3.2.68 \(\int \frac {(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^3} \, dx\) [168]

Optimal. Leaf size=204 \[ -\frac {b^2 e n^2 r}{8 x^2}-\frac {b e n (2 a+b n) r}{8 x^2}-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2} \]

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

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Rubi [A]
time = 0.14, antiderivative size = 204, 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 = {2342, 2341, 2413, 12, 14} \begin {gather*} -\frac {e r \left (2 a^2+2 a b n+b^2 n^2\right )}{8 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {b e r (2 a+b n) \log \left (c x^n\right )}{4 x^2}-\frac {b e n r (2 a+b n)}{8 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b^2 e n^2 r}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(b^2*e*n^2*r)/x^2 - (b*e*n*(2*a + b*n)*r)/(8*x^2) - (e*(2*a^2 + 2*a*b*n + b^2*n^2)*r)/(8*x^2) - (b^2*e*n*
r*Log[c*x^n])/(4*x^2) - (b*e*(2*a + b*n)*r*Log[c*x^n])/(4*x^2) - (b^2*e*r*Log[c*x^n]^2)/(4*x^2) - (b^2*n^2*(d
+ e*Log[f*x^r]))/(4*x^2) - (b*n*(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*x^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 \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-(e r) \int \frac {-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 )}{4 x^3} \, dx\\ &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {1}{4} (e r) \int \frac {-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 )}{x^3} \, dx\\ &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {1}{4} (e r) \int \left (\frac {-2 a^2-2 a b n-b^2 n^2}{x^3}-\frac {2 b (2 a+b n) \log \left (c x^n\right )}{x^3}-\frac {2 b^2 \log ^2\left (c x^n\right )}{x^3}\right ) \, dx\\ &=-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac {1}{2} \left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^3} \, dx+\frac {1}{2} (b e (2 a+b n) r) \int \frac {\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b e n (2 a+b n) r}{8 x^2}-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac {1}{2} \left (b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b^2 e n^2 r}{8 x^2}-\frac {b e n (2 a+b n) r}{8 x^2}-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 151, normalized size = 0.74 \begin {gather*} -\frac {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+e (2 a+b n) \log \left (f x^r\right )\right )}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(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 + e*(2*a + b*n)*Log[f*x^r]))/x^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.43, size = 8407, normalized size = 41.21

method	result	size
risch	\(\text {Expression too large to display}\)	\(8407\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2*(d+e*ln(f*x^r))/x^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.31, size = 230, normalized size = 1.13 \begin {gather*} -\frac {1}{4} \, b^{2} {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} e \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} e \log \left (c x^{n}\right ) - \frac {1}{4} \, b^{2} d {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {1}{8} \, b^{2} {\left (\frac {{\left (2 \, r \log \left (x\right ) + 3 \, r + 2 \, \log \left (f\right )\right )} n^{2}}{x^{2}} + \frac {4 \, n {\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{2}}\right )} e - \frac {a b n {\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} e}{2 \, x^{2}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b d n}{2 \, x^{2}} - \frac {a^{2} r e}{4 \, x^{2}} - \frac {a b d \log \left (c x^{n}\right )}{x^{2}} - \frac {a^{2} e \log \left (f x^{r}\right )}{2 \, x^{2}} - \frac {a^{2} d}{2 \, x^{2}} \end {gather*}

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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 334, normalized size = 1.64 \begin {gather*} -\frac {4 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 4 \, a b d n + 4 \, a^{2} d + {\left (3 \, b^{2} n^{2} + 4 \, a b n + 2 \, a^{2}\right )} r e + 2 \, {\left (b^{2} r e + 2 \, b^{2} d\right )} \log \left (c\right )^{2} + 2 \, {\left (4 \, b^{2} n r e \log \left (c\right ) + 2 \, b^{2} n^{2} e \log \left (f\right ) + 2 \, b^{2} d n^{2} + {\left (3 \, b^{2} n^{2} + 4 \, a b n\right )} r e\right )} \log \left (x\right )^{2} + 4 \, {\left (b^{2} d n + 2 \, a b d + {\left (b^{2} n + a b\right )} r e\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} e \log \left (c\right )^{2} + 2 \, {\left (b^{2} n + 2 \, a b\right )} e \log \left (c\right ) + {\left (b^{2} n^{2} + 2 \, a b n + 2 \, a^{2}\right )} e\right )} \log \left (f\right ) + 2 \, {\left (2 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 4 \, a b d n + {\left (3 \, b^{2} n^{2} + 4 \, a b n + 2 \, a^{2}\right )} r e + 4 \, {\left (b^{2} d n + {\left (b^{2} n + a b\right )} r e\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} n e \log \left (c\right ) + {\left (b^{2} n^{2} + 2 \, a b n\right )} e\right )} \log \left (f\right )\right )} \log \left (x\right )}{8 \, x^{2}} \end {gather*}

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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.25, size = 320, normalized size = 1.57 \begin {gather*} - \frac {a^{2} d}{2 x^{2}} - \frac {a^{2} e r}{4 x^{2}} - \frac {a^{2} e \log {\left (f x^{r} \right )}}{2 x^{2}} - \frac {a b d n}{2 x^{2}} - \frac {a b d \log {\left (c x^{n} \right )}}{x^{2}} - \frac {a b e n r}{2 x^{2}} - \frac {a b e n \log {\left (f x^{r} \right )}}{2 x^{2}} - \frac {a b e r \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {a b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{x^{2}} - \frac {b^{2} d n^{2}}{4 x^{2}} - \frac {b^{2} d n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} d \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} - \frac {3 b^{2} e n^{2} r}{8 x^{2}} - \frac {b^{2} e n^{2} \log {\left (f x^{r} \right )}}{4 x^{2}} - \frac {b^{2} e n r \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} e n \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{2 x^{2}} - \frac {b^{2} e r \log {\left (c x^{n} \right )}^{2}}{4 x^{2}} - \frac {b^{2} e \log {\left (c x^{n} \right )}^{2} \log {\left (f x^{r} \right )}}{2 x^{2}} \end {gather*}

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**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (195) = 390\).
time = 5.27, size = 403, normalized size = 1.98 \begin {gather*} -\frac {4 \, b^{2} n^{2} r e \log \left (x\right )^{3} + 6 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 8 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + 4 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e \log \left (x\right ) + 8 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + 4 \, b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 4 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 8 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + 4 \, b^{2} d n^{2} \log \left (x\right )^{2} + 8 \, a b n r e \log \left (x\right )^{2} + 3 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + 2 \, b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 4 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + 4 \, b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 4 \, b^{2} d n^{2} \log \left (x\right ) + 8 \, a b n r e \log \left (x\right ) + 8 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 8 \, a b r e \log \left (c\right ) \log \left (x\right ) + 8 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 4 \, b^{2} d n \log \left (c\right ) + 4 \, a b r e \log \left (c\right ) + 4 \, b^{2} d \log \left (c\right )^{2} + 4 \, a b n e \log \left (f\right ) + 8 \, a b e \log \left (c\right ) \log \left (f\right ) + 8 \, a b d n \log \left (x\right ) + 4 \, a^{2} r e \log \left (x\right ) + 4 \, a b d n + 2 \, a^{2} r e + 8 \, a b d \log \left (c\right ) + 4 \, a^{2} e \log \left (f\right ) + 4 \, a^{2} d}{8 \, x^{2}} \end {gather*}

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^3,x, algorithm="giac")

[Out]

-1/8*(4*b^2*n^2*r*e*log(x)^3 + 6*b^2*n^2*r*e*log(x)^2 + 8*b^2*n*r*e*log(c)*log(x)^2 + 4*b^2*n^2*e*log(f)*log(x
)^2 + 6*b^2*n^2*r*e*log(x) + 8*b^2*n*r*e*log(c)*log(x) + 4*b^2*r*e*log(c)^2*log(x) + 4*b^2*n^2*e*log(f)*log(x)
+ 8*b^2*n*e*log(c)*log(f)*log(x) + 4*b^2*d*n^2*log(x)^2 + 8*a*b*n*r*e*log(x)^2 + 3*b^2*n^2*r*e + 4*b^2*n*r*e*
log(c) + 2*b^2*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 4*b^2*n*e*log(c)*log(f) + 4*b^2*e*log(c)^2*log(f) + 4*b^2*d
*n^2*log(x) + 8*a*b*n*r*e*log(x) + 8*b^2*d*n*log(c)*log(x) + 8*a*b*r*e*log(c)*log(x) + 8*a*b*n*e*log(f)*log(x)
+ 2*b^2*d*n^2 + 4*a*b*n*r*e + 4*b^2*d*n*log(c) + 4*a*b*r*e*log(c) + 4*b^2*d*log(c)^2 + 4*a*b*n*e*log(f) + 8*a
*b*e*log(c)*log(f) + 8*a*b*d*n*log(x) + 4*a^2*r*e*log(x) + 4*a*b*d*n + 2*a^2*r*e + 8*a*b*d*log(c) + 4*a^2*e*lo
g(f) + 4*a^2*d)/x^2

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Mupad [B]
time = 4.08, size = 186, normalized size = 0.91 \begin {gather*} -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {a\,b\,e}{x^2}+\frac {b^2\,e\,n}{2\,x^2}\right )+\frac {a^2\,e}{2\,x^2}+\frac {b^2\,e\,n^2}{4\,x^2}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{2\,x^2}+\frac {a\,b\,e\,n}{2\,x^2}\right )-\frac {\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}}{x^2}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (2\,d+e\,r\right )}{4\,x^2}-\frac {b\,\ln \left (c\,x^n\right )\,\left (2\,a\,d+b\,d\,n+a\,e\,r+b\,e\,n\,r\right )}{2\,x^2} \end {gather*}

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^3,x)

[Out]

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

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