*c^2 + 20*a^4*c^3)*e^5)*x - (2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a
^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (
b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c -
8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a
^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c
^2)*e^5)*x)*log(c*x^2 + b*x + a) + 2*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*
c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)
*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^
3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c -
8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*log(e*x + d))/((a^2*b^4*c
^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^6 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e + 3*(a^2*b^6*c - 7
*a^3*b^4*c^2 + 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^4*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b*c^3)*d
^3*e^3 + 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^2*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*
c^2)*d*e^5 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*e^6 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6 - 3*(b^5*c^4
- 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*e + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e^2 - (b^7*c
^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^3 + 3*(a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*
a^4*c^5)*d^2*e^4 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^5 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^
5*c^4)*e^6)*x^4 + 2*((b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6 - 3*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d
^5*e + 3*(b^7*c^2 - 7*a*b^5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^4*e^2 - (b^8*c - 2*a*b^6*c^2 - 32*a^2*b^4*c^
3 + 96*a^3*b^2*c^4)*d^3*e^3 + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^4 - 3*(a^2*b^6*
c - 8*a^3*b^4*c^2 + 16*a^4*b^2*c^3)*d*e^5 + (a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^6)*x^3 + ((b^6*c^3 -
6*a*b^4*c^4 + 32*a^3*c^6)*d^6 - 3*(b^7*c^2 - 6*a*b^5*c^3 + 32*a^3*b*c^5)*d^5*e + 3*(b^8*c - 5*a*b^6*c^2 - 6*a^
2*b^4*c^3 + 32*a^3*b^2*c^4 + 32*a^4*c^5)*d^4*e^2 - (b^9 - 36*a^2*b^5*c^2 + 32*a^3*b^3*c^3 + 192*a^4*b*c^4)*d^3
*e^3 + 3*(a*b^8 - 5*a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^4*b^2*c^3 + 32*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 6*a^3*b^5*
c + 32*a^5*b*c^3)*d*e^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^6)*x^2 + 2*((a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a
^3*b*c^5)*d^6 - 3*(a*b^6*c^2 - 8*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^5*e + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*
c^3 + 16*a^4*b*c^4)*d^4*e^2 - (a*b^8 - 2*a^2*b^6*c - 32*a^3*b^4*c^2 + 96*a^4*b^2*c^3)*d^3*e^3 + 3*(a^2*b^7 - 7
*a^3*b^5*c + 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2*e^4 - 3*(a^3*b^6 - 8*a^4*b^4*c + 16*a^5*b^2*c^2)*d*e^5 + (a^4*b
^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^6)*x), -1/2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^5 - (3*b^5*c^2 - 23*a*b
^3*c^3 + 44*a^2*b*c^4)*d^4*e + (3*b^6*c - 18*a*b^4*c^2 + 8*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*e^2 - (b^7 + a*b^5*c
- 50*a^2*b^3*c^2 + 120*a^3*b*c^3)*d^2*e^3 + (4*a*b^6 - 27*a^2*b^4*c + 32*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^4 - (3*
a^2*b^5 - 23*a^3*b^3*c + 44*a^4*b*c^2)*e^5 - 2*((b^2*c^5 - 4*a*c^6)*d^4*e - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e^2 +
2*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^2*e^3 - (b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)*d*e^4 + (a*b^4*c^2 - 7*a
^2*b^2*c^3 + 12*a^3*c^4)*e^5)*x^3 - (3*(b^3*c^4 - 4*a*b*c^5)*d^4*e - 8*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^3
*e^2 + 9*(b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)*d^2*e^3 - 4*(b^6*c - 6*a*b^4*c^2 + 6*a^2*b^2*c^3 + 8*a^3*c^4)*d
*e^4 + (4*a*b^5*c - 29*a^2*b^3*c^2 + 52*a^3*b*c^3)*e^5)*x^2 + 2*(2*a^2*c^4*d^4*e - 4*a^2*b*c^3*d^3*e^2 + 12*a^
3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2)*d*e^4 - (a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4
*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*c^3 - 6*a*b*c^4)*d*e^4 - (b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x
^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12*a*b*c^4*d^2*e^3 + 2*(b^4*c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*
a*b^3*c^2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*e - 4*(b^3*c^3 + 2*a*b*c^4)*d^3*e^2 + 12*(a*b^2
*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*b^3*c^2 - 12*a^2*b*c^3)*d*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 +
12*a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*d^4*e - 4*a*b^2*c^3*d^3*e^2 + 12*a^2*b*c^3*d^2*e^3 + 2*(a*b^4*c - 6*a^2*b^
2*c^2)*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*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)) - 2*((b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 3*(b^5*c^2 - 6*a*b^3*c^3 + 8*a^2*b*c^4)
*d^3*e^2 + 3*(b^6*c - 6*a*b^4*c^2 + 6*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e^3 - (b^7 - 4*a*b^5*c - 10*a^2*b^3*c^2 + 4
0*a^3*b*c^3)*d*e^4 + (a*b^6 - 6*a^2*b^4*c + 3*a^3*b^2*c^2 + 20*a^4*c^3)*e^5)*x - (2*(a^2*b^4*c - 8*a^3*b^2*c^2
+ 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*
e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b
^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c
+ 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^
3*b^2*c^2)*e^5)*x)*log(c*x^2 + b*x + a) + 2*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a
^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2
*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*
e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b
^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*log(e*x + d))/((a^
2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^6 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e + 3*(a^2*b^
6*c - 7*a^3*b^4*c^2 + 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^4*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b
*c^3)*d^3*e^3 + 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^2*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16
*a^6*b*c^2)*d*e^5 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*e^6 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6 - 3*(
b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*e + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e^2 -
(b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^3 + 3*(a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^
4 + 16*a^4*c^5)*d^2*e^4 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^5 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3
+ 16*a^5*c^4)*e^6)*x^4 + 2*((b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6 - 3*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2
*c^5)*d^5*e + 3*(b^7*c^2 - 7*a*b^5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^4*e^2 - (b^8*c - 2*a*b^6*c^2 - 32*a^2
*b^4*c^3 + 96*a^3*b^2*c^4)*d^3*e^3 + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^4 - 3*(a
^2*b^6*c - 8*a^3*b^4*c^2 + 16*a^4*b^2*c^3)*d*e^5 + (a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^6)*x^3 + ((b^6
*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*d^6 - 3*(b^7*c^2 - 6*a*b^5*c^3 + 32*a^3*b*c^5)*d^5*e + 3*(b^8*c - 5*a*b^6*c^2
- 6*a^2*b^4*c^3 + 32*a^3*b^2*c^4 + 32*a^4*c^5)*d^4*e^2 - (b^9 - 36*a^2*b^5*c^2 + 32*a^3*b^3*c^3 + 192*a^4*b*c
^4)*d^3*e^3 + 3*(a*b^8 - 5*a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^4*b^2*c^3 + 32*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 6*a
^3*b^5*c + 32*a^5*b*c^3)*d*e^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^6)*x^2 + 2*((a*b^5*c^3 - 8*a^2*b^3*c^4
+ 16*a^3*b*c^5)*d^6 - 3*(a*b^6*c^2 - 8*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^5*e + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a
^3*b^3*c^3 + 16*a^4*b*c^4)*d^4*e^2 - (a*b^8 - 2*a^2*b^6*c - 32*a^3*b^4*c^2 + 96*a^4*b^2*c^3)*d^3*e^3 + 3*(a^2*
b^7 - 7*a^3*b^5*c + 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2*e^4 - 3*(a^3*b^6 - 8*a^4*b^4*c + 16*a^5*b^2*c^2)*d*e^5 +
(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^6)*x)]

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giac [B]  time = 0.28, size = 1117, normalized size = 2.81 \begin {gather*} \frac {{\left (2 \, c d e^{4} - b e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\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 )}} - \frac {{\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac {{\left (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} d^{6} - 4 \, a c^{4} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 12 \, a b c^{3} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - 9 \, a b^{2} c^{2} d^{4} e^{2} - 12 \, a^{2} c^{3} d^{4} e^{2} - b^{5} d^{3} e^{3} - 2 \, a b^{3} c d^{3} e^{3} + 24 \, a^{2} b c^{2} d^{3} e^{3} + 3 \, a b^{4} d^{2} e^{4} - 9 \, a^{2} b^{2} c d^{2} e^{4} - 12 \, a^{3} c^{2} d^{2} e^{4} - 3 \, a^{2} b^{3} d e^{5} + 12 \, a^{3} b c d e^{5} + a^{3} b^{2} e^{6} - 4 \, a^{4} c e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 11 \, a b c^{3} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - 6 \, a b^{2} c^{2} d^{3} e^{2} - 16 \, a^{2} c^{3} d^{3} e^{2} - b^{5} d^{2} e^{3} - 5 \, a b^{3} c d^{2} e^{3} + 30 \, a^{2} b c^{2} d^{2} e^{3} + 4 \, a b^{4} d e^{4} - 11 \, a^{2} b^{2} c d e^{4} - 12 \, a^{3} c^{2} d e^{4} - 3 \, a^{2} b^{3} e^{5} + 11 \, a^{3} b c e^{5} - 2 \, {\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{2} c^{3} d^{2} e^{3} - 2 \, a c^{4} d^{2} e^{3} - b^{3} c^{2} d e^{4} + 2 \, a b c^{3} d e^{4} + a b^{2} c^{2} e^{5} - 3 \, a^{2} c^{3} e^{5}\right )} x^{3} - {\left (3 \, b c^{4} d^{4} e - 8 \, b^{2} c^{3} d^{3} e^{2} + 8 \, a c^{4} d^{3} e^{2} + 9 \, b^{3} c^{2} d^{2} e^{3} - 18 \, a b c^{3} d^{2} e^{3} - 4 \, b^{4} c d e^{4} + 8 \, a b^{2} c^{2} d e^{4} + 8 \, a^{2} c^{3} d e^{4} + 4 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} - 2 \, {\left (b^{2} c^{3} d^{4} e - a c^{4} d^{4} e - 3 \, b^{3} c^{2} d^{3} e^{2} + 6 \, a b c^{3} d^{3} e^{2} + 3 \, b^{4} c d^{2} e^{3} - 6 \, a b^{2} c^{2} d^{2} e^{3} - 6 \, a^{2} c^{3} d^{2} e^{3} - b^{5} d e^{4} + 10 \, a^{2} b c^{2} d e^{4} + a b^{4} e^{5} - 2 \, a^{2} b^{2} c e^{5} - 5 \, a^{3} c^{2} e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*c*d*e^4 - b*e^5)*log(c*x^2 + b*x + a)/(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^5 - b*e^
6)*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*
d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 12*a
*c^3*d^2*e^3 + 2*b^3*c*d*e^4 - 12*a*b*c^2*d*e^4 - b^4*e^5 + 6*a*b^2*c*e^5 - 6*a^2*c^2*e^5)*arctan((2*c*x + b)/
sqrt(-b^2 + 4*a*c))/((b^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b
^2*c^2*d^4*e^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^2*e^4
- 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*
sqrt(-b^2 + 4*a*c)) - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 - 3*b^3*c^2*d^4*e + 11*a*b*c^3*d^4*e + 3*b^4*c*d^3*e^2 -
6*a*b^2*c^2*d^3*e^2 - 16*a^2*c^3*d^3*e^2 - b^5*d^2*e^3 - 5*a*b^3*c*d^2*e^3 + 30*a^2*b*c^2*d^2*e^3 + 4*a*b^4*d*
e^4 - 11*a^2*b^2*c*d*e^4 - 12*a^3*c^2*d*e^4 - 3*a^2*b^3*e^5 + 11*a^3*b*c*e^5 - 2*(c^5*d^4*e - 2*b*c^4*d^3*e^2
+ 2*b^2*c^3*d^2*e^3 - 2*a*c^4*d^2*e^3 - b^3*c^2*d*e^4 + 2*a*b*c^3*d*e^4 + a*b^2*c^2*e^5 - 3*a^2*c^3*e^5)*x^3 -
(3*b*c^4*d^4*e - 8*b^2*c^3*d^3*e^2 + 8*a*c^4*d^3*e^2 + 9*b^3*c^2*d^2*e^3 - 18*a*b*c^3*d^2*e^3 - 4*b^4*c*d*e^4
+ 8*a*b^2*c^2*d*e^4 + 8*a^2*c^3*d*e^4 + 4*a*b^3*c*e^5 - 13*a^2*b*c^2*e^5)*x^2 - 2*(b^2*c^3*d^4*e - a*c^4*d^4*
e - 3*b^3*c^2*d^3*e^2 + 6*a*b*c^3*d^3*e^2 + 3*b^4*c*d^2*e^3 - 6*a*b^2*c^2*d^2*e^3 - 6*a^2*c^3*d^2*e^3 - b^5*d*
e^4 + 10*a^2*b*c^2*d*e^4 + a*b^4*e^5 - 2*a^2*b^2*c*e^5 - 5*a^3*c^2*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(c*x^2 +
b*x + a)^2*(b^2 - 4*a*c))

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maple [B]  time = 0.08, size = 3233, normalized size = 8.14 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*a^2-2/(a*e^2-b*d*e+c*d^2
)^3*e/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^4-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^
3*e^4/(4*a*c-b^2)*x^3*a*b*d-6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*a*b*c^3*d^3-4/(a*e^2-b*d
*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^4/(4*a*c-b^2)*x^2*a*b^2*d+9/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^3/(4
*a*c-b^2)*x^2*a*b*d^2+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^3/(4*a*c-b^2)*x^3*a*d^2+6/(a*e^2-b*d*e+c*d
^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a^2*c^3*d^2+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x*a*
c^4*d^4+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^3*b*d^3+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b
*x+a)^2*c*e^4/(4*a*c-b^2)*x^2*b^4*d-9/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^3/(4*a*c-b^2)*x^2*b^3*d^2-
4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^4/(4*a*c-b^2)*x^2*a^2*d-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*
c*e^5/(4*a*c-b^2)*x^2*a*b^3-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^2*a*d^3+1/2/(a*e^2-b
*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^2*c^3*d^5-5/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*
b^3*c*d^2*e^3+13/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b^2)*x^2*a^2*b+1/(a*e^2-b*d*e+c*d^2)^3
/(c*x^2+b*x+a)^2*c^2*e^4/(4*a*c-b^2)*x^3*b^3*d-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^3/(4*a*c-b^2)*x^3
*b^2*d^2-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^2*c^2*d^3*e^2+11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x
^2+b*x+a)^2/(4*a*c-b^2)*a*b*c^3*d^4*e-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b^2)*x^3*a*b^2+2/
(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^2*b^2*c+12/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)^(3/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^2*d-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*b^4
*c*d^2+3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*b^3*c^2*d^3-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*
x+a)^2*e/(4*a*c-b^2)*x*b^2*c^3*d^4-11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^2*c*d*e^4+15/(
a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b*c^2*d^2*e^3-6/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)^(3/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*c-1/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d
-12/(a*e^2-b*d*e+c*d^2)^3*e^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^3*d^2-2/(a*e^2-b*d*e+c
*d^2)^3*e^4/(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*e^2-b*d*e+c*d^2)^3*e^2/(4*a*c-b
^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^3*d^3-2/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)*c*ln(c*x^2+b*x
+a)*a*b+4/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*a*d-10/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)
^2*e^4/(4*a*c-b^2)*x*a^2*b*c^2*d+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a*b^2*c^2*d^2+4/(a*
e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^2/(4*a*c-b^2)*x^2*b^2*d^3-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c
^4*e/(4*a*c-b^2)*x^2*b*d^4+11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*b*c*e^5-6/(a*e^2-b*d*e+c
*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*c^2*d*e^4-8/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*c^3*
d^3*e^2+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^4*d*e^4+3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)^2/(4*a*c-b^2)*b^4*c*d^3*e^2-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^3*c^2*d^4*e+3/(a*e^2-b*d
*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^5/(4*a*c-b^2)*x^3*a^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^5*e/(4*a*c-b
^2)*x^3*d^4-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^3*e^5-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b
*x+a)^2/(4*a*c-b^2)*a*c^4*d^5-1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^5*d^2*e^3+e^5/(a*e^2-b*d
*e+c*d^2)^3*ln(e*x+d)*b+1/2/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3+1/(a*e^2-b*d*e+c*d^2)^3*
e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c*d+5/(a*e
^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^3*c^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c
-b^2)*x*a*b^4+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*b^5*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((2*c*x+b)/(e*x+d)/(c*x^2+b*x+a)^3,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 = 5.46, size = 2461, normalized size = 6.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((-(4*a*c - b^2)^3)^(1/2) - b^3 + 4*a*b*c + 8*a*c^2*x - 2*b^2*c*x)*(b^7*e^5 + b^4*e^5*(-(4*a*c - b^2)^3)^(
1/2) - 64*a^3*b*c^3*e^5 + 128*a^3*c^4*d*e^4 - 2*c^4*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 48*a^2*b^3*c^2*e^5 + 6*a^
2*c^2*e^5*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^5*c*e^5 - 2*b^6*c*d*e^4 - 6*a*b^2*c*e^5*(-(4*a*c - b^2)^3)^(1/2) +
24*a*b^4*c^2*d*e^4 - 2*b^3*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 96*a^2*b^2*c^3*d*e^4 - 12*a*c^3*d^2*e^3*(-(4*a*
c - b^2)^3)^(1/2) + 4*b*c^3*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2)))/(2*
(64*a^3*c^6*d^6 - a^3*b^6*e^6 + 64*a^6*c^3*e^6 - b^6*c^3*d^6 + b^9*d^3*e^3 + 12*a*b^4*c^4*d^6 + 12*a^4*b^4*c*e
^6 - 3*a*b^8*d^2*e^4 + 3*a^2*b^7*d*e^5 + 3*b^7*c^2*d^5*e - 3*b^8*c*d^4*e^2 - 48*a^2*b^2*c^5*d^6 - 48*a^5*b^2*c
^2*e^6 + 192*a^4*c^5*d^4*e^2 + 192*a^5*c^4*d^2*e^4 - 108*a^2*b^4*c^3*d^4*e^2 - 24*a^2*b^5*c^2*d^3*e^3 + 48*a^3
*b^2*c^4*d^4*e^2 + 224*a^3*b^3*c^3*d^3*e^3 - 108*a^3*b^4*c^2*d^2*e^4 + 48*a^4*b^2*c^3*d^2*e^4 - 36*a*b^5*c^3*d
^5*e - 6*a*b^7*c*d^3*e^3 - 192*a^3*b*c^5*d^5*e - 36*a^3*b^5*c*d*e^5 - 192*a^5*b*c^3*d*e^5 + 33*a*b^6*c^2*d^4*e
^2 + 144*a^2*b^3*c^4*d^5*e + 33*a^2*b^6*c*d^2*e^4 - 384*a^4*b*c^4*d^3*e^3 + 144*a^4*b^3*c^2*d*e^5)) - ((3*a*b^
3*e^3 + 4*a*c^3*d^3 - b^4*d*e^2 - b^2*c^2*d^3 + 12*a^2*c^2*d*e^2 - 11*a^2*b*c*e^3 + 2*b^3*c*d^2*e - 7*a*b*c^2*
d^2*e)/(2*(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d
*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2)) - (x*(5*a^2*c^2*e^3 - b^4*e^3 -
b^2*c^2*d^2*e + 2*a*b^2*c*e^3 + a*c^3*d^2*e + 2*b^3*c*d*e^2 - 5*a*b*c^2*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 -
a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e
- 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) + (x^2*(4*b^3*c*e^3 - 5*b^2*c^2*d*e^2 - 13*a*b*c^2*e^3 + 8*a*c^3*d*e^2
+ 3*b*c^3*d^2*e))/(2*(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2
+ 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2)) + (e*x^3*(c^4*d^2 -
3*a*c^3*e^2 + b^2*c^2*e^2 - b*c^3*d*e))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 +
8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2))/(
x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + (log(d + e*x)*(b*e^5 - 2*c*d*e^4))/(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(b^3 + (-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c - 8*a*c^2*x + 2*b^2*c*x)*((b^7*e
^5)/2 - (b^4*e^5*(-(4*a*c - b^2)^3)^(1/2))/2 - 32*a^3*b*c^3*e^5 + 64*a^3*c^4*d*e^4 + c^4*d^4*e*(-(4*a*c - b^2)
^3)^(1/2) + 24*a^2*b^3*c^2*e^5 - 3*a^2*c^2*e^5*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^5*c*e^5 - b^6*c*d*e^4 + 3*a*b^
2*c*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^4*c^2*d*e^4 + b^3*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 48*a^2*b^2*c^3*
d*e^4 + 6*a*c^3*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*b*c^3*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b*c^2*d*e^4*
(-(4*a*c - b^2)^3)^(1/2)))/(64*a^3*c^6*d^6 - a^3*b^6*e^6 + 64*a^6*c^3*e^6 - b^6*c^3*d^6 + b^9*d^3*e^3 + 12*a*b
^4*c^4*d^6 + 12*a^4*b^4*c*e^6 - 3*a*b^8*d^2*e^4 + 3*a^2*b^7*d*e^5 + 3*b^7*c^2*d^5*e - 3*b^8*c*d^4*e^2 - 48*a^2
*b^2*c^5*d^6 - 48*a^5*b^2*c^2*e^6 + 192*a^4*c^5*d^4*e^2 + 192*a^5*c^4*d^2*e^4 - 108*a^2*b^4*c^3*d^4*e^2 - 24*a
^2*b^5*c^2*d^3*e^3 + 48*a^3*b^2*c^4*d^4*e^2 + 224*a^3*b^3*c^3*d^3*e^3 - 108*a^3*b^4*c^2*d^2*e^4 + 48*a^4*b^2*c
^3*d^2*e^4 - 36*a*b^5*c^3*d^5*e - 6*a*b^7*c*d^3*e^3 - 192*a^3*b*c^5*d^5*e - 36*a^3*b^5*c*d*e^5 - 192*a^5*b*c^3
*d*e^5 + 33*a*b^6*c^2*d^4*e^2 + 144*a^2*b^3*c^4*d^5*e + 33*a^2*b^6*c*d^2*e^4 - 384*a^4*b*c^4*d^3*e^3 + 144*a^4
*b^3*c^2*d*e^5)

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

[Out]

Timed out

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