3.163 \(\int x^2 (a+b \log (c x^n))^2 (d+e \log (f x^r)) \, dx\)

Optimal. Leaf size=207 \[ -\frac{1}{81} e r x^3 \left (9 a^2-6 a b n+2 b^2 n^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{2}{27} b e r x^3 (3 a-b n) \log \left (c x^n\right )+\frac{2}{81} b e n r x^3 (3 a-b n)-\frac{1}{9} b^2 e r x^3 \log ^2\left (c x^n\right )+\frac{2}{27} b^2 e n r x^3 \log \left (c x^n\right )+\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{81} b^2 e n^2 r x^3 \]

[Out]

(-2*b^2*e*n^2*r*x^3)/81 + (2*b*e*n*(3*a - b*n)*r*x^3)/81 - (e*(9*a^2 - 6*a*b*n + 2*b^2*n^2)*r*x^3)/81 + (2*b^2
*e*n*r*x^3*Log[c*x^n])/27 - (2*b*e*(3*a - b*n)*r*x^3*Log[c*x^n])/27 - (b^2*e*r*x^3*Log[c*x^n]^2)/9 + (2*b^2*n^
2*x^3*(d + e*Log[f*x^r]))/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/9 + (x^3*(a + b*Log[c*x^n])^2
*(d + e*Log[f*x^r]))/3

________________________________________________________________________________________

Rubi [A]  time = 0.20276, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2305, 2304, 2366, 12, 14} \[ -\frac{1}{81} e r x^3 \left (9 a^2-6 a b n+2 b^2 n^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{2}{27} b e r x^3 (3 a-b n) \log \left (c x^n\right )+\frac{2}{81} b e n r x^3 (3 a-b n)-\frac{1}{9} b^2 e r x^3 \log ^2\left (c x^n\right )+\frac{2}{27} b^2 e n r x^3 \log \left (c x^n\right )+\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{81} b^2 e n^2 r x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*e*n^2*r*x^3)/81 + (2*b*e*n*(3*a - b*n)*r*x^3)/81 - (e*(9*a^2 - 6*a*b*n + 2*b^2*n^2)*r*x^3)/81 + (2*b^2
*e*n*r*x^3*Log[c*x^n])/27 - (2*b*e*(3*a - b*n)*r*x^3*Log[c*x^n])/27 - (b^2*e*r*x^3*Log[c*x^n]^2)/9 + (2*b^2*n^
2*x^3*(d + e*Log[f*x^r]))/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/9 + (x^3*(a + b*Log[c*x^n])^2
*(d + e*Log[f*x^r]))/3

Rule 2305

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]

Rule 2304

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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right ) \, dx &=\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-(e r) \int \frac{1}{27} x^2 \left (2 b^2 n^2-6 b n \left (a+b \log \left (c x^n\right )\right )+9 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{27} (e r) \int x^2 \left (2 b^2 n^2-6 b n \left (a+b \log \left (c x^n\right )\right )+9 \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{27} (e r) \int \left (\left (9 a^2-6 a b n+2 b^2 n^2\right ) x^2-6 b (-3 a+b n) x^2 \log \left (c x^n\right )+9 b^2 x^2 \log ^2\left (c x^n\right )\right ) \, dx\\ &=-\frac{1}{81} e \left (9 a^2-6 a b n+2 b^2 n^2\right ) r x^3+\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{3} \left (b^2 e r\right ) \int x^2 \log ^2\left (c x^n\right ) \, dx-\frac{1}{9} (2 b e (3 a-b n) r) \int x^2 \log \left (c x^n\right ) \, dx\\ &=\frac{2}{81} b e n (3 a-b n) r x^3-\frac{1}{81} e \left (9 a^2-6 a b n+2 b^2 n^2\right ) r x^3-\frac{2}{27} b e (3 a-b n) r x^3 \log \left (c x^n\right )-\frac{1}{9} b^2 e r x^3 \log ^2\left (c x^n\right )+\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{9} \left (2 b^2 e n r\right ) \int x^2 \log \left (c x^n\right ) \, dx\\ &=-\frac{2}{81} b^2 e n^2 r x^3+\frac{2}{81} b e n (3 a-b n) r x^3-\frac{1}{81} e \left (9 a^2-6 a b n+2 b^2 n^2\right ) r x^3+\frac{2}{27} b^2 e n r x^3 \log \left (c x^n\right )-\frac{2}{27} b e (3 a-b n) r x^3 \log \left (c x^n\right )-\frac{1}{9} b^2 e r x^3 \log ^2\left (c x^n\right )+\frac{2}{27} b^2 n^2 x^3 \left (d+e \log \left (f x^r\right )\right )-\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.142707, size = 157, normalized size = 0.76 \[ \frac{1}{27} x^3 \left (e \left (9 a^2-6 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+9 a^2 d-3 a^2 e r+2 b \log \left (c x^n\right ) \left ((9 a e-3 b e n) \log \left (f x^r\right )+9 a d-3 a e r-3 b d n+2 b e n r\right )-6 a b d n+4 a b e n r+3 b^2 \log ^2\left (c x^n\right ) \left (3 d+3 e \log \left (f x^r\right )-e r\right )+2 b^2 d n^2-2 b^2 e n^2 r\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(9*a^2*d - 6*a*b*d*n + 2*b^2*d*n^2 - 3*a^2*e*r + 4*a*b*e*n*r - 2*b^2*e*n^2*r + e*(9*a^2 - 6*a*b*n + 2*b^2
*n^2)*Log[f*x^r] + 3*b^2*Log[c*x^n]^2*(3*d - e*r + 3*e*Log[f*x^r]) + 2*b*Log[c*x^n]*(9*a*d - 3*b*d*n - 3*a*e*r
+ 2*b*e*n*r + (9*a*e - 3*b*e*n)*Log[f*x^r])))/27

________________________________________________________________________________________

Maple [C]  time = 0.522, size = 9271, normalized size = 44.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))^2*(d+e*ln(f*x^r)),x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [A]  time = 1.23961, size = 338, normalized size = 1.63 \begin{align*} \frac{1}{3} \, b^{2} d x^{3} \log \left (c x^{n}\right )^{2} - \frac{2}{9} \, a b d n x^{3} - \frac{1}{9} \, a^{2} e r x^{3} + \frac{2}{3} \, a b d x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a^{2} e x^{3} \log \left (f x^{r}\right ) + \frac{1}{3} \, a^{2} d x^{3} - \frac{1}{9} \,{\left (r x^{3} - 3 \, x^{3} \log \left (f x^{r}\right )\right )} b^{2} e \log \left (c x^{n}\right )^{2} + \frac{2}{27} \,{\left ({\left (2 \, r - 3 \, \log \left (f\right )\right )} x^{3} - 3 \, x^{3} \log \left (x^{r}\right )\right )} a b e n - \frac{2}{9} \,{\left (r x^{3} - 3 \, x^{3} \log \left (f x^{r}\right )\right )} a b e \log \left (c x^{n}\right ) + \frac{2}{27} \,{\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d - \frac{2}{27} \,{\left ({\left ({\left (r - \log \left (f\right )\right )} x^{3} - x^{3} \log \left (x^{r}\right )\right )} n^{2} -{\left ({\left (2 \, r - 3 \, \log \left (f\right )\right )} x^{3} - 3 \, x^{3} \log \left (x^{r}\right )\right )} n \log \left (c x^{n}\right )\right )} b^{2} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^2*(d+e*log(f*x^r)),x, algorithm="maxima")

[Out]

1/3*b^2*d*x^3*log(c*x^n)^2 - 2/9*a*b*d*n*x^3 - 1/9*a^2*e*r*x^3 + 2/3*a*b*d*x^3*log(c*x^n) + 1/3*a^2*e*x^3*log(
f*x^r) + 1/3*a^2*d*x^3 - 1/9*(r*x^3 - 3*x^3*log(f*x^r))*b^2*e*log(c*x^n)^2 + 2/27*((2*r - 3*log(f))*x^3 - 3*x^
3*log(x^r))*a*b*e*n - 2/9*(r*x^3 - 3*x^3*log(f*x^r))*a*b*e*log(c*x^n) + 2/27*(n^2*x^3 - 3*n*x^3*log(c*x^n))*b^
2*d - 2/27*(((r - log(f))*x^3 - x^3*log(x^r))*n^2 - ((2*r - 3*log(f))*x^3 - 3*x^3*log(x^r))*n*log(c*x^n))*b^2*
e

________________________________________________________________________________________

Fricas [B]  time = 0.931913, size = 907, normalized size = 4.38 \begin{align*} \frac{1}{3} \, b^{2} e n^{2} r x^{3} \log \left (x\right )^{3} - \frac{1}{9} \,{\left (b^{2} e r - 3 \, b^{2} d\right )} x^{3} \log \left (c\right )^{2} - \frac{2}{27} \,{\left (3 \, b^{2} d n - 9 \, a b d -{\left (2 \, b^{2} e n - 3 \, a b e\right )} r\right )} x^{3} \log \left (c\right ) + \frac{1}{27} \,{\left (2 \, b^{2} d n^{2} - 6 \, a b d n + 9 \, a^{2} d -{\left (2 \, b^{2} e n^{2} - 4 \, a b e n + 3 \, a^{2} e\right )} r\right )} x^{3} + \frac{1}{3} \,{\left (2 \, b^{2} e n r x^{3} \log \left (c\right ) + b^{2} e n^{2} x^{3} \log \left (f\right ) +{\left (b^{2} d n^{2} -{\left (b^{2} e n^{2} - 2 \, a b e n\right )} r\right )} x^{3}\right )} \log \left (x\right )^{2} + \frac{1}{27} \,{\left (9 \, b^{2} e x^{3} \log \left (c\right )^{2} - 6 \,{\left (b^{2} e n - 3 \, a b e\right )} x^{3} \log \left (c\right ) +{\left (2 \, b^{2} e n^{2} - 6 \, a b e n + 9 \, a^{2} e\right )} x^{3}\right )} \log \left (f\right ) + \frac{1}{9} \,{\left (3 \, b^{2} e r x^{3} \log \left (c\right )^{2} + 2 \,{\left (3 \, b^{2} d n -{\left (2 \, b^{2} e n - 3 \, a b e\right )} r\right )} x^{3} \log \left (c\right ) -{\left (2 \, b^{2} d n^{2} - 6 \, a b d n -{\left (2 \, b^{2} e n^{2} - 4 \, a b e n + 3 \, a^{2} e\right )} r\right )} x^{3} + 2 \,{\left (3 \, b^{2} e n x^{3} \log \left (c\right ) -{\left (b^{2} e n^{2} - 3 \, a b e n\right )} x^{3}\right )} \log \left (f\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*e*n^2*r*x^3*log(x)^3 - 1/9*(b^2*e*r - 3*b^2*d)*x^3*log(c)^2 - 2/27*(3*b^2*d*n - 9*a*b*d - (2*b^2*e*n -
3*a*b*e)*r)*x^3*log(c) + 1/27*(2*b^2*d*n^2 - 6*a*b*d*n + 9*a^2*d - (2*b^2*e*n^2 - 4*a*b*e*n + 3*a^2*e)*r)*x^3
+ 1/3*(2*b^2*e*n*r*x^3*log(c) + b^2*e*n^2*x^3*log(f) + (b^2*d*n^2 - (b^2*e*n^2 - 2*a*b*e*n)*r)*x^3)*log(x)^2
+ 1/27*(9*b^2*e*x^3*log(c)^2 - 6*(b^2*e*n - 3*a*b*e)*x^3*log(c) + (2*b^2*e*n^2 - 6*a*b*e*n + 9*a^2*e)*x^3)*log
(f) + 1/9*(3*b^2*e*r*x^3*log(c)^2 + 2*(3*b^2*d*n - (2*b^2*e*n - 3*a*b*e)*r)*x^3*log(c) - (2*b^2*d*n^2 - 6*a*b*
d*n - (2*b^2*e*n^2 - 4*a*b*e*n + 3*a^2*e)*r)*x^3 + 2*(3*b^2*e*n*x^3*log(c) - (b^2*e*n^2 - 3*a*b*e*n)*x^3)*log(
f))*log(x)

________________________________________________________________________________________

Sympy [B]  time = 110.905, size = 654, normalized size = 3.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))**2*(d+e*ln(f*x**r)),x)

[Out]

a**2*d*x**3/3 + a**2*e*r*x**3*log(x)/3 - a**2*e*r*x**3/9 + a**2*e*x**3*log(f)/3 + 2*a*b*d*n*x**3*log(x)/3 - 2*
a*b*d*n*x**3/9 + 2*a*b*d*x**3*log(c)/3 + 2*a*b*e*n*r*x**3*log(x)**2/3 - 4*a*b*e*n*r*x**3*log(x)/9 + 4*a*b*e*n*
r*x**3/27 + 2*a*b*e*n*x**3*log(f)*log(x)/3 - 2*a*b*e*n*x**3*log(f)/9 + 2*a*b*e*r*x**3*log(c)*log(x)/3 - 2*a*b*
e*r*x**3*log(c)/9 + 2*a*b*e*x**3*log(c)*log(f)/3 + b**2*d*n**2*x**3*log(x)**2/3 - 2*b**2*d*n**2*x**3*log(x)/9
+ 2*b**2*d*n**2*x**3/27 + 2*b**2*d*n*x**3*log(c)*log(x)/3 - 2*b**2*d*n*x**3*log(c)/9 + b**2*d*x**3*log(c)**2/3
+ b**2*e*n**2*r*x**3*log(x)**3/3 - b**2*e*n**2*r*x**3*log(x)**2/3 + 2*b**2*e*n**2*r*x**3*log(x)/9 - 2*b**2*e*
n**2*r*x**3/27 + b**2*e*n**2*x**3*log(f)*log(x)**2/3 - 2*b**2*e*n**2*x**3*log(f)*log(x)/9 + 2*b**2*e*n**2*x**3
*log(f)/27 + 2*b**2*e*n*r*x**3*log(c)*log(x)**2/3 - 4*b**2*e*n*r*x**3*log(c)*log(x)/9 + 4*b**2*e*n*r*x**3*log(
c)/27 + 2*b**2*e*n*x**3*log(c)*log(f)*log(x)/3 - 2*b**2*e*n*x**3*log(c)*log(f)/9 + b**2*e*r*x**3*log(c)**2*log
(x)/3 - b**2*e*r*x**3*log(c)**2/9 + b**2*e*x**3*log(c)**2*log(f)/3

________________________________________________________________________________________

Giac [B]  time = 1.30142, size = 683, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*n^2*r*x^3*e*log(x)^3 - 1/3*b^2*n^2*r*x^3*e*log(x)^2 + 2/3*b^2*n*r*x^3*e*log(c)*log(x)^2 + 1/3*b^2*n^2*
x^3*e*log(f)*log(x)^2 + 2/9*b^2*n^2*r*x^3*e*log(x) - 4/9*b^2*n*r*x^3*e*log(c)*log(x) + 1/3*b^2*r*x^3*e*log(c)^
2*log(x) - 2/9*b^2*n^2*x^3*e*log(f)*log(x) + 2/3*b^2*n*x^3*e*log(c)*log(f)*log(x) + 1/3*b^2*d*n^2*x^3*log(x)^2
+ 2/3*a*b*n*r*x^3*e*log(x)^2 - 2/27*b^2*n^2*r*x^3*e + 4/27*b^2*n*r*x^3*e*log(c) - 1/9*b^2*r*x^3*e*log(c)^2 +
2/27*b^2*n^2*x^3*e*log(f) - 2/9*b^2*n*x^3*e*log(c)*log(f) + 1/3*b^2*x^3*e*log(c)^2*log(f) - 2/9*b^2*d*n^2*x^3*
log(x) - 4/9*a*b*n*r*x^3*e*log(x) + 2/3*b^2*d*n*x^3*log(c)*log(x) + 2/3*a*b*r*x^3*e*log(c)*log(x) + 2/3*a*b*n*
x^3*e*log(f)*log(x) + 2/27*b^2*d*n^2*x^3 + 4/27*a*b*n*r*x^3*e - 2/9*b^2*d*n*x^3*log(c) - 2/9*a*b*r*x^3*e*log(c
) + 1/3*b^2*d*x^3*log(c)^2 - 2/9*a*b*n*x^3*e*log(f) + 2/3*a*b*x^3*e*log(c)*log(f) + 2/3*a*b*d*n*x^3*log(x) + 1
/3*a^2*r*x^3*e*log(x) - 2/9*a*b*d*n*x^3 - 1/9*a^2*r*x^3*e + 2/3*a*b*d*x^3*log(c) + 1/3*a^2*x^3*e*log(f) + 1/3*
a^2*d*x^3