\(\int x^3 (a+b \log (c x^n))^2 \, dx\) [50]

Optimal result

Integrand size = 16, antiderivative size = 52 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{32} b^2 n^2 x^4-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \]

[Out]

1/32*b^2*n^2*x^4-1/8*b*n*x^4*(a+b*ln(c*x^n))+1/4*x^4*(a+b*ln(c*x^n))^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{32} b^2 n^2 x^4 \]

[In]

Int[x^3*(a + b*Log[c*x^n])^2,x]

[Out]

(b^2*n^2*x^4)/32 - (b*n*x^4*(a + b*Log[c*x^n]))/8 + (x^4*(a + b*Log[c*x^n])^2)/4

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \frac {1}{32} b^2 n^2 x^4-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{32} x^4 \left (b^2 n^2-4 b n \left (a+b \log \left (c x^n\right )\right )+8 \left (a+b \log \left (c x^n\right )\right )^2\right ) \]

[In]

Integrate[x^3*(a + b*Log[c*x^n])^2,x]

[Out]

(x^4*(b^2*n^2 - 4*b*n*(a + b*Log[c*x^n]) + 8*(a + b*Log[c*x^n])^2))/32

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.40

method	result	size
parallelrisch	\(\frac {x^{4} \ln \left (c \,x^{n}\right )^{2} b^{2}}{4}-\frac {\ln \left (c \,x^{n}\right ) x^{4} b^{2} n}{8}+\frac {b^{2} n^{2} x^{4}}{32}+\frac {\ln \left (c \,x^{n}\right ) x^{4} a b}{2}-\frac {a b n \,x^{4}}{8}+\frac {a^{2} x^{4}}{4}\)	\(73\)
risch	\(\frac {x^{4} b^{2} \ln \left (x^{n}\right )^{2}}{4}+\frac {b \,x^{4} \left (-2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 b \ln \left (c \right )-b n +4 a \right ) \ln \left (x^{n}\right )}{8}+\frac {x^{4} \left (8 a^{2}+2 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-8 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-8 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-8 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-8 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-8 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+4 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+b^{2} n^{2}+16 \ln \left (c \right ) a b +8 \ln \left (c \right )^{2} b^{2}-4 b^{2} \ln \left (c \right ) n -4 a b n -2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+8 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}\right )}{32}\)	\(691\)

[In]

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

[Out]

1/4*x^4*ln(c*x^n)^2*b^2-1/8*ln(c*x^n)*x^4*b^2*n+1/32*b^2*n^2*x^4+1/2*ln(c*x^n)*x^4*a*b-1/8*a*b*n*x^4+1/4*a^2*x
^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (46) = 92\).

Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.96 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} n^{2} x^{4} \log \left (x\right )^{2} + \frac {1}{4} \, b^{2} x^{4} \log \left (c\right )^{2} - \frac {1}{8} \, {\left (b^{2} n - 4 \, a b\right )} x^{4} \log \left (c\right ) + \frac {1}{32} \, {\left (b^{2} n^{2} - 4 \, a b n + 8 \, a^{2}\right )} x^{4} + \frac {1}{8} \, {\left (4 \, b^{2} n x^{4} \log \left (c\right ) - {\left (b^{2} n^{2} - 4 \, a b n\right )} x^{4}\right )} \log \left (x\right ) \]

[In]

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

[Out]

1/4*b^2*n^2*x^4*log(x)^2 + 1/4*b^2*x^4*log(c)^2 - 1/8*(b^2*n - 4*a*b)*x^4*log(c) + 1/32*(b^2*n^2 - 4*a*b*n + 8
*a^2)*x^4 + 1/8*(4*b^2*n*x^4*log(c) - (b^2*n^2 - 4*a*b*n)*x^4)*log(x)

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.50 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {a^{2} x^{4}}{4} - \frac {a b n x^{4}}{8} + \frac {a b x^{4} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} n^{2} x^{4}}{32} - \frac {b^{2} n x^{4} \log {\left (c x^{n} \right )}}{8} + \frac {b^{2} x^{4} \log {\left (c x^{n} \right )}^{2}}{4} \]

[In]

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

[Out]

a**2*x**4/4 - a*b*n*x**4/8 + a*b*x**4*log(c*x**n)/2 + b**2*n**2*x**4/32 - b**2*n*x**4*log(c*x**n)/8 + b**2*x**
4*log(c*x**n)**2/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac {1}{8} \, a b n x^{4} + \frac {1}{2} \, a b x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{32} \, {\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} \]

[In]

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

[Out]

1/4*b^2*x^4*log(c*x^n)^2 - 1/8*a*b*n*x^4 + 1/2*a*b*x^4*log(c*x^n) + 1/4*a^2*x^4 + 1/32*(n^2*x^4 - 4*n*x^4*log(
c*x^n))*b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (46) = 92\).

Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.13 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} n^{2} x^{4} \log \left (x\right )^{2} - \frac {1}{8} \, b^{2} n^{2} x^{4} \log \left (x\right ) + \frac {1}{2} \, b^{2} n x^{4} \log \left (c\right ) \log \left (x\right ) + \frac {1}{32} \, b^{2} n^{2} x^{4} - \frac {1}{8} \, b^{2} n x^{4} \log \left (c\right ) + \frac {1}{4} \, b^{2} x^{4} \log \left (c\right )^{2} + \frac {1}{2} \, a b n x^{4} \log \left (x\right ) - \frac {1}{8} \, a b n x^{4} + \frac {1}{2} \, a b x^{4} \log \left (c\right ) + \frac {1}{4} \, a^{2} x^{4} \]

[In]

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

[Out]

1/4*b^2*n^2*x^4*log(x)^2 - 1/8*b^2*n^2*x^4*log(x) + 1/2*b^2*n*x^4*log(c)*log(x) + 1/32*b^2*n^2*x^4 - 1/8*b^2*n
*x^4*log(c) + 1/4*b^2*x^4*log(c)^2 + 1/2*a*b*n*x^4*log(x) - 1/8*a*b*n*x^4 + 1/2*a*b*x^4*log(c) + 1/4*a^2*x^4

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.17 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=x^4\,\left (\frac {a^2}{4}-\frac {a\,b\,n}{8}+\frac {b^2\,n^2}{32}\right )+\frac {x^4\,\ln \left (c\,x^n\right )\,\left (a\,b-\frac {b^2\,n}{4}\right )}{2}+\frac {b^2\,x^4\,{\ln \left (c\,x^n\right )}^2}{4} \]

[In]

int(x^3*(a + b*log(c*x^n))^2,x)

[Out]

x^4*(a^2/4 + (b^2*n^2)/32 - (a*b*n)/8) + (x^4*log(c*x^n)*(a*b - (b^2*n)/4))/2 + (b^2*x^4*log(c*x^n)^2)/4