3.72 \(\int x^3 \log (d (a+b x+c x^2)^n) \, dx\)

Optimal. Leaf size=167 \[ -\frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {b n x \left (b^2-3 a c\right )}{4 c^3}-\frac {n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 167, 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} \[ -\frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac {n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac {b n x \left (b^2-3 a c\right )}{4 c^3}-\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[d*(a + b*x + c*x^2)^n],x]

[Out]

(b*(b^2 - 3*a*c)*n*x)/(4*c^3) - ((b^2 - 2*a*c)*n*x^2)/(8*c^2) + (b*n*x^3)/(12*c) - (n*x^4)/8 - (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*c^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a
+ b*x + c*x^2])/(8*c^4) + (x^4*Log[d*(a + b*x + c*x^2)^n])/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 x^3 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{4} n \int \frac {x^4 (b+2 c x)}{a+b x+c x^2} \, dx\\ &=\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{4} n \int \left (-\frac {b \left (b^2-3 a c\right )}{c^3}+\frac {\left (b^2-2 a c\right ) x}{c^2}-\frac {b x^2}{c}+2 x^3+\frac {a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {n \int \frac {a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{4 c^3}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\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 c^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 c^4}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}-\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 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\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 c^4}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}-\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 c^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 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 151, normalized size = 0.90 \[ \frac {-3 n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log (a+x (b+c x))-6 b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+c n x \left (2 b c \left (c x^2-9 a\right )-3 c^2 x \left (c x^2-2 a\right )+6 b^3-3 b^2 c x\right )+6 c^4 x^4 \log \left (d (a+x (b+c x))^n\right )}{24 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*n*x*(6*b^3 - 3*b^2*c*x + 2*b*c*(-9*a + c*x^2) - 3*c^2*x*(-2*a + c*x^2)) - 6*b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*a*
c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 3*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + x*(b + c*x)] + 6*c^4*x
^4*Log[d*(a + x*(b + c*x))^n])/(24*c^4)

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fricas [A]  time = 0.48, size = 364, normalized size = 2.18 \[ \left [-\frac {3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \relax (d) - 2 \, b c^{3} n x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 3 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} n x - 3 \, {\left (2 \, c^{4} n x^{4} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}, -\frac {3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \relax (d) - 2 \, b c^{3} n x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 6 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} n x - 3 \, {\left (2 \, c^{4} n x^{4} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 176, normalized size = 1.05 \[ \frac {1}{4} \, n x^{4} \log \left (c x^{2} + b x + a\right ) - \frac {1}{8} \, {\left (n - 2 \, \log \relax (d)\right )} x^{4} + \frac {b n x^{3}}{12 \, c} - \frac {{\left (b^{2} n - 2 \, a c n\right )} x^{2}}{8 \, c^{2}} + \frac {{\left (b^{3} n - 3 \, a b c n\right )} x}{4 \, c^{3}} - \frac {{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{8 \, c^{4}} + \frac {{\left (b^{5} n - 6 \, a b^{3} c n + 8 \, a^{2} b c^{2} n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.61, size = 1146, normalized size = 6.86 \[ -\frac {i \pi \,x^{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 )}{8}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{8}+\frac {i \pi \,x^{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}}{8}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3}}{8}-\frac {n \,x^{4}}{8}+\frac {x^{4} \ln \relax (d )}{4}+\frac {x^{4} \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )}{4}+\frac {b n \,x^{3}}{12 c}+\frac {a n \,x^{2}}{4 c}-\frac {b^{2} n \,x^{2}}{8 c^{2}}-\frac {a^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{4 c^{2}}-\frac {a^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{4 c^{2}}+\frac {a \,b^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{2 c^{3}}+\frac {a \,b^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{2 c^{3}}-\frac {3 a b n x}{4 c^{2}}-\frac {b^{4} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}-\frac {b^{4} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}+\frac {b^{3} n x}{4 c^{3}}+\frac {\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}-\frac {\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(d*(c*x^2+b*x+a)^n),x)

[Out]

1/4*x^4*ln((c*x^2+b*x+a)^n)+1/8*I*Pi*x^4*csgn(I*d)*csgn(I*d*(c*x^2+b*x+a)^n)^2-1/8*I*Pi*x^4*csgn(I*d*(c*x^2+b*
x+a)^n)^3-1/8*I*Pi*x^4*csgn(I*d)*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)+1/8*I*Pi*x^4*csgn(I*(c*x^2+
b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)^2+1/4*ln(d)*x^4-1/8*n*x^4+1/12*b*n*x^3/c+1/4/c*a*n*x^2-1/8*b^2*n*x^2/c^2-1
/4/c^2*n*ln(8*a^2*b*c^2-6*a*b^3*c+b^5+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x+(-16*a^3*b^2*
c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*a^2+1/2/c^3*n*ln(8*a^2*b*c^2-6*a*b^3*c+b^5+2*(-16*a^3*b^2*c^3+20*a^
2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x+(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*a*b^2-1/8/c^4*n*ln(
8*a^2*b*c^2-6*a*b^3*c+b^5+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x+(-16*a^3*b^2*c^3+20*a^2*b
^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*b^4-1/4/c^2*n*ln(8*a^2*b*c^2-6*a*b^3*c+b^5-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*
a*b^6*c+b^8)^(1/2)*c*x-(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*a^2+1/2/c^3*n*ln(8*a^2*b*c^2-6*
a*b^3*c+b^5-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x-(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6
*c+b^8)^(1/2)*b)*a*b^2-1/8/c^4*n*ln(8*a^2*b*c^2-6*a*b^3*c+b^5-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)
^(1/2)*c*x-(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*b^4-3/4/c^2*a*b*n*x+1/4*b^3*n*x/c^3-1/8/c^4
*n*ln(8*a^2*b*c^2-6*a*b^3*c+b^5+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x+(-16*a^3*b^2*c^3+20
*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b)*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)+1/8/c^4*n*ln(8*a^2*b
*c^2-6*a*b^3*c+b^5-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*c*x-(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-
8*a*b^6*c+b^8)^(1/2)*b)*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)

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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(x^3*log(d*(c*x^2+b*x+a)^n),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 = 0.54, size = 288, normalized size = 1.72 \[ x\,\left (\frac {b\,\left (\frac {b^2\,n}{4\,c^2}-\frac {a\,n}{2\,c}\right )}{c}-\frac {a\,b\,n}{4\,c^2}\right )-\frac {n\,x^4}{8}+\frac {x^4\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{4}-x^2\,\left (\frac {b^2\,n}{8\,c^2}-\frac {a\,n}{4\,c}\right )+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (c\,\left (\frac {a\,b^2\,n}{2}-\frac {a\,b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )-\frac {b^4\,n}{8}+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}-\frac {a^2\,c^2\,n}{4}\right )}{c^4}-\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^4\,n}{8}-c\,\left (\frac {a\,b^2\,n}{2}+\frac {a\,b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}+\frac {a^2\,c^2\,n}{4}\right )}{c^4}+\frac {b\,n\,x^3}{12\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*log(d*(a + b*x + c*x^2)^n),x)

[Out]

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

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

[Out]

Timed out

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