3.2.64 \(\int x (a+b \log (c x^n))^2 (d+e \log (f x^r)) \, dx\) [164]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, 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 = {2342, 2341, 2413, 12, 14} \begin {gather*} -\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 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(b^2*e*n^2*r*x^2) + (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*Lo
g[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 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]]

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]

Rule 2413

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])

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*}

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Mathematica [A]
time = 0.07, size = 154, normalized size = 0.75 \begin {gather*} \frac {1}{8} x^2 \left (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 \left (2 a^2-2 a b n+b^2 n^2\right ) \log \left (f x^r\right )+2 b^2 \log ^2\left (c x^n\right ) \left (2 d-e r+2 e \log \left (f x^r\right )\right )-4 b \log \left (c x^n\right ) \left (-2 a d+b d n+a e r-b e n r+(-2 a e+b e n) \log \left (f x^r\right )\right )\right ) \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.37, size = 9262, normalized size = 44.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(9262\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.31, size = 253, normalized size = 1.23 \begin {gather*} \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} r x^{2} e + 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} x^{2} e \log \left (f x^{r}\right ) + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{2} \, {\left ({\left (r - \log \left (f\right )\right )} x^{2} - x^{2} \log \left (x^{r}\right )\right )} a b n e - \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 {gather*}

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*r*x^2*e + 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*x^2*e*log(f*x^r) + 1/2*a^2*d*x^2 + 1/2*((r - log(f))*x^2 - x^2*log(x^r))
*a*b*n*e - 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] Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (195) = 390\).
time = 0.35, size = 405, normalized size = 1.97 \begin {gather*} \frac {1}{2} \, b^{2} n^{2} r x^{2} e \log \left (x\right )^{3} - \frac {1}{8} \, {\left (3 \, b^{2} n^{2} - 4 \, a b n + 2 \, a^{2}\right )} r x^{2} e + \frac {1}{4} \, {\left (b^{2} d n^{2} - 2 \, a b d n + 2 \, a^{2} d\right )} x^{2} - \frac {1}{4} \, {\left (b^{2} r x^{2} e - 2 \, b^{2} d x^{2}\right )} \log \left (c\right )^{2} + \frac {1}{4} \, {\left (4 \, b^{2} n r x^{2} e \log \left (c\right ) + 2 \, b^{2} n^{2} x^{2} e \log \left (f\right ) + 2 \, b^{2} d n^{2} x^{2} - {\left (3 \, b^{2} n^{2} - 4 \, a b n\right )} r x^{2} e\right )} \log \left (x\right )^{2} + \frac {1}{2} \, {\left ({\left (b^{2} n - a b\right )} r x^{2} e - {\left (b^{2} d n - 2 \, a b d\right )} x^{2}\right )} \log \left (c\right ) + \frac {1}{4} \, {\left (2 \, b^{2} x^{2} e \log \left (c\right )^{2} - 2 \, {\left (b^{2} n - 2 \, a b\right )} x^{2} e \log \left (c\right ) + {\left (b^{2} n^{2} - 2 \, a b n + 2 \, a^{2}\right )} x^{2} e\right )} \log \left (f\right ) + \frac {1}{4} \, {\left (2 \, b^{2} r x^{2} e \log \left (c\right )^{2} + {\left (3 \, b^{2} n^{2} - 4 \, a b n + 2 \, a^{2}\right )} r x^{2} e - 2 \, {\left (b^{2} d n^{2} - 2 \, a b d n\right )} x^{2} + 4 \, {\left (b^{2} d n x^{2} - {\left (b^{2} n - a b\right )} r x^{2} e\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} n x^{2} e \log \left (c\right ) - {\left (b^{2} n^{2} - 2 \, a b n\right )} x^{2} e\right )} \log \left (f\right )\right )} \log \left (x\right ) \end {gather*}

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

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Sympy [A]
time = 2.45, size = 318, normalized size = 1.54 \begin {gather*} \frac {a^{2} d x^{2}}{2} - \frac {a^{2} e r x^{2}}{4} + \frac {a^{2} e x^{2} \log {\left (f x^{r} \right )}}{2} - \frac {a b d n x^{2}}{2} + a b d x^{2} \log {\left (c x^{n} \right )} + \frac {a b e n r x^{2}}{2} - \frac {a b e n x^{2} \log {\left (f x^{r} \right )}}{2} - \frac {a b e r x^{2} \log {\left (c x^{n} \right )}}{2} + a b e x^{2} \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )} + \frac {b^{2} d n^{2} x^{2}}{4} - \frac {b^{2} d n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d x^{2} \log {\left (c x^{n} \right )}^{2}}{2} - \frac {3 b^{2} e n^{2} r x^{2}}{8} + \frac {b^{2} e n^{2} x^{2} \log {\left (f x^{r} \right )}}{4} + \frac {b^{2} e n r x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b^{2} e n x^{2} \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{2} - \frac {b^{2} e r x^{2} \log {\left (c x^{n} \right )}^{2}}{4} + \frac {b^{2} e x^{2} \log {\left (c x^{n} \right )}^{2} \log {\left (f x^{r} \right )}}{2} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (195) = 390\).
time = 5.34, size = 497, normalized size = 2.41 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 4.15, size = 187, normalized size = 0.91 \begin {gather*} \ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (a\,b\,e\,x^2-\frac {b^2\,e\,n\,x^2}{2}\right )+\frac {a^2\,e\,x^2}{2}+\frac {b^2\,e\,n^2\,x^2}{4}+\frac {b^2\,e\,x^2\,{\ln \left (c\,x^n\right )}^2}{2}-\frac {a\,b\,e\,n\,x^2}{2}\right )+x^2\,\left (\frac {a^2\,d}{2}+\frac {b^2\,d\,n^2}{4}-\frac {a^2\,e\,r}{4}-\frac {3\,b^2\,e\,n^2\,r}{8}-\frac {a\,b\,d\,n}{2}+\frac {a\,b\,e\,n\,r}{2}\right )+\frac {b^2\,x^2\,{\ln \left (c\,x^n\right )}^2\,\left (2\,d-e\,r\right )}{4}+\frac {b\,x^2\,\ln \left (c\,x^n\right )\,\left (2\,a\,d-b\,d\,n-a\,e\,r+b\,e\,n\,r\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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