\(\int \frac {x^3 (a+b \log (c x^n))}{(d+e x^2)^2} \, dx\) [222]

Optimal result

Integrand size = 23, antiderivative size = 95 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{4 e^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2} \]

[Out]

-1/2*x^2*(a+b*ln(c*x^n))/e/(e*x^2+d)+1/4*b*n*ln(e*x^2+d)/e^2+1/2*(a+b*ln(c*x^n))*ln(1+e*x^2/d)/e^2+1/4*b*n*pol
ylog(2,-e*x^2/d)/e^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {272, 45, 2393, 2373, 266, 2375, 2438} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2}+\frac {b n \log \left (d+e x^2\right )}{4 e^2} \]

[In]

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

[Out]

-1/2*(x^2*(a + b*Log[c*x^n]))/(e*(d + e*x^2)) + (b*n*Log[d + e*x^2])/(4*e^2) + ((a + b*Log[c*x^n])*Log[1 + (e*
x^2)/d])/(2*e^2) + (b*n*PolyLog[2, -((e*x^2)/d)])/(4*e^2)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2373

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

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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}& = \int \left (-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e} \\ & = -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}-\frac {(b n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 e^2}+\frac {(b n) \int \frac {x}{d+e x^2} \, dx}{2 e} \\ & = -\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{4 e^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.38 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\frac {2 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}+2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+\frac {b n \left (-2 e x^2 \log (x)+d \log \left (i \sqrt {d}-\sqrt {e} x\right )+e x^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+d \log \left (i \sqrt {d}+\sqrt {e} x\right )+e x^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+2 d \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 e x^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 d \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 e x^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \left (d+e x^2\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \left (d+e x^2\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{d+e x^2}}{4 e^2} \]

[In]

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

[Out]

((2*d*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^2) + 2*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d + e*x^2] + (b*n*(
-2*e*x^2*Log[x] + d*Log[I*Sqrt[d] - Sqrt[e]*x] + e*x^2*Log[I*Sqrt[d] - Sqrt[e]*x] + d*Log[I*Sqrt[d] + Sqrt[e]*
x] + e*x^2*Log[I*Sqrt[d] + Sqrt[e]*x] + 2*d*Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 2*e*x^2*Log[x]*Log[1 - (I*
Sqrt[e]*x)/Sqrt[d]] + 2*d*Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + 2*e*x^2*Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]
] + 2*(d + e*x^2)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 2*(d + e*x^2)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]))/(d
+ e*x^2))/(4*e^2)

Maple [C] (warning: unable to verify)

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

Time = 0.47 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.21

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) d}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \ln \left (e \,x^{2}+d \right )}{4 e^{2}}-\frac {b n \ln \left (x \right )}{2 e^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {d}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{2 e^{2}}\right )\)	\(305\)

[In]

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

[Out]

1/2*b*ln(x^n)*d/e^2/(e*x^2+d)+1/2*b*ln(x^n)/e^2*ln(e*x^2+d)+1/2*b*n/e^2*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1
/2))+1/2*b*n/e^2*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/e^2*ln(x)*ln(e*x^2+d)+1/2*b*n/e^2*dilog((-e
*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n/e^2*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/4*b*n*ln(e*x^2+d)/e^2-1/2*
b*n/e^2*ln(x)+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(1/2*d/e^2/(e*x^2+d)+1/2/e^2*ln(e*x^2+d))

Fricas [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

[In]

integrate(x**3*(a+b*ln(c*x**n))/(e*x**2+d)**2,x)

[Out]

Integral(x**3*(a + b*log(c*x**n))/(d + e*x**2)**2, x)

Maxima [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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