\(\int \frac {(a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [117]

Optimal result

Integrand size = 20, antiderivative size = 203 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {b^2 n^2}{3 d^2 e (d+e x)}-\frac {b^2 n^2 \log (x)}{3 d^3 e}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac {2 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {b^2 n^2 \log (d+e x)}{d^3 e}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^3 e} \]

[Out]

-1/3*b^2*n^2/d^2/e/(e*x+d)-1/3*b^2*n^2*ln(x)/d^3/e+1/3*b*n*(a+b*ln(c*x^n))/d/e/(e*x+d)^2-2/3*b*n*x*(a+b*ln(c*x
^n))/d^3/(e*x+d)-2/3*b*n*ln(1+d/e/x)*(a+b*ln(c*x^n))/d^3/e-1/3*(a+b*ln(c*x^n))^2/e/(e*x+d)^3+b^2*n^2*ln(e*x+d)
/d^3/e+2/3*b^2*n^2*polylog(2,-d/e/x)/d^3/e

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2356, 2389, 2379, 2438, 2351, 31, 46} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {2 b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^3 e}-\frac {b^2 n^2 \log (x)}{3 d^3 e}+\frac {b^2 n^2 \log (d+e x)}{d^3 e}-\frac {b^2 n^2}{3 d^2 e (d+e x)} \]

[In]

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

[Out]

-1/3*(b^2*n^2)/(d^2*e*(d + e*x)) - (b^2*n^2*Log[x])/(3*d^3*e) + (b*n*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) -
(2*b*n*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)) - (2*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^3*e) - (a +
b*Log[c*x^n])^2/(3*e*(d + e*x)^3) + (b^2*n^2*Log[d + e*x])/(d^3*e) + (2*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*d^
3*e)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 d}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 d e} \\ & = \frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d^2}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d^2 e}-\frac {\left (b^2 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 d e} \\ & = \frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac {2 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {\left (2 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{3 d^3}+\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{3 d^3 e}-\frac {\left (b^2 n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 d e} \\ & = -\frac {b^2 n^2}{3 d^2 e (d+e x)}-\frac {b^2 n^2 \log (x)}{3 d^3 e}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e (d+e x)^2}-\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)}-\frac {2 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {b^2 n^2 \log (d+e x)}{d^3 e}+\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{3 d^3 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}+\frac {2 b n \left (\frac {a+b \log \left (c x^n\right )}{2 d (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{d^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac {b n \left (\frac {1}{d (d+e x)}+\frac {\log (x)}{d^2}-\frac {\log (d+e x)}{d^2}\right )}{2 d}-\frac {b n \left (\frac {\log (x)}{d}-\frac {\log (d+e x)}{d}\right )}{d^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {d+e x}{d}\right )}{d^3}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}\right )}{3 e} \]

[In]

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

[Out]

-1/3*(a + b*Log[c*x^n])^2/(e*(d + e*x)^3) + (2*b*n*((a + b*Log[c*x^n])/(2*d*(d + e*x)^2) + (a + b*Log[c*x^n])/
(d^2*(d + e*x)) + (a + b*Log[c*x^n])^2/(2*b*d^3*n) - (b*n*(1/(d*(d + e*x)) + Log[x]/d^2 - Log[d + e*x]/d^2))/(
2*d) - (b*n*(Log[x]/d - Log[d + e*x]/d))/d^2 - ((a + b*Log[c*x^n])*Log[(d + e*x)/d])/d^3 - (b*n*PolyLog[2, -((
e*x)/d)])/d^3))/(3*e)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.51 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.44

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{3 e \left (e x +d \right )^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{3 e \,d^{3}}+\frac {2 b^{2} n \ln \left (x^{n}\right )}{3 e \,d^{2} \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right )}{3 e d \left (e x +d \right )^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{3 e \,d^{3}}-\frac {b^{2} n^{2}}{3 d^{2} e \left (e x +d \right )}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{d^{3} e}-\frac {b^{2} n^{2} \ln \left (x \right )}{d^{3} e}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{3 e \,d^{3}}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{3 e \,d^{3}}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{3 e \,d^{3}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{3 e \left (e x +d \right )^{3}}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d^{3}}+\frac {1}{d^{2} \left (e x +d \right )}+\frac {1}{2 d \left (e x +d \right )^{2}}+\frac {\ln \left (x \right )}{d^{3}}\right )}{3 e}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2}}{12 \left (e x +d \right )^{3} e}\)	\(495\)

[In]

int((a+b*ln(c*x^n))^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*b^2*ln(x^n)^2/e/(e*x+d)^3-2/3*b^2/e*n*ln(x^n)/d^3*ln(e*x+d)+2/3*b^2*n*ln(x^n)/e/d^2/(e*x+d)+1/3*b^2*n*ln(
x^n)/e/d/(e*x+d)^2+2/3*b^2/e*n*ln(x^n)/d^3*ln(x)-1/3*b^2*n^2/d^2/e/(e*x+d)+b^2*n^2*ln(e*x+d)/d^3/e-b^2*n^2*ln(
x)/d^3/e-1/3*b^2/e*n^2/d^3*ln(x)^2+2/3*b^2/e*n^2/d^3*ln(e*x+d)*ln(-e*x/d)+2/3*b^2/e*n^2/d^3*dilog(-e*x/d)+(-I*
b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I
*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-1/3*ln(x^n)/e/(e*x+d)^3+1/3/e*n*(-1/d^3*ln(e*x+d)+1/d^2/(e*x+d)+1/2/d
/(e*x+d)^2+1/d^3*ln(x)))-1/12*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*
b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2/(e*x+d)^3/e

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)
, x)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]

[In]

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

[Out]

Integral((a + b*log(c*x**n))**2/(d + e*x)**4, x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

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

[Out]

1/3*a*b*n*((2*e*x + 3*d)/(d^2*e^3*x^2 + 2*d^3*e^2*x + d^4*e) - 2*log(e*x + d)/(d^3*e) + 2*log(x)/(d^3*e)) - 1/
3*b^2*(log(x^n)^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 3*integrate(1/3*(3*e*x*log(c)^2 + 2*(d*n + (
e*n + 3*e*log(c))*x)*log(x^n))/(e^5*x^5 + 4*d*e^4*x^4 + 6*d^2*e^3*x^3 + 4*d^3*e^2*x^2 + d^4*e*x), x)) - 2/3*a*
b*log(c*x^n)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/3*a^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^
3*e)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

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

[Out]

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