3.1.89 \(\int \frac {\log (d (a+b x+c x^2)^n)}{(d+e x)^3} \, dx\) [89]

Optimal. Leaf size=259 \[ \frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2} \]

[Out]

1/2*(-b*e+2*c*d)*n/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)-1/2*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*n*ln(e*x+d)/e/(a*e^2-
b*d*e+c*d^2)^2+1/4*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*n*ln(c*x^2+b*x+a)/e/(a*e^2-b*d*e+c*d^2)^2-1/2*ln(d*(c*x
^2+b*x+a)^n)/e/(e*x+d)^2+1/2*(-b*e+2*c*d)*n*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^(1/2)/(a*e^2-b*
d*e+c*d^2)^2

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Rubi [A]
time = 0.25, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2605, 814, 648, 632, 212, 642} \begin {gather*} \frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 e \left (a e^2-b d e+c d^2\right )^2}-\frac {n \log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{2 e \left (a e^2-b d e+c d^2\right )^2}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac {n (2 c d-b e)}{2 e (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^3} \, dx &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \frac {b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{2 e}\\ &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \left (\frac {e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\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)}+\frac {-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \frac {-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{a+b x+c x^2} \, dx}{2 e \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{4 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{4 e \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) n\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 215, normalized size = 0.83 \begin {gather*} \frac {\frac {n (d+e x) \left (2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )-2 \sqrt {b^2-4 a c} e (-2 c d+b e) (d+e x) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x) \log (d+e x)+\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x) \log (a+x (b+c x))\right )}{\left (c d^2+e (-b d+a e)\right )^2}-2 \log \left (d (a+x (b+c x))^n\right )}{4 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((n*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)) - 2*Sqrt[b^2 - 4*a*c]*e*(-2*c*d + b*e)*(d + e*x)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x)*Log[d + e*x] + (2*
c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x)*Log[a + x*(b + c*x)]))/(c*d^2 + e*(-(b*d) + a*e))^2 - 2*Log[d
*(a + x*(b + c*x))^n])/(4*e*(d + e*x)^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 14679, normalized size = 56.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(14679\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^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. 658 vs. \(2 (248) = 496\).
time = 1.34, size = 1337, normalized size = 5.16 \begin {gather*} \left [\frac {4 \, c^{2} d^{4} n - 2 \, a b n x e^{4} - {\left (2 \, c d^{3} n e - b n x^{2} e^{4} + 2 \, {\left (c d n x^{2} - b d n x\right )} e^{3} + {\left (4 \, c d^{2} n x - b d^{2} n\right )} e^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (a b d n - {\left (b^{2} + 2 \, a c\right )} d n x\right )} e^{3} - 2 \, {\left (3 \, b c d^{2} n x - {\left (b^{2} + 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - 3 \, b c d^{3} n\right )} e + {\left ({\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} e^{4} - 2 \, {\left (b c d n x^{2} - 2 \, a b d n - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x - {\left (b^{2} + 6 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x + b c d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c^{2} d^{4} n + {\left (b^{2} - 2 \, a c\right )} n x^{2} e^{4} - 2 \, {\left (b c d n x^{2} - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x + {\left (b^{2} - 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - b c d^{3} n\right )} e\right )} \log \left (x e + d\right ) - 2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (d\right )}{4 \, {\left (c^{2} d^{6} e + a^{2} x^{2} e^{7} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{6} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{4} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{3} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e^{2}\right )}}, \frac {4 \, c^{2} d^{4} n - 2 \, a b n x e^{4} + 2 \, {\left (2 \, c d^{3} n e - b n x^{2} e^{4} + 2 \, {\left (c d n x^{2} - b d n x\right )} e^{3} + {\left (4 \, c d^{2} n x - b d^{2} n\right )} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (a b d n - {\left (b^{2} + 2 \, a c\right )} d n x\right )} e^{3} - 2 \, {\left (3 \, b c d^{2} n x - {\left (b^{2} + 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - 3 \, b c d^{3} n\right )} e + {\left ({\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} e^{4} - 2 \, {\left (b c d n x^{2} - 2 \, a b d n - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x - {\left (b^{2} + 6 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x + b c d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c^{2} d^{4} n + {\left (b^{2} - 2 \, a c\right )} n x^{2} e^{4} - 2 \, {\left (b c d n x^{2} - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x + {\left (b^{2} - 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - b c d^{3} n\right )} e\right )} \log \left (x e + d\right ) - 2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (d\right )}{4 \, {\left (c^{2} d^{6} e + a^{2} x^{2} e^{7} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{6} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{4} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{3} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(ln(d*(c*x**2+b*x+a)**n)/(e*x+d)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (248) = 496\).
time = 3.35, size = 887, normalized size = 3.42 \begin {gather*} \frac {{\left (2 \, c^{2} d^{2} n - 2 \, b c d n e + b^{2} n e^{2} - 2 \, a c n e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{4 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )}} - \frac {{\left (2 \, b^{2} c d n - 8 \, a c^{2} d n - b^{3} n e + 4 \, a b c n e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c^{2} d^{2} n x^{2} e^{2} \log \left (x e + d\right ) + 4 \, c^{2} d^{3} n x e \log \left (x e + d\right ) - 2 \, c^{2} d^{3} n x e + c^{2} d^{4} n \log \left (c x^{2} + b x + a\right ) - 2 \, b c d^{3} n e \log \left (c x^{2} + b x + a\right ) + 2 \, c^{2} d^{4} n \log \left (x e + d\right ) - 2 \, b c d n x^{2} e^{3} \log \left (x e + d\right ) - 4 \, b c d^{2} n x e^{2} \log \left (x e + d\right ) - 2 \, b c d^{3} n e \log \left (x e + d\right ) - 2 \, c^{2} d^{4} n + 3 \, b c d^{2} n x e^{2} + 3 \, b c d^{3} n e + b^{2} d^{2} n e^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, a c d^{2} n e^{2} \log \left (c x^{2} + b x + a\right ) + b^{2} n x^{2} e^{4} \log \left (x e + d\right ) - 2 \, a c n x^{2} e^{4} \log \left (x e + d\right ) + 2 \, b^{2} d n x e^{3} \log \left (x e + d\right ) - 4 \, a c d n x e^{3} \log \left (x e + d\right ) + b^{2} d^{2} n e^{2} \log \left (x e + d\right ) - 2 \, a c d^{2} n e^{2} \log \left (x e + d\right ) + c^{2} d^{4} \log \left (d\right ) - 2 \, b c d^{3} e \log \left (d\right ) - b^{2} d n x e^{3} - 2 \, a c d n x e^{3} - b^{2} d^{2} n e^{2} - 2 \, a c d^{2} n e^{2} - 2 \, a b d n e^{3} \log \left (c x^{2} + b x + a\right ) + b^{2} d^{2} e^{2} \log \left (d\right ) + 2 \, a c d^{2} e^{2} \log \left (d\right ) + a b n x e^{4} + a b d n e^{3} + a^{2} n e^{4} \log \left (c x^{2} + b x + a\right ) - 2 \, a b d e^{3} \log \left (d\right ) + a^{2} e^{4} \log \left (d\right )}{2 \, {\left (c^{2} d^{4} x^{2} e^{3} + 2 \, c^{2} d^{5} x e^{2} + c^{2} d^{6} e - 2 \, b c d^{3} x^{2} e^{4} - 4 \, b c d^{4} x e^{3} - 2 \, b c d^{5} e^{2} + b^{2} d^{2} x^{2} e^{5} + 2 \, a c d^{2} x^{2} e^{5} + 2 \, b^{2} d^{3} x e^{4} + 4 \, a c d^{3} x e^{4} + b^{2} d^{4} e^{3} + 2 \, a c d^{4} e^{3} - 2 \, a b d x^{2} e^{6} - 4 \, a b d^{2} x e^{5} - 2 \, a b d^{3} e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + a^{2} d^{2} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.74, size = 1715, normalized size = 6.62 \begin {gather*} \frac {\ln \left (3\,b^2\,c^3\,d^4-12\,a\,c^4\,d^4-2\,b^5\,e^4\,x-12\,a^3\,c^2\,e^4-2\,a\,b^4\,e^4+2\,b^4\,e^4\,x\,\sqrt {b^2-4\,a\,c}+6\,c^4\,d^4\,x\,\sqrt {b^2-4\,a\,c}+11\,a^2\,b^2\,c\,e^4-2\,b^3\,c^2\,d^3\,e+b^4\,c\,d^2\,e^2+40\,a^2\,c^3\,d^2\,e^2+2\,a\,b^3\,e^4\,\sqrt {b^2-4\,a\,c}+3\,b\,c^3\,d^4\,\sqrt {b^2-4\,a\,c}+8\,a\,b\,c^3\,d^3\,e+6\,a\,b^3\,c\,d\,e^3+12\,a\,b^3\,c\,e^4\,x-32\,a\,c^4\,d^3\,e\,x+8\,b^4\,c\,d\,e^3\,x-5\,a^2\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}-16\,a\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}-24\,a^2\,b\,c^2\,d\,e^3-16\,a^2\,b\,c^2\,e^4\,x+32\,a^2\,c^3\,d\,e^3\,x+8\,b^2\,c^3\,d^3\,e\,x+16\,a^2\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^2\,c^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+b^3\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}-14\,a\,b^2\,c^2\,d^2\,e^2-12\,b^3\,c^2\,d^2\,e^2\,x+14\,a\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-20\,a\,c^3\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}+14\,b^2\,c^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,e^4\,x\,\sqrt {b^2-4\,a\,c}-12\,b\,c^3\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}-8\,b^3\,c\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}+48\,a\,b\,c^3\,d^2\,e^2\,x-40\,a\,b^2\,c^2\,d\,e^3\,x+20\,a\,b\,c^2\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (e\,\left (\frac {c\,d\,n\,\sqrt {b^2-4\,a\,c}}{2}-\frac {b\,c\,d\,n}{2}\right )-e^2\,\left (\frac {a\,c\,n}{2}-\frac {b^2\,n}{4}+\frac {b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )+\frac {c^2\,d^2\,n}{2}\right )}{a^2\,e^5-2\,a\,b\,d\,e^4+2\,a\,c\,d^2\,e^3+b^2\,d^2\,e^3-2\,b\,c\,d^3\,e^2+c^2\,d^4\,e}-\frac {\ln \left (d+e\,x\right )\,\left (e^2\,\left (b^2\,n-2\,a\,c\,n\right )+2\,c^2\,d^2\,n-2\,b\,c\,d\,e\,n\right )}{2\,a^2\,e^5-4\,a\,b\,d\,e^4+4\,a\,c\,d^2\,e^3+2\,b^2\,d^2\,e^3-4\,b\,c\,d^3\,e^2+2\,c^2\,d^4\,e}+\frac {\ln \left (2\,a\,b^4\,e^4+12\,a\,c^4\,d^4+2\,b^5\,e^4\,x+12\,a^3\,c^2\,e^4-3\,b^2\,c^3\,d^4+2\,b^4\,e^4\,x\,\sqrt {b^2-4\,a\,c}+6\,c^4\,d^4\,x\,\sqrt {b^2-4\,a\,c}-11\,a^2\,b^2\,c\,e^4+2\,b^3\,c^2\,d^3\,e-b^4\,c\,d^2\,e^2-40\,a^2\,c^3\,d^2\,e^2+2\,a\,b^3\,e^4\,\sqrt {b^2-4\,a\,c}+3\,b\,c^3\,d^4\,\sqrt {b^2-4\,a\,c}-8\,a\,b\,c^3\,d^3\,e-6\,a\,b^3\,c\,d\,e^3-12\,a\,b^3\,c\,e^4\,x+32\,a\,c^4\,d^3\,e\,x-8\,b^4\,c\,d\,e^3\,x-5\,a^2\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}-16\,a\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}+24\,a^2\,b\,c^2\,d\,e^3+16\,a^2\,b\,c^2\,e^4\,x-32\,a^2\,c^3\,d\,e^3\,x-8\,b^2\,c^3\,d^3\,e\,x+16\,a^2\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^2\,c^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+b^3\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+14\,a\,b^2\,c^2\,d^2\,e^2+12\,b^3\,c^2\,d^2\,e^2\,x+14\,a\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-20\,a\,c^3\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}+14\,b^2\,c^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,e^4\,x\,\sqrt {b^2-4\,a\,c}-12\,b\,c^3\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}-8\,b^3\,c\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}-48\,a\,b\,c^3\,d^2\,e^2\,x+40\,a\,b^2\,c^2\,d\,e^3\,x+20\,a\,b\,c^2\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (e^2\,\left (\frac {b^2\,n}{4}-\frac {a\,c\,n}{2}+\frac {b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )-e\,\left (\frac {c\,d\,n\,\sqrt {b^2-4\,a\,c}}{2}+\frac {b\,c\,d\,n}{2}\right )+\frac {c^2\,d^2\,n}{2}\right )}{a^2\,e^5-2\,a\,b\,d\,e^4+2\,a\,c\,d^2\,e^3+b^2\,d^2\,e^3-2\,b\,c\,d^3\,e^2+c^2\,d^4\,e}-\frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}-\frac {n\,\left (b\,e-2\,c\,d\right )}{\left (2\,x\,e^2+2\,d\,e\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(3*b^2*c^3*d^4 - 12*a*c^4*d^4 - 2*b^5*e^4*x - 12*a^3*c^2*e^4 - 2*a*b^4*e^4 + 2*b^4*e^4*x*(b^2 - 4*a*c)^(1/
2) + 6*c^4*d^4*x*(b^2 - 4*a*c)^(1/2) + 11*a^2*b^2*c*e^4 - 2*b^3*c^2*d^3*e + b^4*c*d^2*e^2 + 40*a^2*c^3*d^2*e^2
+ 2*a*b^3*e^4*(b^2 - 4*a*c)^(1/2) + 3*b*c^3*d^4*(b^2 - 4*a*c)^(1/2) + 8*a*b*c^3*d^3*e + 6*a*b^3*c*d*e^3 + 12*
a*b^3*c*e^4*x - 32*a*c^4*d^3*e*x + 8*b^4*c*d*e^3*x - 5*a^2*b*c*e^4*(b^2 - 4*a*c)^(1/2) - 16*a*c^3*d^3*e*(b^2 -
4*a*c)^(1/2) - 24*a^2*b*c^2*d*e^3 - 16*a^2*b*c^2*e^4*x + 32*a^2*c^3*d*e^3*x + 8*b^2*c^3*d^3*e*x + 16*a^2*c^2*
d*e^3*(b^2 - 4*a*c)^(1/2) - 2*b^2*c^2*d^3*e*(b^2 - 4*a*c)^(1/2) + b^3*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) + 6*a^2*c^
2*e^4*x*(b^2 - 4*a*c)^(1/2) - 14*a*b^2*c^2*d^2*e^2 - 12*b^3*c^2*d^2*e^2*x + 14*a*b*c^2*d^2*e^2*(b^2 - 4*a*c)^(
1/2) - 20*a*c^3*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + 14*b^2*c^2*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) - 10*a*b^2*c*d*e^3*(b
^2 - 4*a*c)^(1/2) - 8*a*b^2*c*e^4*x*(b^2 - 4*a*c)^(1/2) - 12*b*c^3*d^3*e*x*(b^2 - 4*a*c)^(1/2) - 8*b^3*c*d*e^3
*x*(b^2 - 4*a*c)^(1/2) + 48*a*b*c^3*d^2*e^2*x - 40*a*b^2*c^2*d*e^3*x + 20*a*b*c^2*d*e^3*x*(b^2 - 4*a*c)^(1/2))
*(e*((c*d*n*(b^2 - 4*a*c)^(1/2))/2 - (b*c*d*n)/2) - e^2*((a*c*n)/2 - (b^2*n)/4 + (b*n*(b^2 - 4*a*c)^(1/2))/4)
+ (c^2*d^2*n)/2))/(a^2*e^5 + c^2*d^4*e + b^2*d^2*e^3 - 2*a*b*d*e^4 + 2*a*c*d^2*e^3 - 2*b*c*d^3*e^2) - (log(d +
e*x)*(e^2*(b^2*n - 2*a*c*n) + 2*c^2*d^2*n - 2*b*c*d*e*n))/(2*a^2*e^5 + 2*c^2*d^4*e + 2*b^2*d^2*e^3 - 4*a*b*d*
e^4 + 4*a*c*d^2*e^3 - 4*b*c*d^3*e^2) + (log(2*a*b^4*e^4 + 12*a*c^4*d^4 + 2*b^5*e^4*x + 12*a^3*c^2*e^4 - 3*b^2*
c^3*d^4 + 2*b^4*e^4*x*(b^2 - 4*a*c)^(1/2) + 6*c^4*d^4*x*(b^2 - 4*a*c)^(1/2) - 11*a^2*b^2*c*e^4 + 2*b^3*c^2*d^3
*e - b^4*c*d^2*e^2 - 40*a^2*c^3*d^2*e^2 + 2*a*b^3*e^4*(b^2 - 4*a*c)^(1/2) + 3*b*c^3*d^4*(b^2 - 4*a*c)^(1/2) -
8*a*b*c^3*d^3*e - 6*a*b^3*c*d*e^3 - 12*a*b^3*c*e^4*x + 32*a*c^4*d^3*e*x - 8*b^4*c*d*e^3*x - 5*a^2*b*c*e^4*(b^2
- 4*a*c)^(1/2) - 16*a*c^3*d^3*e*(b^2 - 4*a*c)^(1/2) + 24*a^2*b*c^2*d*e^3 + 16*a^2*b*c^2*e^4*x - 32*a^2*c^3*d*
e^3*x - 8*b^2*c^3*d^3*e*x + 16*a^2*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) - 2*b^2*c^2*d^3*e*(b^2 - 4*a*c)^(1/2) + b^3*c
*d^2*e^2*(b^2 - 4*a*c)^(1/2) + 6*a^2*c^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 14*a*b^2*c^2*d^2*e^2 + 12*b^3*c^2*d^2*e^2
*x + 14*a*b*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) - 20*a*c^3*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + 14*b^2*c^2*d^2*e^2*x*(b
^2 - 4*a*c)^(1/2) - 10*a*b^2*c*d*e^3*(b^2 - 4*a*c)^(1/2) - 8*a*b^2*c*e^4*x*(b^2 - 4*a*c)^(1/2) - 12*b*c^3*d^3*
e*x*(b^2 - 4*a*c)^(1/2) - 8*b^3*c*d*e^3*x*(b^2 - 4*a*c)^(1/2) - 48*a*b*c^3*d^2*e^2*x + 40*a*b^2*c^2*d*e^3*x +
20*a*b*c^2*d*e^3*x*(b^2 - 4*a*c)^(1/2))*(e^2*((b^2*n)/4 - (a*c*n)/2 + (b*n*(b^2 - 4*a*c)^(1/2))/4) - e*((c*d*n
*(b^2 - 4*a*c)^(1/2))/2 + (b*c*d*n)/2) + (c^2*d^2*n)/2))/(a^2*e^5 + c^2*d^4*e + b^2*d^2*e^3 - 2*a*b*d*e^4 + 2*
a*c*d^2*e^3 - 2*b*c*d^3*e^2) - log(d*(a + b*x + c*x^2)^n)/(2*e*(d^2 + e^2*x^2 + 2*d*e*x)) - (n*(b*e - 2*c*d))/
((2*d*e + 2*e^2*x)*(a*e^2 + c*d^2 - b*d*e))

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