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

Optimal. Leaf size=206 \[ -\frac{1}{8} e r x^2 \left (2 a^2-2 a b n+b^2 n^2\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{4} b e r x^2 (2 a-b n) \log \left (c x^n\right )+\frac{1}{8} b e n r x^2 (2 a-b n)-\frac{1}{4} b^2 e r x^2 \log ^2\left (c x^n\right )+\frac{1}{4} b^2 e n r x^2 \log \left (c x^n\right )+\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{8} b^2 e n^2 r x^2 \]

[Out]

-(b^2*e*n^2*r*x^2)/8 + (b*e*n*(2*a - b*n)*r*x^2)/8 - (e*(2*a^2 - 2*a*b*n + b^2*n^2)*r*x^2)/8 + (b^2*e*n*r*x^2*
Log[c*x^n])/4 - (b*e*(2*a - b*n)*r*x^2*Log[c*x^n])/4 - (b^2*e*r*x^2*Log[c*x^n]^2)/4 + (b^2*n^2*x^2*(d + e*Log[
f*x^r]))/4 - (b*n*x^2*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/2 + (x^2*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))
/2

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Rubi [A]  time = 0.165645, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2305, 2304, 2366, 12, 14} \[ -\frac{1}{8} e r x^2 \left (2 a^2-2 a b n+b^2 n^2\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-\frac{1}{4} b e r x^2 (2 a-b n) \log \left (c x^n\right )+\frac{1}{8} b e n r x^2 (2 a-b n)-\frac{1}{4} b^2 e r x^2 \log ^2\left (c x^n\right )+\frac{1}{4} b^2 e n r x^2 \log \left (c x^n\right )+\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{8} b^2 e n^2 r x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*e*n^2*r*x^2)/8 + (b*e*n*(2*a - b*n)*r*x^2)/8 - (e*(2*a^2 - 2*a*b*n + b^2*n^2)*r*x^2)/8 + (b^2*e*n*r*x^2*
Log[c*x^n])/4 - (b*e*(2*a - b*n)*r*x^2*Log[c*x^n])/4 - (b^2*e*r*x^2*Log[c*x^n]^2)/4 + (b^2*n^2*x^2*(d + e*Log[
f*x^r]))/4 - (b*n*x^2*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/2 + (x^2*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))
/2

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 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right ) \, dx &=\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \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}{4} x \left (2 a^2 \left (1+\frac{b n (-2 a+b n)}{2 a^2}\right )-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \, dx\\ &=\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{4} (e r) \int x \left (2 a^2 \left (1+\frac{b n (-2 a+b n)}{2 a^2}\right )-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \, dx\\ &=\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{4} (e r) \int \left (\left (2 a^2-2 a b n+b^2 n^2\right ) x-2 b (-2 a+b n) x \log \left (c x^n\right )+2 b^2 x \log ^2\left (c x^n\right )\right ) \, dx\\ &=-\frac{1}{8} e \left (2 a^2-2 a b n+b^2 n^2\right ) r x^2+\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} \left (b^2 e r\right ) \int x \log ^2\left (c x^n\right ) \, dx-\frac{1}{2} (b e (2 a-b n) r) \int x \log \left (c x^n\right ) \, dx\\ &=\frac{1}{8} b e n (2 a-b n) r x^2-\frac{1}{8} e \left (2 a^2-2 a b n+b^2 n^2\right ) r x^2-\frac{1}{4} b e (2 a-b n) r x^2 \log \left (c x^n\right )-\frac{1}{4} b^2 e r x^2 \log ^2\left (c x^n\right )+\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} \left (b^2 e n r\right ) \int x \log \left (c x^n\right ) \, dx\\ &=-\frac{1}{8} b^2 e n^2 r x^2+\frac{1}{8} b e n (2 a-b n) r x^2-\frac{1}{8} e \left (2 a^2-2 a b n+b^2 n^2\right ) r x^2+\frac{1}{4} b^2 e n r x^2 \log \left (c x^n\right )-\frac{1}{4} b e (2 a-b n) r x^2 \log \left (c x^n\right )-\frac{1}{4} b^2 e r x^2 \log ^2\left (c x^n\right )+\frac{1}{4} b^2 n^2 x^2 \left (d+e \log \left (f x^r\right )\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )+\frac{1}{2} x^2 \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 = 0.130786, size = 154, normalized size = 0.75 \[ \frac{1}{8} x^2 \left (2 e \left (2 a^2-2 a b n+b^2 n^2\right ) \log \left (f x^r\right )+4 a^2 d-2 a^2 e r-4 b \log \left (c x^n\right ) \left ((b e n-2 a e) \log \left (f x^r\right )-2 a d+a e r+b d n-b e n r\right )-4 a b d n+4 a b e n r+2 b^2 \log ^2\left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )-e r\right )+2 b^2 d n^2-3 b^2 e n^2 r\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(4*a^2*d - 4*a*b*d*n + 2*b^2*d*n^2 - 2*a^2*e*r + 4*a*b*e*n*r - 3*b^2*e*n^2*r + 2*e*(2*a^2 - 2*a*b*n + b^2
*n^2)*Log[f*x^r] + 2*b^2*Log[c*x^n]^2*(2*d - e*r + 2*e*Log[f*x^r]) - 4*b*Log[c*x^n]*(-2*a*d + b*d*n + a*e*r -
b*e*n*r + (-2*a*e + b*e*n)*Log[f*x^r])))/8

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Maple [C]  time = 0.52, size = 9262, normalized size = 45. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]  time = 1.21365, size = 333, normalized size = 1.62 \begin{align*} \frac{1}{2} \, b^{2} d x^{2} \log \left (c x^{n}\right )^{2} - \frac{1}{2} \, a b d n x^{2} - \frac{1}{4} \, a^{2} e r x^{2} + a b d x^{2} \log \left (c x^{n}\right ) - \frac{1}{4} \,{\left (r x^{2} - 2 \, x^{2} \log \left (f x^{r}\right )\right )} b^{2} e \log \left (c x^{n}\right )^{2} + \frac{1}{2} \, a^{2} e x^{2} \log \left (f x^{r}\right ) + \frac{1}{2} \,{\left ({\left (r - \log \left (f\right )\right )} x^{2} - x^{2} \log \left (x^{r}\right )\right )} a b e n + \frac{1}{2} \, a^{2} d x^{2} - \frac{1}{2} \,{\left (r x^{2} - 2 \, x^{2} \log \left (f x^{r}\right )\right )} a b e \log \left (c x^{n}\right ) + \frac{1}{4} \,{\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d - \frac{1}{8} \,{\left ({\left ({\left (3 \, r - 2 \, \log \left (f\right )\right )} x^{2} - 2 \, x^{2} \log \left (x^{r}\right )\right )} n^{2} - 4 \,{\left ({\left (r - \log \left (f\right )\right )} x^{2} - x^{2} \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*(a+b*log(c*x^n))^2*(d+e*log(f*x^r)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.832257, size = 892, normalized size = 4.33 \begin{align*} \frac{1}{2} \, b^{2} e n^{2} r x^{2} \log \left (x\right )^{3} - \frac{1}{4} \,{\left (b^{2} e r - 2 \, b^{2} d\right )} x^{2} \log \left (c\right )^{2} - \frac{1}{2} \,{\left (b^{2} d n - 2 \, a b d -{\left (b^{2} e n - a b e\right )} r\right )} x^{2} \log \left (c\right ) + \frac{1}{8} \,{\left (2 \, b^{2} d n^{2} - 4 \, a b d n + 4 \, a^{2} d -{\left (3 \, b^{2} e n^{2} - 4 \, a b e n + 2 \, a^{2} e\right )} r\right )} x^{2} + \frac{1}{4} \,{\left (4 \, b^{2} e n r x^{2} \log \left (c\right ) + 2 \, b^{2} e n^{2} x^{2} \log \left (f\right ) +{\left (2 \, b^{2} d n^{2} -{\left (3 \, b^{2} e n^{2} - 4 \, a b e n\right )} r\right )} x^{2}\right )} \log \left (x\right )^{2} + \frac{1}{4} \,{\left (2 \, b^{2} e x^{2} \log \left (c\right )^{2} - 2 \,{\left (b^{2} e n - 2 \, a b e\right )} x^{2} \log \left (c\right ) +{\left (b^{2} e n^{2} - 2 \, a b e n + 2 \, a^{2} e\right )} x^{2}\right )} \log \left (f\right ) + \frac{1}{4} \,{\left (2 \, b^{2} e r x^{2} \log \left (c\right )^{2} + 4 \,{\left (b^{2} d n -{\left (b^{2} e n - a b e\right )} r\right )} x^{2} \log \left (c\right ) -{\left (2 \, b^{2} d n^{2} - 4 \, a b d n -{\left (3 \, b^{2} e n^{2} - 4 \, a b e n + 2 \, a^{2} e\right )} r\right )} x^{2} + 2 \,{\left (2 \, b^{2} e n x^{2} \log \left (c\right ) -{\left (b^{2} e n^{2} - 2 \, a b e n\right )} x^{2}\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*(a+b*log(c*x^n))^2*(d+e*log(f*x^r)),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 36.6781, size = 600, normalized size = 2.91 \begin{align*} \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e r x^{2} \log{\left (x \right )}}{2} - \frac{a^{2} e r x^{2}}{4} + \frac{a^{2} e x^{2} \log{\left (f \right )}}{2} + a b d n x^{2} \log{\left (x \right )} - \frac{a b d n x^{2}}{2} + a b d x^{2} \log{\left (c \right )} + a b e n r x^{2} \log{\left (x \right )}^{2} - a b e n r x^{2} \log{\left (x \right )} + \frac{a b e n r x^{2}}{2} + a b e n x^{2} \log{\left (f \right )} \log{\left (x \right )} - \frac{a b e n x^{2} \log{\left (f \right )}}{2} + a b e r x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{a b e r x^{2} \log{\left (c \right )}}{2} + a b e x^{2} \log{\left (c \right )} \log{\left (f \right )} + \frac{b^{2} d n^{2} x^{2} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} d n^{2} x^{2} \log{\left (x \right )}}{2} + \frac{b^{2} d n^{2} x^{2}}{4} + b^{2} d n x^{2} \log{\left (c \right )} \log{\left (x \right )} - \frac{b^{2} d n x^{2} \log{\left (c \right )}}{2} + \frac{b^{2} d x^{2} \log{\left (c \right )}^{2}}{2} + \frac{b^{2} e n^{2} r x^{2} \log{\left (x \right )}^{3}}{2} - \frac{3 b^{2} e n^{2} r x^{2} \log{\left (x \right )}^{2}}{4} + \frac{3 b^{2} e n^{2} r x^{2} \log{\left (x \right )}}{4} - \frac{3 b^{2} e n^{2} r x^{2}}{8} + \frac{b^{2} e n^{2} x^{2} \log{\left (f \right )} \log{\left (x \right )}^{2}}{2} - \frac{b^{2} e n^{2} x^{2} \log{\left (f \right )} \log{\left (x \right )}}{2} + \frac{b^{2} e n^{2} x^{2} \log{\left (f \right )}}{4} + b^{2} e n r x^{2} \log{\left (c \right )} \log{\left (x \right )}^{2} - b^{2} e n r x^{2} \log{\left (c \right )} \log{\left (x \right )} + \frac{b^{2} e n r x^{2} \log{\left (c \right )}}{2} + b^{2} e n x^{2} \log{\left (c \right )} \log{\left (f \right )} \log{\left (x \right )} - \frac{b^{2} e n x^{2} \log{\left (c \right )} \log{\left (f \right )}}{2} + \frac{b^{2} e r x^{2} \log{\left (c \right )}^{2} \log{\left (x \right )}}{2} - \frac{b^{2} e r x^{2} \log{\left (c \right )}^{2}}{4} + \frac{b^{2} e x^{2} \log{\left (c \right )}^{2} \log{\left (f \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*x**2/2 + a**2*e*r*x**2*log(x)/2 - a**2*e*r*x**2/4 + a**2*e*x**2*log(f)/2 + a*b*d*n*x**2*log(x) - a*b*d*
n*x**2/2 + a*b*d*x**2*log(c) + a*b*e*n*r*x**2*log(x)**2 - a*b*e*n*r*x**2*log(x) + a*b*e*n*r*x**2/2 + a*b*e*n*x
**2*log(f)*log(x) - a*b*e*n*x**2*log(f)/2 + a*b*e*r*x**2*log(c)*log(x) - a*b*e*r*x**2*log(c)/2 + a*b*e*x**2*lo
g(c)*log(f) + b**2*d*n**2*x**2*log(x)**2/2 - b**2*d*n**2*x**2*log(x)/2 + b**2*d*n**2*x**2/4 + b**2*d*n*x**2*lo
g(c)*log(x) - b**2*d*n*x**2*log(c)/2 + b**2*d*x**2*log(c)**2/2 + b**2*e*n**2*r*x**2*log(x)**3/2 - 3*b**2*e*n**
2*r*x**2*log(x)**2/4 + 3*b**2*e*n**2*r*x**2*log(x)/4 - 3*b**2*e*n**2*r*x**2/8 + b**2*e*n**2*x**2*log(f)*log(x)
**2/2 - b**2*e*n**2*x**2*log(f)*log(x)/2 + b**2*e*n**2*x**2*log(f)/4 + b**2*e*n*r*x**2*log(c)*log(x)**2 - b**2
*e*n*r*x**2*log(c)*log(x) + b**2*e*n*r*x**2*log(c)/2 + b**2*e*n*x**2*log(c)*log(f)*log(x) - b**2*e*n*x**2*log(
c)*log(f)/2 + b**2*e*r*x**2*log(c)**2*log(x)/2 - b**2*e*r*x**2*log(c)**2/4 + b**2*e*x**2*log(c)**2*log(f)/2

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Giac [B]  time = 1.33374, size = 671, normalized size = 3.26 \begin{align*} \frac{1}{2} \, b^{2} n^{2} r x^{2} e \log \left (x\right )^{3} - \frac{3}{4} \, b^{2} n^{2} r x^{2} e \log \left (x\right )^{2} + b^{2} n r x^{2} e \log \left (c\right ) \log \left (x\right )^{2} + \frac{1}{2} \, b^{2} n^{2} x^{2} e \log \left (f\right ) \log \left (x\right )^{2} + \frac{3}{4} \, b^{2} n^{2} r x^{2} e \log \left (x\right ) - b^{2} n r x^{2} e \log \left (c\right ) \log \left (x\right ) + \frac{1}{2} \, b^{2} r x^{2} e \log \left (c\right )^{2} \log \left (x\right ) - \frac{1}{2} \, b^{2} n^{2} x^{2} e \log \left (f\right ) \log \left (x\right ) + b^{2} n x^{2} e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + \frac{1}{2} \, b^{2} d n^{2} x^{2} \log \left (x\right )^{2} + a b n r x^{2} e \log \left (x\right )^{2} - \frac{3}{8} \, b^{2} n^{2} r x^{2} e + \frac{1}{2} \, b^{2} n r x^{2} e \log \left (c\right ) - \frac{1}{4} \, b^{2} r x^{2} e \log \left (c\right )^{2} + \frac{1}{4} \, b^{2} n^{2} x^{2} e \log \left (f\right ) - \frac{1}{2} \, b^{2} n x^{2} e \log \left (c\right ) \log \left (f\right ) + \frac{1}{2} \, b^{2} x^{2} e \log \left (c\right )^{2} \log \left (f\right ) - \frac{1}{2} \, b^{2} d n^{2} x^{2} \log \left (x\right ) - a b n r x^{2} e \log \left (x\right ) + b^{2} d n x^{2} \log \left (c\right ) \log \left (x\right ) + a b r x^{2} e \log \left (c\right ) \log \left (x\right ) + a b n x^{2} e \log \left (f\right ) \log \left (x\right ) + \frac{1}{4} \, b^{2} d n^{2} x^{2} + \frac{1}{2} \, a b n r x^{2} e - \frac{1}{2} \, b^{2} d n x^{2} \log \left (c\right ) - \frac{1}{2} \, a b r x^{2} e \log \left (c\right ) + \frac{1}{2} \, b^{2} d x^{2} \log \left (c\right )^{2} - \frac{1}{2} \, a b n x^{2} e \log \left (f\right ) + a b x^{2} e \log \left (c\right ) \log \left (f\right ) + a b d n x^{2} \log \left (x\right ) + \frac{1}{2} \, a^{2} r x^{2} e \log \left (x\right ) - \frac{1}{2} \, a b d n x^{2} - \frac{1}{4} \, a^{2} r x^{2} e + a b d x^{2} \log \left (c\right ) + \frac{1}{2} \, a^{2} x^{2} e \log \left (f\right ) + \frac{1}{2} \, a^{2} d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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