Integrand size = 27, antiderivative size = 388 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=-\frac {b e n}{6 d f x^2}+\frac {b e^2 n}{3 d^2 f x}+\frac {b e^3 n \log (x)}{3 d^3 f}-\frac {b e g n \log (x)}{d f^2}-\frac {b e^3 n \log (d+e x)}{3 d^3 f}+\frac {b e g n \log (d+e x)}{d f^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 f x^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}} \] Output:
-1/6*b*e*n/d/f/x^2+1/3*b*e^2*n/d^2/f/x+1/3*b*e^3*n*ln(x)/d^3/f-b*e*g*n*ln( x)/d/f^2-1/3*b*e^3*n*ln(e*x+d)/d^3/f+b*e*g*n*ln(e*x+d)/d/f^2-1/3*(a+b*ln(c *(e*x+d)^n))/f/x^3+g*(a+b*ln(c*(e*x+d)^n))/f^2/x+1/2*g^(3/2)*(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)))/(-f)^(5/2)- 1/2*g^(3/2)*(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)))/(-f)^(5/2)-1/2*b*g^(3/2)*n*polylog(2,-g^(1/2)*(e*x+d)/(e*(-f )^(1/2)-d*g^(1/2)))/(-f)^(5/2)+1/2*b*g^(3/2)*n*polylog(2,g^(1/2)*(e*x+d)/( e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(5/2)
Time = 0.41 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\frac {1}{6} \left (-\frac {6 b e g n (\log (x)-\log (d+e x))}{d f^2}-\frac {b e n \left (d (d-2 e x)-2 e^2 x^2 \log (x)+2 e^2 x^2 \log (d+e x)\right )}{d^3 f x^2}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f x^3}+\frac {6 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {3 g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {3 g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {3 b g^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {3 b g^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}\right ) \] Input:
Integrate[(a + b*Log[c*(d + e*x)^n])/(x^4*(f + g*x^2)),x]
Output:
((-6*b*e*g*n*(Log[x] - Log[d + e*x]))/(d*f^2) - (b*e*n*(d*(d - 2*e*x) - 2* e^2*x^2*Log[x] + 2*e^2*x^2*Log[d + e*x]))/(d^3*f*x^2) - (2*(a + b*Log[c*(d + e*x)^n]))/(f*x^3) + (6*g*(a + b*Log[c*(d + e*x)^n]))/(f^2*x) + (3*g^(3/ 2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2) - (3*g^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(S qrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(-f)^(5/2) - (3*b*g^(3/2) *n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(5/2) + (3*b*g^(3/2)*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])] )/(-f)^(5/2))/6
Time = 1.12 (sec) , antiderivative size = 388, 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.
| \(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (f+g x^2\right )}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{f x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 (-f)^{5/2}}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 f x^3}+\frac {b e^3 n \log (x)}{3 d^3 f}-\frac {b e^3 n \log (d+e x)}{3 d^3 f}+\frac {b e^2 n}{3 d^2 f x}-\frac {b e g n \log (x)}{d f^2}+\frac {b e g n \log (d+e x)}{d f^2}-\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 (-f)^{5/2}}-\frac {b e n}{6 d f x^2}\) |
Input:
Int[(a + b*Log[c*(d + e*x)^n])/(x^4*(f + g*x^2)),x]
Output:
-1/6*(b*e*n)/(d*f*x^2) + (b*e^2*n)/(3*d^2*f*x) + (b*e^3*n*Log[x])/(3*d^3*f ) - (b*e*g*n*Log[x])/(d*f^2) - (b*e^3*n*Log[d + e*x])/(3*d^3*f) + (b*e*g*n *Log[d + e*x])/(d*f^2) - (a + b*Log[c*(d + e*x)^n])/(3*f*x^3) + (g*(a + b* Log[c*(d + e*x)^n]))/(f^2*x) + (g^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e* (Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2)) - (g^(3/ 2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(5/2)) - (b*g^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x ))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)) + (b*g^(3/2)*n*PolyLog[2, (S qrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2))
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.67 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.54
| method | result | size |
| risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{3 f \,x^{3}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g}{f^{2} x}-\frac {b \,g^{2} \arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {g f}}\right ) n \ln \left (e x +d \right )}{f^{2} \sqrt {g f}}+\frac {b \,g^{2} \arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {g f}}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{f^{2} \sqrt {g f}}+\frac {b n \,g^{2} \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 f^{2} \sqrt {-g f}}-\frac {b n \,g^{2} \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 f^{2} \sqrt {-g f}}+\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {e \sqrt {-g f}-g \left (e x +d \right )+d g}{e \sqrt {-g f}+d g}\right )}{2 f^{2} \sqrt {-g f}}-\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {e \sqrt {-g f}+g \left (e x +d \right )-d g}{e \sqrt {-g f}-d g}\right )}{2 f^{2} \sqrt {-g f}}-\frac {b e n g \ln \left (e x \right )}{f^{2} d}+\frac {b e g n \ln \left (e x +d \right )}{d \,f^{2}}+\frac {b \,e^{3} n \ln \left (e x \right )}{3 f \,d^{3}}+\frac {b \,e^{2} n}{3 d^{2} f x}-\frac {b e n}{6 d f \,x^{2}}-\frac {b \,e^{3} n \ln \left (e x +d \right )}{3 d^{3} f}+\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 {1}{3 f \,x^{3}}+\frac {g}{f^{2} x}+\frac {g^{2} \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{f^{2} \sqrt {g f}}\right )\) | \(598\) |
Input:
int((a+b*ln(c*(e*x+d)^n))/x^4/(g*x^2+f),x,method=_RETURNVERBOSE)
Output:
-1/3*b*ln((e*x+d)^n)/f/x^3+b*ln((e*x+d)^n)*g/f^2/x-b*g^2/f^2/(g*f)^(1/2)*a rctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(g*f)^(1/2))*n*ln(e*x+d)+b*g^2/f^2/(g*f)^( 1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(g*f)^(1/2))*ln((e*x+d)^n)+1/2*b*n*g ^2/f^2*ln(e*x+d)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^ (1/2)+d*g))-1/2*b*n*g^2/f^2*ln(e*x+d)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)+g*(e *x+d)-d*g)/(e*(-g*f)^(1/2)-d*g))+1/2*b*n*g^2/f^2/(-g*f)^(1/2)*dilog((e*(-g *f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))-1/2*b*n*g^2/f^2/(-g*f)^(1/2 )*dilog((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f)^(1/2)-d*g))-b*e*n/f^2*g/d *ln(e*x)+b*e*g*n*ln(e*x+d)/d/f^2+1/3*b*e^3*n/f/d^3*ln(e*x)+1/3*b*e^2*n/d^2 /f/x-1/6*b*e*n/d/f/x^2-1/3*b*e^3*n*ln(e*x+d)/d^3/f+(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)*(-1/3/f/x^3+g/f^2/x+g^2/f^2/(g*f)^(1/2)*arctan( g*x/(g*f)^(1/2)))
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/x^4/(g*x^2+f),x, algorithm="fricas")
Output:
integral((b*log((e*x + d)^n*c) + a)/(g*x^6 + f*x^4), x)
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(e*x+d)**n))/x**4/(g*x**2+f),x)
Output:
Timed out
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/x^4/(g*x^2+f),x, algorithm="maxima")
Output:
1/3*a*(3*g^2*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f^2) + (3*g*x^2 - f)/(f^2*x^ 3)) + b*integrate((log((e*x + d)^n) + log(c))/(g*x^6 + f*x^4), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/x^4/(g*x^2+f),x, algorithm="giac")
Output:
integrate((b*log((e*x + d)^n*c) + a)/((g*x^2 + f)*x^4), x)
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x^4\,\left (g\,x^2+f\right )} \,d x \] Input:
int((a + b*log(c*(d + e*x)^n))/(x^4*(f + g*x^2)),x)
Output:
int((a + b*log(c*(d + e*x)^n))/(x^4*(f + g*x^2)), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 \left (f+g x^2\right )} \, dx=\frac {6 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a \,d^{3} g \,x^{3}-6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{g \,x^{4}+f \,x^{2}}d x \right ) b \,d^{3} f^{2} g \,x^{3}-2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{3} f^{2}-2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{3} f^{2} x^{3}+2 \,\mathrm {log}\left (x \right ) b \,e^{3} f^{2} n \,x^{3}-2 a \,d^{3} f^{2}+6 a \,d^{3} f g \,x^{2}-b \,d^{2} e \,f^{2} n x +2 b d \,e^{2} f^{2} n \,x^{2}}{6 d^{3} f^{3} x^{3}} \] Input:
int((a+b*log(c*(e*x+d)^n))/x^4/(g*x^2+f),x)
Output:
(6*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*d**3*g*x**3 - 6*int(log ((d + e*x)**n*c)/(f*x**2 + g*x**4),x)*b*d**3*f**2*g*x**3 - 2*log((d + e*x) **n*c)*b*d**3*f**2 - 2*log((d + e*x)**n*c)*b*e**3*f**2*x**3 + 2*log(x)*b*e **3*f**2*n*x**3 - 2*a*d**3*f**2 + 6*a*d**3*f*g*x**2 - b*d**2*e*f**2*n*x + 2*b*d*e**2*f**2*n*x**2)/(6*d**3*f**3*x**3)