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

Optimal. Leaf size=303 \[ \frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {(2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 c d-b e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

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Rubi [A]  time = 0.48, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {800, 634, 618, 206, 628} \begin {gather*} \frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {(2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac {2 c d-b e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {e \left (-2 c^2 d^2-b^2 e^2+2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {e (2 c d-b e) \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {3 b^3 c d e^2-b^4 e^3+b c^2 d \left (c d^2-9 a e^2\right )-b^2 c e \left (3 c d^2-4 a e^2\right )+2 a c^2 e \left (3 c d^2-a e^2\right )+c (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {2 c d-b e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {3 b^3 c d e^2-b^4 e^3+b c^2 d \left (c d^2-9 a e^2\right )-b^2 c e \left (3 c d^2-4 a e^2\right )+2 a c^2 e \left (3 c d^2-a e^2\right )+c (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 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 c d-b e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (\left (b^2-4 a c\right ) e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left ((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 {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 {2 c d-b e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (\left (b^2-4 a c\right ) e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+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 c d-b e}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\sqrt {b^2-4 a c} e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (c d^2-b d e+a e^2\right )^3}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \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 = 0.37, size = 268, normalized size = 0.88 \begin {gather*} \frac {\frac {2 \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )}{d+e x}-2 (2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )+(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log (a+x (b+c x))+2 e \sqrt {4 a c-b^2} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+\frac {(2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}}{2 \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)^3*(a + b*x + c*x^2)),x]

[Out]

(((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (2*(c*d^2 + e*(-(b*d) + a*e))*(2*c^2*d^2 + b^2*e^2
- 2*c*e*(b*d + a*e)))/(d + e*x) + 2*Sqrt[-b^2 + 4*a*c]*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*ArcTan[(b
+ 2*c*x)/Sqrt[-b^2 + 4*a*c]] - 2*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Log[d + e*x] + (2*c*d -
b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Log[a + x*(b + c*x)])/(2*(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)^3 \left (a+b x+c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 14.97, size = 1961, normalized size = 6.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(6*c^3*d^5 - 13*b*c^2*d^4*e - a^2*b*e^5 + 2*(5*b^2*c + 2*a*c^2)*d^3*e^2 - 3*(b^3 + 2*a*b*c)*d^2*e^3 + 2*(
2*a*b^2 - a^2*c)*d*e^4 - (3*c^2*d^4*e - 3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (
b^2 - a*c)*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + (b^2 - a*c)*d*e^4)*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*(2*c^3*d^4*e - 4*b*c^2*d^3*e
^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4 + (a*b^2 - 2*a^2*c)*e^5)*x + (2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)
*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a
*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*lo
g(c*x^2 + b*x + a) - 2*(2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2
*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c
^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*log(e*x + d))/(c^3*d^8 - 3*b*c^2*d^7*e -
3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^
4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3
*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*
c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), 1/2*(6*c^3*d^5 - 13*b*c^2*d^4*e
- a^2*b*e^5 + 2*(5*b^2*c + 2*a*c^2)*d^3*e^2 - 3*(b^3 + 2*a*b*c)*d^2*e^3 + 2*(2*a*b^2 - a^2*c)*d*e^4 + 2*(3*c^2
*d^4*e - 3*b*c*d^3*e^2 + (b^2 - a*c)*d^2*e^3 + (3*c^2*d^2*e^3 - 3*b*c*d*e^4 + (b^2 - a*c)*e^5)*x^2 + 2*(3*c^2*
d^3*e^2 - 3*b*c*d^2*e^3 + (b^2 - a*c)*d*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2
- 4*a*c)) + 2*(2*c^3*d^4*e - 4*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4 + (a*b^2 - 2*a^2*c)*e^5)*x + (2*c^
3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e
^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*
c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x)*log(c*x^2 + b*x + a) - 2*(2*c^3*d^5 - 3*b*c^2*d^4*e + 3*(b^2*c - 2*a*
c^2)*d^3*e^2 - (b^3 - 3*a*b*c)*d^2*e^3 + (2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 -
3*a*b*c)*e^5)*x^2 + 2*(2*c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c - 2*a*c^2)*d^2*e^3 - (b^3 - 3*a*b*c)*d*e^4)*x
)*log(e*x + d))/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 +
6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b
^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*
e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^
3*e^5)*x)]

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giac [B]  time = 0.20, size = 710, normalized size = 2.34 \begin {gather*} \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\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^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\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 (3 \, b^{2} c^{2} d^{2} e - 12 \, a c^{3} d^{2} e - 3 \, b^{3} c d e^{2} + 12 \, a b c^{2} d e^{2} + b^{4} e^{3} - 5 \, a b^{2} c e^{3} + 4 \, a^{2} c^{2} e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\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 {6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 10 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - 3 \, b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 4 \, a b^{2} d e^{4} - 2 \, a^{2} c d e^{4} - a^{2} b e^{5} + 2 \, {\left (2 \, c^{3} d^{4} e - 4 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4} + a b^{2} e^{5} - 2 \, a^{2} c e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*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^3*d^3*e - 3*b*c^2*d^2*e^2 + 3*b^2*c*d*e^3 - 6*a*c^2*d*e^3
- b^3*e^4 + 3*a*b*c*e^4)*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) - (3*b^2*c^2*d^2*
e - 12*a*c^3*d^2*e - 3*b^3*c*d*e^2 + 12*a*b*c^2*d*e^2 + b^4*e^3 - 5*a*b^2*c*e^3 + 4*a^2*c^2*e^3)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/((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)) + 1/2*(6*c^3*d^5
- 13*b*c^2*d^4*e + 10*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - 3*b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 4*a*b^2*d*e^4 - 2*a^
2*c*d*e^4 - a^2*b*e^5 + 2*(2*c^3*d^4*e - 4*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4 + a*b^2*e^5 - 2*a^2*c*e
^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(x*e + d)^2)

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maple [B]  time = 0.06, size = 1033, normalized size = 3.41 \begin {gather*} -\frac {4 a^{2} c^{2} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}+\frac {5 a \,b^{2} c \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {12 a b \,c^{2} d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}+\frac {12 a \,c^{3} d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {b^{4} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}+\frac {3 b^{3} c d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {3 b^{2} c^{2} d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {4 a c -b^{2}}}-\frac {3 a b c \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {3 a b c \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {6 a \,c^{2} d \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {3 a \,c^{2} d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {b^{3} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {b^{3} e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {3 b^{2} c d \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {3 b^{2} c d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {3 b \,c^{2} d^{2} e \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {3 b \,c^{2} d^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {2 c^{3} d^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}+\frac {c^{3} d^{3} \ln \left (c \,x^{2}+b x +a \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {2 a c \,e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}+\frac {b^{2} e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 b c d e}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}+\frac {2 c^{2} d^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {b e}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{2}}+\frac {c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))*a*b*c^2*d*e^2+6/(a*e^2-b*d*e+c
*d^2)^3*ln(e*x+d)*c^2*a*d*e^2-3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*b*c*e^3-3/2/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x
^2+b*x+a)*b*d^2*e-4/(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^2*c^2*e^3-3/
(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b^2*c*d*e^2+3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*c^2*d^2*e-3/(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^2*d^2*e+5/(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^2*c*e^3+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))*a*c^3*d^2*e+3/(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*c*d*e^2-2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*b*c*d*e+3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*a*b*e^3-3/(a*
e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*a*d*e^2+3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*b^2*d*e^2+1/(a*e^2-
b*d*e+c*d^2)^3*ln(e*x+d)*b^3*e^3-2/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^3*d^3+1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*b^2
*e^2+2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*c^2*d^2-1/2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2*b*e+1/(a*e^2-b*d*e+c*d^2)/(e*x+
d)^2*c*d-1/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*b^3*e^3+1/(a*e^2-b*d*e+c*d^2)^3*c^3*ln(c*x^2+b*x+a)*d^3-2/(
a*e^2-b*d*e+c*d^2)^2/(e*x+d)*a*c*e^2-1/(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^4*e^3

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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)^3/(c*x^2+b*x+a),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 = 10.98, size = 2608, normalized size = 8.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(2*a*e^5*(b^2 - 4*a*c)^(5/2) + 32*a*b^5*e^5 - 192*a*c^5*d^5 + 32*b^6*e^5*x + 48*b^2*c^4*d^5 + 18*b^3*e^5*x
*(b^2 - 4*a*c)^(3/2) + 3*b^5*e^5*x*(b^2 - 4*a*c)^(1/2) - 96*c^5*d^5*x*(b^2 - 4*a*c)^(1/2) - 208*a^2*b^3*c*e^5
+ 320*a^3*b*c^2*e^5 - 704*a^3*c^3*d*e^4 - 48*b^3*c^3*d^4*e - 16*b^5*c*d^2*e^3 - 64*a^3*c^3*e^5*x + 1152*a^2*c^
4*d^3*e^2 + 48*b^4*c^2*d^3*e^2 + 33*b*d*e^4*(b^2 - 4*a*c)^(5/2) + 11*b*e^5*x*(b^2 - 4*a*c)^(5/2) + 24*a*b^2*e^
5*(b^2 - 4*a*c)^(3/2) + 6*a*b^4*e^5*(b^2 - 4*a*c)^(1/2) - 48*b*c^4*d^5*(b^2 - 4*a*c)^(1/2) - 18*b^3*d*e^4*(b^2
- 4*a*c)^(3/2) - 15*b^5*d*e^4*(b^2 - 4*a*c)^(1/2) - 44*c*d^2*e^3*(b^2 - 4*a*c)^(5/2) - 72*c^3*d^4*e*(b^2 - 4*
a*c)^(3/2) - 22*c*d*e^4*x*(b^2 - 4*a*c)^(5/2) + 192*a*b*c^4*d^4*e - 128*a*b^4*c*d*e^4 - 120*b^3*c^2*d^3*e^2*(b
^2 - 4*a*c)^(1/2) - 224*a*b^4*c*e^5*x - 576*a*c^5*d^4*e*x - 160*b^5*c*d*e^4*x + 144*b^2*c^4*d^4*e*x + 72*b*c^2
*d^3*e^2*(b^2 - 4*a*c)^(3/2) + 120*b^2*c^3*d^4*e*(b^2 - 4*a*c)^(1/2) + 60*b^4*c*d^2*e^3*(b^2 - 4*a*c)^(1/2) -
144*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(3/2) - 480*a*b^2*c^3*d^3*e^2 + 320*a*b^3*c^2*d^2*e^3 - 1024*a^2*b*c^3*d^2*e^3
+ 688*a^2*b^2*c^2*d*e^4 + 400*a^2*b^2*c^2*e^5*x + 1408*a^2*c^4*d^2*e^3*x - 288*b^3*c^3*d^3*e^2*x + 304*b^4*c^
2*d^2*e^3*x + 216*b*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(3/2) - 1568*a*b^2*c^3*d^2*e^3*x - 240*b^2*c^3*d^3*e^2*x*(b^2
- 4*a*c)^(1/2) + 120*b^3*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(1/2) + 240*b*c^4*d^4*e*x*(b^2 - 4*a*c)^(1/2) - 108*b^2*c
*d*e^4*x*(b^2 - 4*a*c)^(3/2) - 30*b^4*c*d*e^4*x*(b^2 - 4*a*c)^(1/2) + 1152*a*b*c^4*d^3*e^2*x + 992*a*b^3*c^2*d
*e^4*x - 1408*a^2*b*c^3*d*e^4*x)*(e^2*((3*b^2*c*d)/2 - 3*a*c^2*d + (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) - e^3*(b^3
/2 + (b^2*(b^2 - 4*a*c)^(1/2))/2 - (3*a*b*c)/2 - (a*c*(b^2 - 4*a*c)^(1/2))/2) + c^3*d^3 - e*((3*b*c^2*d^2)/2 +
(3*c^2*d^2*(b^2 - 4*a*c)^(1/2))/2)))/(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(32*a*b^5*e^5 - 2*a*
e^5*(b^2 - 4*a*c)^(5/2) - 192*a*c^5*d^5 + 32*b^6*e^5*x + 48*b^2*c^4*d^5 - 18*b^3*e^5*x*(b^2 - 4*a*c)^(3/2) - 3
*b^5*e^5*x*(b^2 - 4*a*c)^(1/2) + 96*c^5*d^5*x*(b^2 - 4*a*c)^(1/2) - 208*a^2*b^3*c*e^5 + 320*a^3*b*c^2*e^5 - 70
4*a^3*c^3*d*e^4 - 48*b^3*c^3*d^4*e - 16*b^5*c*d^2*e^3 - 64*a^3*c^3*e^5*x + 1152*a^2*c^4*d^3*e^2 + 48*b^4*c^2*d
^3*e^2 - 33*b*d*e^4*(b^2 - 4*a*c)^(5/2) - 11*b*e^5*x*(b^2 - 4*a*c)^(5/2) - 24*a*b^2*e^5*(b^2 - 4*a*c)^(3/2) -
6*a*b^4*e^5*(b^2 - 4*a*c)^(1/2) + 48*b*c^4*d^5*(b^2 - 4*a*c)^(1/2) + 18*b^3*d*e^4*(b^2 - 4*a*c)^(3/2) + 15*b^5
*d*e^4*(b^2 - 4*a*c)^(1/2) + 44*c*d^2*e^3*(b^2 - 4*a*c)^(5/2) + 72*c^3*d^4*e*(b^2 - 4*a*c)^(3/2) + 22*c*d*e^4*
x*(b^2 - 4*a*c)^(5/2) + 192*a*b*c^4*d^4*e - 128*a*b^4*c*d*e^4 + 120*b^3*c^2*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 224*
a*b^4*c*e^5*x - 576*a*c^5*d^4*e*x - 160*b^5*c*d*e^4*x + 144*b^2*c^4*d^4*e*x - 72*b*c^2*d^3*e^2*(b^2 - 4*a*c)^(
3/2) - 120*b^2*c^3*d^4*e*(b^2 - 4*a*c)^(1/2) - 60*b^4*c*d^2*e^3*(b^2 - 4*a*c)^(1/2) + 144*c^3*d^3*e^2*x*(b^2 -
4*a*c)^(3/2) - 480*a*b^2*c^3*d^3*e^2 + 320*a*b^3*c^2*d^2*e^3 - 1024*a^2*b*c^3*d^2*e^3 + 688*a^2*b^2*c^2*d*e^4
+ 400*a^2*b^2*c^2*e^5*x + 1408*a^2*c^4*d^2*e^3*x - 288*b^3*c^3*d^3*e^2*x + 304*b^4*c^2*d^2*e^3*x - 216*b*c^2*
d^2*e^3*x*(b^2 - 4*a*c)^(3/2) - 1568*a*b^2*c^3*d^2*e^3*x + 240*b^2*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(1/2) - 120*b^3
*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(1/2) - 240*b*c^4*d^4*e*x*(b^2 - 4*a*c)^(1/2) + 108*b^2*c*d*e^4*x*(b^2 - 4*a*c)^(
3/2) + 30*b^4*c*d*e^4*x*(b^2 - 4*a*c)^(1/2) + 1152*a*b*c^4*d^3*e^2*x + 992*a*b^3*c^2*d*e^4*x - 1408*a^2*b*c^3*
d*e^4*x)*(e^2*(3*a*c^2*d - (3*b^2*c*d)/2 + (3*b*c*d*(b^2 - 4*a*c)^(1/2))/2) + e^3*(b^3/2 - (b^2*(b^2 - 4*a*c)^
(1/2))/2 - (3*a*b*c)/2 + (a*c*(b^2 - 4*a*c)^(1/2))/2) - c^3*d^3 + e*((3*b*c^2*d^2)/2 - (3*c^2*d^2*(b^2 - 4*a*c
)^(1/2))/2)))/(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) - ((a*b*e^3 - 3*b^2*d*e^2 - 6*c^2*d^3 + 2*a*c*d*e^
2 + 7*b*c*d^2*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)) - (x*(b^2*e
^3 + 2*c^2*d^2*e - 2*a*c*e^3 - 2*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))/(d^2 + e^2*x^2 + 2*d*e*x) + (log(d + e*x)*(e^2*(6*a*c^2*d - 3*b^2*c*d) + e^3*(b^3 - 3*a*b*c) - 2
*c^3*d^3 + 3*b*c^2*d^2*e))/(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)

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

[Out]

Timed out

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