\(\int \frac {x^3 (a+b \log (c x^n))}{d+e x^2} \, dx\) [211]

Optimal result

Integrand size = 23, antiderivative size = 83 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {b n x^2}{4 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.63 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {b e n x^2-2 e x^2 \left (a+b \log \left (c x^n\right )\right )+2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+2 b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 b d n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 e^2} \] Input:

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

Output:

-1/4*(b*e*n*x^2 - 2*e*x^2*(a + b*Log[c*x^n]) + 2*d*(a + b*Log[c*x^n])*Log[ 
1 + (Sqrt[e]*x)/Sqrt[-d]] + 2*d*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/( 
-d)^(3/2)] + 2*b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]] + 2*b*d*n*PolyLog[2, 
 (d*Sqrt[e]*x)/(-d)^(3/2)])/e^2
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2793

\(\displaystyle \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2}-\frac {b n x^2}{4 e}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{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] && Integer 
Q[r]))
 

Maple [C] (warning: unable to verify)

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

Time = 0.54 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.42

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e}-\frac {b \ln \left (x^{n}\right ) d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}-\frac {b n \,x^{2}}{4 e}+\frac {b n d \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x^{2}}{2 e}-\frac {d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}\right )\)	\(284\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 18.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.43 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=- \frac {a d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right )}{2 e} + \frac {a x^{2}}{2 e} + \frac {b d n \left (\begin {cases} \frac {x^{2}}{2 d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{2 e} - \frac {b d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 e} - \frac {b n x^{2}}{4 e} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e} \] Input:

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

Output:

-a*d*Piecewise((x**2/d, Eq(e, 0)), (log(d + e*x**2)/e, True))/(2*e) + a*x* 
*2/(2*e) + b*d*n*Piecewise((x**2/(2*d), Eq(e, 0)), (Piecewise((-polylog(2, 
 e*x**2*exp_polar(I*pi)/d)/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log( 
x) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, Abs(x) < 1), (-log(d)*log(1/x 
) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, 1/Abs(x) < 1), (-meijerg(((), 
(1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)* 
log(d) - polylog(2, e*x**2*exp_polar(I*pi)/d)/2, True))/e, True))/(2*e) - 
b*d*Piecewise((x**2/d, Eq(e, 0)), (log(d + e*x**2)/e, True))*log(c*x**n)/( 
2*e) - b*n*x**2/(4*e) + b*x**2*log(c*x**n)/(2*e)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+d x}d x \right ) b \,d^{2} n -2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d n -2 \mathrm {log}\left (x^{n} c \right )^{2} b d +2 \,\mathrm {log}\left (x^{n} c \right ) b e n \,x^{2}+2 a e n \,x^{2}-b e \,n^{2} x^{2}}{4 e^{2} n} \] Input:

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

Output:

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