3.169 \(\int \frac {(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^4} \, dx\)

Optimal. Leaf size=205 \[ -\frac {e r \left (9 a^2+6 a b n+2 b^2 n^2\right )}{81 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b e r (3 a+b n) \log \left (c x^n\right )}{27 x^3}-\frac {2 b e n r (3 a+b n)}{81 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b^2 e n^2 r}{81 x^3} \]

[Out]

-2/81*b^2*e*n^2*r/x^3-2/81*b*e*n*(b*n+3*a)*r/x^3-1/81*e*(2*b^2*n^2+6*a*b*n+9*a^2)*r/x^3-2/27*b^2*e*n*r*ln(c*x^
n)/x^3-2/27*b*e*(b*n+3*a)*r*ln(c*x^n)/x^3-1/9*b^2*e*r*ln(c*x^n)^2/x^3-2/27*b^2*n^2*(d+e*ln(f*x^r))/x^3-2/9*b*n
*(a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x^3-1/3*(a+b*ln(c*x^n))^2*(d+e*ln(f*x^r))/x^3

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Rubi [A]  time = 0.21, antiderivative size = 205, normalized size of antiderivative = 1.00, 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} \[ -\frac {e r \left (9 a^2+6 a b n+2 b^2 n^2\right )}{81 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {2 b e r (3 a+b n) \log \left (c x^n\right )}{27 x^3}-\frac {2 b e n r (3 a+b n)}{81 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b^2 e n^2 r}{81 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-(e r) \int \frac {-9 a^2 \left (1+\frac {2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{27 x^4} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{27} (e r) \int \frac {-9 a^2 \left (1+\frac {2 b n (3 a+b n)}{9 a^2}\right )-6 b (3 a+b n) \log \left (c x^n\right )-9 b^2 \log ^2\left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{27} (e r) \int \left (\frac {-9 a^2-6 a b n-2 b^2 n^2}{x^4}-\frac {6 b (3 a+b n) \log \left (c x^n\right )}{x^4}-\frac {9 b^2 \log ^2\left (c x^n\right )}{x^4}\right ) \, dx\\ &=-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac {1}{3} \left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^4} \, dx+\frac {1}{9} (2 b e (3 a+b n) r) \int \frac {\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac {1}{9} \left (2 b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {2 b^2 e n^2 r}{81 x^3}-\frac {2 b e n (3 a+b n) r}{81 x^3}-\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) r}{81 x^3}-\frac {2 b^2 e n r \log \left (c x^n\right )}{27 x^3}-\frac {2 b e (3 a+b n) r \log \left (c x^n\right )}{27 x^3}-\frac {b^2 e r \log ^2\left (c x^n\right )}{9 x^3}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 155, normalized size = 0.76 \[ -\frac {e \left (9 a^2+6 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+9 a^2 d+3 a^2 e r+2 b \log \left (c x^n\right ) \left (3 e (3 a+b n) \log \left (f x^r\right )+9 a d+3 a e r+3 b d n+2 b e n r\right )+6 a b d n+4 a b e n r+3 b^2 \log ^2\left (c x^n\right ) \left (3 d+3 e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+2 b^2 e n^2 r}{27 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*(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 + 3*e*(3*a + b*n)*Log[f*x^r]))/x^3

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fricas [A]  time = 0.69, size = 329, normalized size = 1.60 \[ -\frac {9 \, b^{2} e n^{2} r \log \relax (x)^{3} + 2 \, b^{2} d n^{2} + 6 \, a b d n + 9 \, a^{2} d + 3 \, {\left (b^{2} e r + 3 \, b^{2} d\right )} \log \relax (c)^{2} + 9 \, {\left (2 \, b^{2} e n r \log \relax (c) + b^{2} e n^{2} \log \relax (f) + b^{2} d n^{2} + {\left (b^{2} e n^{2} + 2 \, a b e n\right )} r\right )} \log \relax (x)^{2} + {\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \, {\left (3 \, b^{2} d n + 9 \, a b d + {\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \relax (c) + {\left (2 \, b^{2} e n^{2} + 9 \, b^{2} e \log \relax (c)^{2} + 6 \, a b e n + 9 \, a^{2} e + 6 \, {\left (b^{2} e n + 3 \, a b e\right )} \log \relax (c)\right )} \log \relax (f) + 3 \, {\left (3 \, b^{2} e r \log \relax (c)^{2} + 2 \, b^{2} d n^{2} + 6 \, a b d n + {\left (2 \, b^{2} e n^{2} + 4 \, a b e n + 3 \, a^{2} e\right )} r + 2 \, {\left (3 \, b^{2} d n + {\left (2 \, b^{2} e n + 3 \, a b e\right )} r\right )} \log \relax (c) + 2 \, {\left (b^{2} e n^{2} + 3 \, b^{2} e n \log \relax (c) + 3 \, a b e n\right )} \log \relax (f)\right )} \log \relax (x)}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^4,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.39, size = 403, normalized size = 1.97 \[ -\frac {9 \, b^{2} n^{2} r e \log \relax (x)^{3} + 9 \, b^{2} n^{2} r e \log \relax (x)^{2} + 18 \, b^{2} n r e \log \relax (c) \log \relax (x)^{2} + 9 \, b^{2} n^{2} e \log \relax (f) \log \relax (x)^{2} + 6 \, b^{2} n^{2} r e \log \relax (x) + 12 \, b^{2} n r e \log \relax (c) \log \relax (x) + 9 \, b^{2} r e \log \relax (c)^{2} \log \relax (x) + 6 \, b^{2} n^{2} e \log \relax (f) \log \relax (x) + 18 \, b^{2} n e \log \relax (c) \log \relax (f) \log \relax (x) + 9 \, b^{2} d n^{2} \log \relax (x)^{2} + 18 \, a b n r e \log \relax (x)^{2} + 2 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \relax (c) + 3 \, b^{2} r e \log \relax (c)^{2} + 2 \, b^{2} n^{2} e \log \relax (f) + 6 \, b^{2} n e \log \relax (c) \log \relax (f) + 9 \, b^{2} e \log \relax (c)^{2} \log \relax (f) + 6 \, b^{2} d n^{2} \log \relax (x) + 12 \, a b n r e \log \relax (x) + 18 \, b^{2} d n \log \relax (c) \log \relax (x) + 18 \, a b r e \log \relax (c) \log \relax (x) + 18 \, a b n e \log \relax (f) \log \relax (x) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 6 \, b^{2} d n \log \relax (c) + 6 \, a b r e \log \relax (c) + 9 \, b^{2} d \log \relax (c)^{2} + 6 \, a b n e \log \relax (f) + 18 \, a b e \log \relax (c) \log \relax (f) + 18 \, a b d n \log \relax (x) + 9 \, a^{2} r e \log \relax (x) + 6 \, a b d n + 3 \, a^{2} r e + 18 \, a b d \log \relax (c) + 9 \, a^{2} e \log \relax (f) + 9 \, a^{2} d}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^4,x, algorithm="giac")

[Out]

-1/27*(9*b^2*n^2*r*e*log(x)^3 + 9*b^2*n^2*r*e*log(x)^2 + 18*b^2*n*r*e*log(c)*log(x)^2 + 9*b^2*n^2*e*log(f)*log
(x)^2 + 6*b^2*n^2*r*e*log(x) + 12*b^2*n*r*e*log(c)*log(x) + 9*b^2*r*e*log(c)^2*log(x) + 6*b^2*n^2*e*log(f)*log
(x) + 18*b^2*n*e*log(c)*log(f)*log(x) + 9*b^2*d*n^2*log(x)^2 + 18*a*b*n*r*e*log(x)^2 + 2*b^2*n^2*r*e + 4*b^2*n
*r*e*log(c) + 3*b^2*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 6*b^2*n*e*log(c)*log(f) + 9*b^2*e*log(c)^2*log(f) + 6*
b^2*d*n^2*log(x) + 12*a*b*n*r*e*log(x) + 18*b^2*d*n*log(c)*log(x) + 18*a*b*r*e*log(c)*log(x) + 18*a*b*n*e*log(
f)*log(x) + 2*b^2*d*n^2 + 4*a*b*n*r*e + 6*b^2*d*n*log(c) + 6*a*b*r*e*log(c) + 9*b^2*d*log(c)^2 + 6*a*b*n*e*log
(f) + 18*a*b*e*log(c)*log(f) + 18*a*b*d*n*log(x) + 9*a^2*r*e*log(x) + 6*a*b*d*n + 3*a^2*r*e + 18*a*b*d*log(c)
+ 9*a^2*e*log(f) + 9*a^2*d)/x^3

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maple [C]  time = 0.90, size = 8407, normalized size = 41.01 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)^2*(d+e*ln(f*x^r))/x^4,x)

[Out]

result too large to display

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maxima [A]  time = 0.94, size = 230, normalized size = 1.12 \[ -\frac {1}{9} \, b^{2} e {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right )^{2} - \frac {2}{9} \, a b e {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right ) - \frac {2}{27} \, b^{2} e {\left (\frac {{\left (r \log \relax (x) + r + \log \relax (f)\right )} n^{2}}{x^{3}} + \frac {n {\left (2 \, r + 3 \, \log \relax (f) + 3 \, \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {2}{27} \, b^{2} d {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {2 \, a b e n {\left (2 \, r + 3 \, \log \relax (f) + 3 \, \log \left (x^{r}\right )\right )}}{27 \, x^{3}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d n}{9 \, x^{3}} - \frac {a^{2} e r}{9 \, x^{3}} - \frac {2 \, a b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a^{2} d}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.20, size = 190, normalized size = 0.93 \[ -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {2\,a\,b\,e}{3\,x^3}+\frac {2\,b^2\,e\,n}{9\,x^3}\right )+\frac {a^2\,e}{3\,x^3}+\frac {2\,b^2\,e\,n^2}{27\,x^3}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{3\,x^3}+\frac {2\,a\,b\,e\,n}{9\,x^3}\right )-\frac {\frac {a^2\,d}{3}+\frac {2\,b^2\,d\,n^2}{27}+\frac {a^2\,e\,r}{9}+\frac {2\,b^2\,e\,n^2\,r}{27}+\frac {2\,a\,b\,d\,n}{9}+\frac {4\,a\,b\,e\,n\,r}{27}}{x^3}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (3\,d+e\,r\right )}{9\,x^3}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (9\,a\,d+3\,b\,d\,n+3\,a\,e\,r+2\,b\,e\,n\,r\right )}{27\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^2)/x^4,x)

[Out]

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

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sympy [B]  time = 24.96, size = 656, normalized size = 3.20 \[ - \frac {a^{2} d}{3 x^{3}} - \frac {a^{2} e r \log {\relax (x )}}{3 x^{3}} - \frac {a^{2} e r}{9 x^{3}} - \frac {a^{2} e \log {\relax (f )}}{3 x^{3}} - \frac {2 a b d n \log {\relax (x )}}{3 x^{3}} - \frac {2 a b d n}{9 x^{3}} - \frac {2 a b d \log {\relax (c )}}{3 x^{3}} - \frac {2 a b e n r \log {\relax (x )}^{2}}{3 x^{3}} - \frac {4 a b e n r \log {\relax (x )}}{9 x^{3}} - \frac {4 a b e n r}{27 x^{3}} - \frac {2 a b e n \log {\relax (f )} \log {\relax (x )}}{3 x^{3}} - \frac {2 a b e n \log {\relax (f )}}{9 x^{3}} - \frac {2 a b e r \log {\relax (c )} \log {\relax (x )}}{3 x^{3}} - \frac {2 a b e r \log {\relax (c )}}{9 x^{3}} - \frac {2 a b e \log {\relax (c )} \log {\relax (f )}}{3 x^{3}} - \frac {b^{2} d n^{2} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {2 b^{2} d n^{2} \log {\relax (x )}}{9 x^{3}} - \frac {2 b^{2} d n^{2}}{27 x^{3}} - \frac {2 b^{2} d n \log {\relax (c )} \log {\relax (x )}}{3 x^{3}} - \frac {2 b^{2} d n \log {\relax (c )}}{9 x^{3}} - \frac {b^{2} d \log {\relax (c )}^{2}}{3 x^{3}} - \frac {b^{2} e n^{2} r \log {\relax (x )}^{3}}{3 x^{3}} - \frac {b^{2} e n^{2} r \log {\relax (x )}^{2}}{3 x^{3}} - \frac {2 b^{2} e n^{2} r \log {\relax (x )}}{9 x^{3}} - \frac {2 b^{2} e n^{2} r}{27 x^{3}} - \frac {b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {2 b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}}{9 x^{3}} - \frac {2 b^{2} e n^{2} \log {\relax (f )}}{27 x^{3}} - \frac {2 b^{2} e n r \log {\relax (c )} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {4 b^{2} e n r \log {\relax (c )} \log {\relax (x )}}{9 x^{3}} - \frac {4 b^{2} e n r \log {\relax (c )}}{27 x^{3}} - \frac {2 b^{2} e n \log {\relax (c )} \log {\relax (f )} \log {\relax (x )}}{3 x^{3}} - \frac {2 b^{2} e n \log {\relax (c )} \log {\relax (f )}}{9 x^{3}} - \frac {b^{2} e r \log {\relax (c )}^{2} \log {\relax (x )}}{3 x^{3}} - \frac {b^{2} e r \log {\relax (c )}^{2}}{9 x^{3}} - \frac {b^{2} e \log {\relax (c )}^{2} \log {\relax (f )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2*(d+e*ln(f*x**r))/x**4,x)

[Out]

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

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