3.16.45 \(\int \frac {b+2 c x}{(d+e x) (a+b x+c x^2)^3} \, dx\) [1545]

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

[Out]

1/2*(-(-4*a*c+b^2)*(-b*e+c*d)+c*(-4*a*c+b^2)*e*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^2-1/2*e*(3*b^
2*c*d*e-8*a*c^2*d*e-2*b^3*e^2-b*c*(-7*a*e^2+c*d^2)-2*c*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*x)/(-4*a*c+b^2)/(a*e^
2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)-e*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b*d)-6*a*c^2*e^3*(a*e+2*b*d)+2*b^2*c*e
^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^3-e^4*(-b*e+2*c*d
)*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3+1/2*e^4*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^3

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Rubi [A]
time = 0.71, 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 = {836, 814, 648, 632, 212, 642} \begin {gather*} -\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 \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((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)*(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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 814

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 836

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 ) \text {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 = 0.75, size = 356, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (\frac {-c d+b e+c e x}{\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^2}+\frac {e \left (2 b^3 e^2+b^2 c e (-3 d+2 e x)+2 c^2 \left (c d^2 x+a e (4 d-3 e x)\right )+b c \left (-7 a e^2+c d (d-2 e x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 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 ) \tan ^{-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+e (b d-a e)\right )^3}+\frac {2 e^4 (-2 c d+b e) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^4 (2 c d-b e) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(874\) vs. \(2(387)=774\).
time = 1.52, size = 875, normalized size = 2.20

method	result	size
default	\(\frac {\frac {\frac {c^{2} e \left (3 e^{4} a^{2} c -a \,b^{2} e^{4}-2 a b c d \,e^{3}+2 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}-2 b^{2} c \,d^{2} e^{2}+2 d^{3} e b \,c^{2}-d^{4} c^{3}\right ) x^{3}}{4 a c -b^{2}}+\frac {c e \left (13 c \,e^{4} a^{2} b -8 a^{2} c^{2} d \,e^{3}-4 a \,b^{3} e^{4}-8 a \,b^{2} c d \,e^{3}+18 a b \,c^{2} d^{2} e^{2}-8 a \,c^{3} d^{3} e +4 b^{4} d \,e^{3}-9 b^{3} c \,d^{2} e^{2}+8 b^{2} c^{2} d^{3} e -3 d^{4} b \,c^{3}\right ) x^{2}}{8 a c -2 b^{2}}+\frac {e \left (5 a^{3} c^{2} e^{4}+2 a^{2} b^{2} c \,e^{4}-10 a^{2} b \,c^{2} d \,e^{3}+6 a^{2} c^{3} d^{2} e^{2}-a \,b^{4} e^{4}+6 a \,b^{2} c^{2} d^{2} e^{2}-6 a b \,c^{3} d^{3} e +a \,c^{4} d^{4}+b^{5} d \,e^{3}-3 b^{4} c \,d^{2} e^{2}+3 b^{3} c^{2} d^{3} e -b^{2} c^{3} d^{4}\right ) x}{4 a c -b^{2}}+\frac {11 a^{3} b c \,e^{5}-12 a^{3} c^{2} d \,e^{4}-3 a^{2} b^{3} e^{5}-11 a^{2} b^{2} c d \,e^{4}+30 a^{2} b \,c^{2} d^{2} e^{3}-16 a^{2} c^{3} d^{3} e^{2}+4 a \,b^{4} d \,e^{4}-5 a \,b^{3} c \,d^{2} e^{3}-6 a \,b^{2} c^{2} d^{3} e^{2}+11 a b \,c^{3} d^{4} e -4 a \,c^{4} d^{5}-b^{5} d^{2} e^{3}+3 b^{4} c \,d^{3} e^{2}-3 b^{3} c^{2} d^{4} e +b^{2} c^{3} d^{5}}{8 a c -2 b^{2}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {e \left (\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (3 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 d^{2} e^{2} c^{3} a +b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}+\frac {\left (b e -2 c d \right ) e^{4} \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\)	\(875\)
risch	\(\text {Expression too large to display}\)	\(56735\)

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,method=_RETURNVERBOSE)

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3212 vs. \(2 (394) = 788\).
time = 152.71, size = 6444, normalized size = 16.23 \begin {gather*} \text {Too large to display} \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="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1117 vs. \(2 (394) = 788\).
time = 1.16, 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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Mupad [B]
time = 5.46, size = 2461, normalized size = 6.20 \begin {gather*} \frac {\ln \left (\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c+8\,a\,c^2\,x-2\,b^2\,c\,x\right )\,\left (b^7\,e^5+b^4\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-64\,a^3\,b\,c^3\,e^5+128\,a^3\,c^4\,d\,e^4-2\,c^4\,d^4\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a^2\,b^3\,c^2\,e^5+6\,a^2\,c^2\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a\,b^5\,c\,e^5-2\,b^6\,c\,d\,e^4-6\,a\,b^2\,c\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a\,b^4\,c^2\,d\,e^4-2\,b^3\,c\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-96\,a^2\,b^2\,c^3\,d\,e^4-12\,a\,c^3\,d^2\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+4\,b\,c^3\,d^3\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a\,b\,c^2\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,\left (64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6\right )}-\frac {\frac {-11\,a^2\,b\,c\,e^3+12\,a^2\,c^2\,d\,e^2+3\,a\,b^3\,e^3-7\,a\,b\,c^2\,d^2\,e+4\,a\,c^3\,d^3-b^4\,d\,e^2+2\,b^3\,c\,d^2\,e-b^2\,c^2\,d^3}{2\,\left (4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4\right )}-\frac {x\,\left (5\,a^2\,c^2\,e^3+2\,a\,b^2\,c\,e^3-5\,a\,b\,c^2\,d\,e^2+a\,c^3\,d^2\,e-b^4\,e^3+2\,b^3\,c\,d\,e^2-b^2\,c^2\,d^2\,e\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}+\frac {x^2\,\left (4\,b^3\,c\,e^3-5\,b^2\,c^2\,d\,e^2+3\,b\,c^3\,d^2\,e-13\,a\,b\,c^2\,e^3+8\,a\,c^3\,d\,e^2\right )}{2\,\left (4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4\right )}+\frac {e\,x^3\,\left (b^2\,c^2\,e^2-b\,c^3\,d\,e+c^4\,d^2-3\,a\,c^3\,e^2\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b\,e^5-2\,c\,d\,e^4\right )}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c-8\,a\,c^2\,x+2\,b^2\,c\,x\right )\,\left (\frac {b^7\,e^5}{2}-\frac {b^4\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-32\,a^3\,b\,c^3\,e^5+64\,a^3\,c^4\,d\,e^4+c^4\,d^4\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^3\,c^2\,e^5-3\,a^2\,c^2\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b^5\,c\,e^5-b^6\,c\,d\,e^4+3\,a\,b^2\,c\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a\,b^4\,c^2\,d\,e^4+b^3\,c\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-48\,a^2\,b^2\,c^3\,d\,e^4+6\,a\,c^3\,d^2\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,b\,c^3\,d^3\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b\,c^2\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6} \end {gather*}

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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