3.80 \(\int \frac {\log (d (a+b x+c x^2)^n)}{x^5} \, dx\)

Optimal. Leaf size=190 \[ -\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 a^4}-\frac {b n \left (b^2-3 a c\right )}{4 a^3 x}+\frac {n \left (b^2-2 a c\right )}{8 a^2 x^2}+\frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 a^4}-\frac {n \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{4 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac {b n}{12 a x^3} \]

[Out]

-1/12*b*n/a/x^3+1/8*(-2*a*c+b^2)*n/a^2/x^2-1/4*b*(-3*a*c+b^2)*n/a^3/x-1/4*(2*a^2*c^2-4*a*b^2*c+b^4)*n*ln(x)/a^
4+1/8*(2*a^2*c^2-4*a*b^2*c+b^4)*n*ln(c*x^2+b*x+a)/a^4-1/4*ln(d*(c*x^2+b*x+a)^n)/x^4-1/4*b*(-2*a*c+b^2)*n*arcta
nh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^(1/2)/a^4

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Rubi [A]  time = 0.22, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 a^4}-\frac {n \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{4 a^4}+\frac {n \left (b^2-2 a c\right )}{8 a^2 x^2}-\frac {b n \left (b^2-3 a c\right )}{4 a^3 x}-\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac {b n}{12 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/x^5,x]

[Out]

-(b*n)/(12*a*x^3) + ((b^2 - 2*a*c)*n)/(8*a^2*x^2) - (b*(b^2 - 3*a*c)*n)/(4*a^3*x) - (b*Sqrt[b^2 - 4*a*c]*(b^2
- 2*a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*a^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[x])/(4*a^4)
+ ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + b*x + c*x^2])/(8*a^4) - Log[d*(a + b*x + c*x^2)^n]/(4*x^4)

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 2525

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 )}{x^5} \, dx &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac {1}{4} n \int \frac {b+2 c x}{x^4 \left (a+b x+c x^2\right )} \, dx\\ &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac {1}{4} n \int \left (\frac {b}{a x^4}+\frac {-b^2+2 a c}{a^2 x^3}+\frac {b^3-3 a b c}{a^3 x^2}+\frac {-b^4+4 a b^2 c-2 a^2 c^2}{a^4 x}+\frac {b \left (b^4-5 a b^2 c+5 a^2 c^2\right )+c \left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {b n}{12 a x^3}+\frac {\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac {b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac {n \int \frac {b \left (b^4-5 a b^2 c+5 a^2 c^2\right )+c \left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{4 a^4}\\ &=-\frac {b n}{12 a x^3}+\frac {\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac {b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac {\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{8 a^4}+\frac {\left (\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{8 a^4}\\ &=-\frac {b n}{12 a x^3}+\frac {\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac {b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}+\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac {\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 a^4}\\ &=-\frac {b n}{12 a x^3}+\frac {\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac {b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac {b \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 a^4}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}+\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 a^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.46, size = 172, normalized size = 0.91 \[ -\frac {\frac {n x \left (2 a^3 b-3 a^2 x \left (b^2-2 a c\right )+6 x^3 \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )-3 x^3 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log (a+x (b+c x))+6 b x^3 \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+6 a b x^2 \left (b^2-3 a c\right )\right )}{a^4}+6 \log \left (d (a+x (b+c x))^n\right )}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(a + b*x + c*x^2)^n]/x^5,x]

[Out]

-1/24*((n*x*(2*a^3*b - 3*a^2*(b^2 - 2*a*c)*x + 6*a*b*(b^2 - 3*a*c)*x^2 + 6*b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*a*c)*x
^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^3*Log[x] - 3*(b^4 - 4*a*b^2*c +
2*a^2*c^2)*x^3*Log[a + x*(b + c*x)]))/a^4 + 6*Log[d*(a + x*(b + c*x))^n])/x^4

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fricas [A]  time = 0.55, size = 404, normalized size = 2.13 \[ \left [-\frac {3 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} n x^{4} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} \log \relax (x) + 2 \, a^{3} b n x + 6 \, {\left (a b^{3} - 3 \, a^{2} b c\right )} n x^{3} + 6 \, a^{4} \log \relax (d) - 3 \, {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} n x^{2} - 3 \, {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} - 2 \, a^{4} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, a^{4} x^{4}}, -\frac {6 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} n x^{4} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} \log \relax (x) + 2 \, a^{3} b n x + 6 \, {\left (a b^{3} - 3 \, a^{2} b c\right )} n x^{3} + 6 \, a^{4} \log \relax (d) - 3 \, {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} n x^{2} - 3 \, {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} - 2 \, a^{4} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, a^{4} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^5,x, algorithm="fricas")

[Out]

[-1/24*(3*(b^3 - 2*a*b*c)*sqrt(b^2 - 4*a*c)*n*x^4*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)) + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4*log(x) + 2*a^3*b*n*x + 6*(a*b^3 - 3*a^2
*b*c)*n*x^3 + 6*a^4*log(d) - 3*(a^2*b^2 - 2*a^3*c)*n*x^2 - 3*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4 - 2*a^4*n)*l
og(c*x^2 + b*x + a))/(a^4*x^4), -1/24*(6*(b^3 - 2*a*b*c)*sqrt(-b^2 + 4*a*c)*n*x^4*arctan(-sqrt(-b^2 + 4*a*c)*(
2*c*x + b)/(b^2 - 4*a*c)) + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4*log(x) + 2*a^3*b*n*x + 6*(a*b^3 - 3*a^2*b*c)
*n*x^3 + 6*a^4*log(d) - 3*(a^2*b^2 - 2*a^3*c)*n*x^2 - 3*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4 - 2*a^4*n)*log(c*
x^2 + b*x + a))/(a^4*x^4)]

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giac [A]  time = 0.21, size = 210, normalized size = 1.11 \[ \frac {{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{8 \, a^{4}} - \frac {n \log \left (c x^{2} + b x + a\right )}{4 \, x^{4}} - \frac {{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \relax (x)}{4 \, a^{4}} + \frac {{\left (b^{5} n - 6 \, a b^{3} c n + 8 \, a^{2} b c^{2} n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} a^{4}} - \frac {6 \, b^{3} n x^{3} - 18 \, a b c n x^{3} - 3 \, a b^{2} n x^{2} + 6 \, a^{2} c n x^{2} + 2 \, a^{2} b n x + 6 \, a^{3} \log \relax (d)}{24 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^5,x, algorithm="giac")

[Out]

1/8*(b^4*n - 4*a*b^2*c*n + 2*a^2*c^2*n)*log(c*x^2 + b*x + a)/a^4 - 1/4*n*log(c*x^2 + b*x + a)/x^4 - 1/4*(b^4*n
- 4*a*b^2*c*n + 2*a^2*c^2*n)*log(x)/a^4 + 1/4*(b^5*n - 6*a*b^3*c*n + 8*a^2*b*c^2*n)*arctan((2*c*x + b)/sqrt(-
b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) - 1/24*(6*b^3*n*x^3 - 18*a*b*c*n*x^3 - 3*a*b^2*n*x^2 + 6*a^2*c*n*x^2 +
2*a^2*b*n*x + 6*a^3*log(d))/(a^3*x^4)

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maple [C]  time = 0.88, size = 505, normalized size = 2.66 \[ -\frac {\ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )}{4 x^{4}}-\frac {12 a^{2} c^{2} n \,x^{4} \ln \relax (x )-24 a \,b^{2} c n \,x^{4} \ln \relax (x )+6 b^{4} n \,x^{4} \ln \relax (x )-6 a^{4} x^{4} \RootOf \left (\textit {\_Z}^{2} a^{4}+c^{4} n^{2}+\left (-2 a^{2} c^{2} n +4 a \,b^{2} c n -b^{4} n \right ) \textit {\_Z} \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2} a^{4}+c^{4} n^{2}+\left (-2 a^{2} c^{2} n +4 a \,b^{2} c n -b^{4} n \right ) \textit {\_Z} \right )^{2} a^{7} b -6 a^{3} b \,c^{4} n^{2}+14 a^{2} b^{3} c^{3} n^{2}-7 a \,b^{5} c^{2} n^{2}+b^{7} c \,n^{2}+\left (-5 a^{5} b \,c^{2} n +5 a^{4} b^{3} c n -a^{3} b^{5} n \right ) \RootOf \left (\textit {\_Z}^{2} a^{4}+c^{4} n^{2}+\left (-2 a^{2} c^{2} n +4 a \,b^{2} c n -b^{4} n \right ) \textit {\_Z} \right )+\left (9 a^{2} b^{2} c^{4} n^{2}-6 a \,b^{4} c^{3} n^{2}+b^{6} c^{2} n^{2}+\left (6 a^{7} c -2 a^{6} b^{2}\right ) \RootOf \left (\textit {\_Z}^{2} a^{4}+c^{4} n^{2}+\left (-2 a^{2} c^{2} n +4 a \,b^{2} c n -b^{4} n \right ) \textit {\_Z} \right )^{2}+\left (-6 a^{5} c^{3} n +9 a^{4} b^{2} c^{2} n -2 a^{3} b^{4} c n \right ) \RootOf \left (\textit {\_Z}^{2} a^{4}+c^{4} n^{2}+\left (-2 a^{2} c^{2} n +4 a \,b^{2} c n -b^{4} n \right ) \textit {\_Z} \right )\right ) x \right )-18 a^{2} b c n \,x^{3}+6 a \,b^{3} n \,x^{3}-3 i \pi \,a^{4} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )+3 i \pi \,a^{4} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}+3 i \pi \,a^{4} \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}-3 i \pi \,a^{4} \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3}+6 a^{3} c n \,x^{2}-3 a^{2} b^{2} n \,x^{2}+2 a^{3} b n x +6 a^{4} \ln \relax (d )}{24 a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/x^5,x)

[Out]

-1/4/x^4*ln((c*x^2+b*x+a)^n)-1/24*(12*ln(x)*a^2*c^2*n*x^4-24*ln(x)*a*b^2*c*n*x^4+6*ln(x)*b^4*n*x^4-3*I*Pi*a^4*
csgn(I*d)*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)+3*I*Pi*a^4*csgn(I*d)*csgn(I*d*(c*x^2+b*x+a)^n)^2+3
*I*Pi*a^4*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)^2-3*I*Pi*a^4*csgn(I*d*(c*x^2+b*x+a)^n)^3-6*sum(_R*
ln(((6*a^7*c-2*a^6*b^2)*_R^2+(-6*a^5*c^3*n+9*a^4*b^2*c^2*n-2*a^3*b^4*c*n)*_R+9*a^2*b^2*c^4*n^2-6*a*b^4*c^3*n^2
+b^6*c^2*n^2)*x-a^7*b*_R^2+(-5*a^5*b*c^2*n+5*a^4*b^3*c*n-a^3*b^5*n)*_R-6*a^3*b*c^4*n^2+14*a^2*b^3*c^3*n^2-7*a*
b^5*c^2*n^2+b^7*c*n^2),_R=RootOf(a^4*_Z^2+(-2*a^2*c^2*n+4*a*b^2*c*n-b^4*n)*_Z+c^4*n^2))*a^4*x^4-18*a^2*b*c*n*x
^3+6*a*b^3*n*x^3+6*a^3*c*n*x^2-3*a^2*b^2*n*x^2+2*a^3*b*n*x+6*ln(d)*a^4)/a^4/x^4

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^5,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 = 1.07, size = 627, normalized size = 3.30 \[ \frac {\ln \left (2\,a\,b^6+2\,b^7\,x-12\,a^4\,c^3+2\,a\,b^5\,\sqrt {b^2-4\,a\,c}+2\,b^6\,x\,\sqrt {b^2-4\,a\,c}-15\,a^2\,b^4\,c+31\,a^3\,b^2\,c^2+37\,a^2\,b^3\,c^2\,x-16\,a\,b^5\,c\,x-20\,a^3\,b\,c^3\,x-9\,a^2\,b^3\,c\,\sqrt {b^2-4\,a\,c}+7\,a^3\,b\,c^2\,\sqrt {b^2-4\,a\,c}-6\,a^3\,c^3\,x\,\sqrt {b^2-4\,a\,c}-12\,a\,b^4\,c\,x\,\sqrt {b^2-4\,a\,c}+19\,a^2\,b^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^4\,n}{8}-a\,\left (\frac {b^2\,c\,n}{2}+\frac {b\,c\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}+\frac {a^2\,c^2\,n}{4}\right )}{a^4}-\frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{4\,x^4}-\frac {\ln \relax (x)\,\left (2\,n\,a^2\,c^2-4\,n\,a\,b^2\,c+n\,b^4\right )}{4\,a^4}-\frac {\ln \left (12\,a^4\,c^3-2\,b^7\,x-2\,a\,b^6+2\,a\,b^5\,\sqrt {b^2-4\,a\,c}+2\,b^6\,x\,\sqrt {b^2-4\,a\,c}+15\,a^2\,b^4\,c-31\,a^3\,b^2\,c^2-37\,a^2\,b^3\,c^2\,x+16\,a\,b^5\,c\,x+20\,a^3\,b\,c^3\,x-9\,a^2\,b^3\,c\,\sqrt {b^2-4\,a\,c}+7\,a^3\,b\,c^2\,\sqrt {b^2-4\,a\,c}-6\,a^3\,c^3\,x\,\sqrt {b^2-4\,a\,c}-12\,a\,b^4\,c\,x\,\sqrt {b^2-4\,a\,c}+19\,a^2\,b^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a\,\left (\frac {b^2\,c\,n}{2}-\frac {b\,c\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )-\frac {b^4\,n}{8}+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}-\frac {a^2\,c^2\,n}{4}\right )}{a^4}-\frac {\frac {b\,n}{3\,a}+\frac {n\,x\,\left (2\,a\,c-b^2\right )}{2\,a^2}-\frac {b\,n\,x^2\,\left (3\,a\,c-b^2\right )}{a^3}}{4\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(d*(a + b*x + c*x^2)^n)/x^5,x)

[Out]

(log(2*a*b^6 + 2*b^7*x - 12*a^4*c^3 + 2*a*b^5*(b^2 - 4*a*c)^(1/2) + 2*b^6*x*(b^2 - 4*a*c)^(1/2) - 15*a^2*b^4*c
+ 31*a^3*b^2*c^2 + 37*a^2*b^3*c^2*x - 16*a*b^5*c*x - 20*a^3*b*c^3*x - 9*a^2*b^3*c*(b^2 - 4*a*c)^(1/2) + 7*a^3
*b*c^2*(b^2 - 4*a*c)^(1/2) - 6*a^3*c^3*x*(b^2 - 4*a*c)^(1/2) - 12*a*b^4*c*x*(b^2 - 4*a*c)^(1/2) + 19*a^2*b^2*c
^2*x*(b^2 - 4*a*c)^(1/2))*((b^4*n)/8 - a*((b^2*c*n)/2 + (b*c*n*(b^2 - 4*a*c)^(1/2))/4) + (b^3*n*(b^2 - 4*a*c)^
(1/2))/8 + (a^2*c^2*n)/4))/a^4 - log(d*(a + b*x + c*x^2)^n)/(4*x^4) - (log(x)*(b^4*n + 2*a^2*c^2*n - 4*a*b^2*c
*n))/(4*a^4) - (log(12*a^4*c^3 - 2*b^7*x - 2*a*b^6 + 2*a*b^5*(b^2 - 4*a*c)^(1/2) + 2*b^6*x*(b^2 - 4*a*c)^(1/2)
+ 15*a^2*b^4*c - 31*a^3*b^2*c^2 - 37*a^2*b^3*c^2*x + 16*a*b^5*c*x + 20*a^3*b*c^3*x - 9*a^2*b^3*c*(b^2 - 4*a*c
)^(1/2) + 7*a^3*b*c^2*(b^2 - 4*a*c)^(1/2) - 6*a^3*c^3*x*(b^2 - 4*a*c)^(1/2) - 12*a*b^4*c*x*(b^2 - 4*a*c)^(1/2)
+ 19*a^2*b^2*c^2*x*(b^2 - 4*a*c)^(1/2))*(a*((b^2*c*n)/2 - (b*c*n*(b^2 - 4*a*c)^(1/2))/4) - (b^4*n)/8 + (b^3*n
*(b^2 - 4*a*c)^(1/2))/8 - (a^2*c^2*n)/4))/a^4 - ((b*n)/(3*a) + (n*x*(2*a*c - b^2))/(2*a^2) - (b*n*x^2*(3*a*c -
b^2))/a^3)/(4*x^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(d*(c*x**2+b*x+a)**n)/x**5,x)

[Out]

Timed out

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