\(\int \frac {x^3 (a+b \log (c (d+e x)^n))}{f+g x^2} \, dx\) [257]

Optimal result

Integrand size = 27, antiderivative size = 278 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\frac {b d n x}{2 e g}-\frac {b n x^2}{4 g}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}-\frac {b f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}-\frac {b f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=-\frac {\frac {b g n \left (e x (-2 d+e x)+2 d^2 \log (d+e x)\right )}{e^2}-2 g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )+2 b f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+2 b f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^2} \] Input:

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

Output:

-1/4*((b*g*n*(e*x*(-2*d + e*x) + 2*d^2*Log[d + e*x]))/e^2 - 2*g*x^2*(a + b 
*Log[c*(d + e*x)^n]) + 2*f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - S 
qrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] + 2*f*(a + b*Log[c*(d + e*x)^n])*Log[ 
(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])] + 2*b*f*n*PolyLog[2, 
-((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 2*b*f*n*PolyLog[2, (Sqr 
t[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^2
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2863, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*d*n*x)/(2*e*g) - (b*n*x^2)/(4*g) - (b*d^2*n*Log[d + e*x])/(2*e^2*g) + ( 
x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) - (f*(a + b*Log[c*(d + e*x)^n])*Log[ 
(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) - (f*(a + b* 
Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g] 
)])/(2*g^2) - (b*f*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt 
[g]))])/(2*g^2) - (b*f*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sq 
rt[g])])/(2*g^2)
 

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

Maple [C] (warning: unable to verify)

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

Time = 1.86 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.55

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{2}}{2 g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f \ln \left (g \,x^{2}+f \right )}{2 g^{2}}-\frac {b n \,x^{2}}{4 g}+\frac {b d n x}{2 e g}-\frac {b \,d^{2} n \ln \left (e x +d \right )}{2 e^{2} g}+\frac {b n f \ln \left (e x +d \right ) \ln \left (g \,x^{2}+f \right )}{2 g^{2}}-\frac {b n f \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-g f}-g \left (e x +d \right )+d g}{e \sqrt {-g f}+d g}\right )}{2 g^{2}}-\frac {b n f \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-g f}+g \left (e x +d \right )-d g}{e \sqrt {-g f}-d g}\right )}{2 g^{2}}-\frac {b n f \operatorname {dilog}\left (\frac {e \sqrt {-g f}-g \left (e x +d \right )+d g}{e \sqrt {-g f}+d g}\right )}{2 g^{2}}-\frac {b n f \operatorname {dilog}\left (\frac {e \sqrt {-g f}+g \left (e x +d \right )-d g}{e \sqrt {-g f}-d g}\right )}{2 g^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x^{2}}{2 g}-\frac {f \ln \left (g \,x^{2}+f \right )}{2 g^{2}}\right )\)	\(431\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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