3.1.94 \(\int \frac {x (a+b \log (c x^n))^2}{d+e x} \, dx\) [94]
Optimal. Leaf size=130 \[ -\frac {2 a b n x}{e}+\frac {2 b^2 n^2 x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^2} \]
[Out]
-2*a*b*n*x/e+2*b^2*n^2*x/e-2*b^2*n*x*ln(c*x^n)/e+x*(a+b*ln(c*x^n))^2/e-d*(a+b*ln(c*x^n))^2*ln(1+e*x/d)/e^2-2*b
*d*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/e^2+2*b^2*d*n^2*polylog(3,-e*x/d)/e^2
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Rubi [A]
time = 0.11, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2395, 2333,
2332, 2354, 2421, 6724} \begin {gather*} -\frac {2 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 b^2 d n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 a b n x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {2 b^2 n^2 x}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x*(a + b*Log[c*x^n])^2)/(d + e*x),x]
[Out]
(-2*a*b*n*x)/e + (2*b^2*n^2*x)/e - (2*b^2*n*x*Log[c*x^n])/e + (x*(a + b*Log[c*x^n])^2)/e - (d*(a + b*Log[c*x^n
])^2*Log[1 + (e*x)/d])/e^2 - (2*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^2 + (2*b^2*d*n^2*PolyLog[3,
-((e*x)/d)])/e^2
Rule 2332
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]
Rule 2333
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Rule 2354
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Rule 2395
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))
Rule 2421
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Rule 6724
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e}-\frac {d \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {(2 b d n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^2}-\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac {2 a b n x}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}-\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e}+\frac {\left (2 b^2 d n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^2}\\ &=-\frac {2 a b n x}{e}+\frac {2 b^2 n^2 x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 103, normalized size = 0.79 \begin {gather*} \frac {e x \left (a+b \log \left (c x^n\right )\right )^2-2 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-2 b d n \left (\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x*(a + b*Log[c*x^n])^2)/(d + e*x),x]
[Out]
(e*x*(a + b*Log[c*x^n])^2 - 2*b*e*n*x*(a - b*n + b*Log[c*x^n]) - d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] - 2*b
*d*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/e^2
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.20, size = 2420, normalized size = 18.62
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| method |
result |
size |
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| risch |
\(\text {Expression too large to display}\) |
\(2420\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*ln(c*x^n))^2/(e*x+d),x,method=_RETURNVERBOSE)
[Out]
I*ln(x^n)*d/e^2*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csg
n(I*c)*csgn(I*c*x^n)^2+I*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*n*d/e^2*dilog(-e*x/
d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*b/e*x*ln(x^n)*a-2*b^2*n/e*x*ln(x^n)-a^2*d/e^2*ln(e*x+d)+I/e*x*
ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+a^2/e*x-2*b*d/e^2*n*a-1/e*x*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)^4+1/4*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4+1/4*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*x^n)^2*cs
gn(I*c*x^n)^4+1/2/e*x*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-I/e*x*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)+2*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*b^2*ln(c)+2*b*n*d/e^2*dilog(-e*x/d)*a+I*ln(x^n)*d/e^2*ln(e*x+d)
*b^2*Pi*csgn(I*c*x^n)^3+1/4*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+2*n*d/e^2*dilog
(-e*x/d)*b^2*ln(c)-2*d/e^2*ln(e*x+d)*ln(c)*a*b-1/4/e*x*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4+1/2/e*x*Pi^2*b^2*c
sgn(I*c)*csgn(I*c*x^n)^5+1/2/e*x*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-I*n*d/e^2*dilog(-e*x/d)*b^2*Pi*csgn(I*c*
x^n)^3-2*b^2*d/e^2*ln(x)*dilog(-e*x/d)*n^2+b^2*d/e^2*n^2*ln(x)^2*ln(e*x+d)-b^2*d/e^2*n^2*ln(x)^2*ln(1+e*x/d)-2
*b^2*d/e^2*n^2*ln(x)*polylog(2,-e*x/d)-1/2*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+1/2/e*x*Pi^2*b
^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-1/2*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5-2*b^2*d/e^2*
ln(x)*ln(-e*x/d)*ln(e*x+d)*n^2+2*b*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*a+2/e*x*ln(x^n)*b^2*ln(c)-I*d/e^2*ln(e*x+d)*Pi
*a*b*csgn(I*c)*csgn(I*c*x^n)^2+2*b^2*n*d/e^2*ln(-e*x/d)*ln(e*x+d)*ln(x^n)-d/e^2*ln(e*x+d)*ln(c)^2*b^2-2/e*n*x*
b^2*ln(c)+I*d/e^2*n*b^2*Pi*csgn(I*c*x^n)^3+I/e*n*x*b^2*Pi*csgn(I*c*x^n)^3-b^2*ln(x^n)^2*d/e^2*ln(e*x+d)-I*d/e^
2*n*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*d/e^2*n*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+2/e*x*ln(c)*a*b-2*d/e^2*n*b^
2*ln(c)-I/e*x*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-I/e*x*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^3-I/e*x*Pi*a*b*csgn(I*c*x^n)^3-I
/e*n*x*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/e*n*x*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*d/e^2*ln(e*x+d)*ln(c)*Pi*
b^2*csgn(I*c*x^n)^3+1/e*x*ln(c)^2*b^2+I/e*x*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+I/e*x*Pi*a*b*csgn(I*x^n)*csgn(I*c
*x^n)^2+I/e*x*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)^4+I/e*x*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2-1/4/e*x*Pi^2*b^2*csgn(I*c*x^n)^6-1/2*d/e^2*ln(e*x+d)*Pi^2
*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-1/2*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n
)^3-I*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*b^2*n^2*x/e-1/4/e*x*Pi^2*b^2*c
sgn(I*x^n)^2*csgn(I*c*x^n)^4+I/e*x*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*d/e^2*ln(e*x+d)*Pi*a*b*csgn(I*c*
x^n)^3-2*b*ln(x^n)*d/e^2*ln(e*x+d)*a-2*ln(x^n)*d/e^2*ln(e*x+d)*b^2*ln(c)+2*b^2*n*d/e^2*dilog(-e*x/d)*ln(x^n)+1
/4*d/e^2*ln(e*x+d)*Pi^2*b^2*csgn(I*c*x^n)^6-1/4/e*x*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+b^2*ln(
x^n)^2/e*x+I*d/e^2*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-2*a*b*n*x/e+2*b^2*d*n^2*polylog(
3,-e*x/d)/e^2+I*n*d/e^2*dilog(-e*x/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*ln(x^n)*d/e^2*ln(e*x+d)*b^2*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2-I*n*d/e^2*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*c*x^n)^3-I/e*x*Pi*a*b*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)-I*d/e^2*ln(e*x+d)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+I*n*d/e^2*dilog(-e*x/d)*b^2*Pi*csgn(I*c)
*csgn(I*c*x^n)^2+I*d/e^2*n*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I/e*x*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)-I*ln(x^n)*d/e^2*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I/e*n*x*b^2*Pi*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)+I*d/e^2*ln(e*x+d)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*d/e^2*ln(e*x+d)*ln(c)*Pi*b^2*c
sgn(I*c)*csgn(I*c*x^n)^2-I*d/e^2*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))^2/(e*x+d),x, algorithm="maxima")
[Out]
-(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^2 + integrate((b^2*x*log(x^n)^2 + 2*(b^2*log(c) + a*b)*x*log(x^n) + (b^2
*log(c)^2 + 2*a*b*log(c))*x)/(x*e + d), x)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))^2/(e*x+d),x, algorithm="fricas")
[Out]
integral((b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x)/(x*e + d), x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*ln(c*x**n))**2/(e*x+d),x)
[Out]
Integral(x*(a + b*log(c*x**n))**2/(d + e*x), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*log(c*x^n))^2/(e*x+d),x, algorithm="giac")
[Out]
integrate((b*log(c*x^n) + a)^2*x/(x*e + d), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(a + b*log(c*x^n))^2)/(d + e*x),x)
[Out]
int((x*(a + b*log(c*x^n))^2)/(d + e*x), x)
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