Integrand size = 27, antiderivative size = 369 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=-\frac {a f x}{g^2}+\frac {b f n x}{g^2}-\frac {b d^2 n x}{3 e^2 g}+\frac {b d n x^2}{6 e g}-\frac {b n x^3}{9 g}+\frac {b d^3 n \log (d+e x)}{3 e^3 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {b (-f)^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}+\frac {b (-f)^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}} \] Output:
-a*f*x/g^2+b*f*n*x/g^2-1/3*b*d^2*n*x/e^2/g+1/6*b*d*n*x^2/e/g-1/9*b*n*x^3/g +1/3*b*d^3*n*ln(e*x+d)/e^3/g-b*f*(e*x+d)*ln(c*(e*x+d)^n)/e/g^2+1/3*x^3*(a+ b*ln(c*(e*x+d)^n))/g+1/2*(-f)^(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)))/g^(5/2)-1/2*(-f)^(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)))/g^(5/2)-1/2 *b*(-f)^(3/2)*n*polylog(2,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/g^(5/ 2)+1/2*b*(-f)^(3/2)*n*polylog(2,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/ g^(5/2)
Time = 0.36 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.92 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\frac {-18 a f \sqrt {g} x+18 b f \sqrt {g} n x-\frac {b g^{3/2} n \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 d^3 \log (d+e x)\right )}{e^3}-\frac {18 b f \sqrt {g} (d+e x) \log \left (c (d+e x)^n\right )}{e}+6 g^{3/2} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+9 (-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+9 \sqrt {-f} f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )-9 b (-f)^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+9 b (-f)^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{18 g^{5/2}} \] Input:
Integrate[(x^4*(a + b*Log[c*(d + e*x)^n]))/(f + g*x^2),x]
Output:
(-18*a*f*Sqrt[g]*x + 18*b*f*Sqrt[g]*n*x - (b*g^(3/2)*n*(e*x*(6*d^2 - 3*d*e *x + 2*e^2*x^2) - 6*d^3*Log[d + e*x]))/e^3 - (18*b*f*Sqrt[g]*(d + e*x)*Log [c*(d + e*x)^n])/e + 6*g^(3/2)*x^3*(a + b*Log[c*(d + e*x)^n]) + 9*(-f)^(3/ 2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] + 9*Sqrt[-f]*f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])] - 9*b*(-f)^(3/2)*n*PolyLog[2, -((Sqr t[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 9*b*(-f)^(3/2)*n*PolyLog[2, ( Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(18*g^(5/2))
Time = 1.12 (sec) , antiderivative size = 369, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}-\frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b d^3 n \log (d+e x)}{3 e^3 g}-\frac {b d^2 n x}{3 e^2 g}-\frac {b (-f)^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}+\frac {b (-f)^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 g^{5/2}}+\frac {b d n x^2}{6 e g}+\frac {b f n x}{g^2}-\frac {b n x^3}{9 g}\) |
Input:
Int[(x^4*(a + b*Log[c*(d + e*x)^n]))/(f + g*x^2),x]
Output:
-((a*f*x)/g^2) + (b*f*n*x)/g^2 - (b*d^2*n*x)/(3*e^2*g) + (b*d*n*x^2)/(6*e* g) - (b*n*x^3)/(9*g) + (b*d^3*n*Log[d + e*x])/(3*e^3*g) - (b*f*(d + e*x)*L og[c*(d + e*x)^n])/(e*g^2) + (x^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((-f )^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[ -f] + d*Sqrt[g])])/(2*g^(5/2)) - ((-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Lo g[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(5/2)) - (b*( -f)^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/( 2*g^(5/2)) + (b*(-f)^(3/2)*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(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 = 2.37 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.64
| method | result | size |
| risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{3}}{3 g}+\frac {b \,d^{3} \ln \left (\left (e x +d \right )^{n}\right )}{3 e^{3} g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f x}{g^{2}}-\frac {b d f \ln \left (\left (e x +d \right )^{n}\right )}{e \,g^{2}}-\frac {b \,f^{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 )}{g^{2} \sqrt {g f}}+\frac {b \,f^{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 )}{g^{2} \sqrt {g f}}-\frac {b n \,x^{3}}{9 g}+\frac {b d n \,x^{2}}{6 e g}-\frac {b \,d^{2} n x}{3 e^{2} g}-\frac {11 b \,d^{3} n}{18 e^{3} g}+\frac {b f n x}{g^{2}}+\frac {b d f n}{e \,g^{2}}+\frac {b n \,f^{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 g^{2} \sqrt {-g f}}-\frac {b n \,f^{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 g^{2} \sqrt {-g f}}+\frac {b n \,f^{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 g^{2} \sqrt {-g f}}-\frac {b n \,f^{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 g^{2} \sqrt {-g 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 {\frac {1}{3} g \,x^{3}-f x}{g^{2}}+\frac {f^{2} \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{g^{2} \sqrt {g f}}\right )\) | \(606\) |
Input:
int(x^4*(a+b*ln(c*(e*x+d)^n))/(g*x^2+f),x,method=_RETURNVERBOSE)
Output:
1/3*b*ln((e*x+d)^n)/g*x^3+1/3*b/e^3/g*d^3*ln((e*x+d)^n)-b*ln((e*x+d)^n)/g^ 2*f*x-b/e/g^2*d*f*ln((e*x+d)^n)-b*f^2/g^2/(g*f)^(1/2)*arctan(1/2*(2*g*(e*x +d)-2*d*g)/e/(g*f)^(1/2))*n*ln(e*x+d)+b*f^2/g^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)-1/9*b*n*x^3/g+1/6*b*d*n*x^2/ e/g-1/3*b*d^2*n*x/e^2/g-11/18*b*d^3*n/e^3/g+b*f*n*x/g^2+b*d*f*n/e/g^2+1/2* b*n*f^2/g^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*f^2/g^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*f^2/g^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*f^2/g^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*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)*c sgn(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*c sgn(I*c*(e*x+d)^n)^2*csgn(I*c)+b*ln(c)+a)*(1/g^2*(1/3*g*x^3-f*x)+f^2/g^2/( g*f)^(1/2)*arctan(g*x/(g*f)^(1/2)))
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{g x^{2} + f} \,d x } \] Input:
integrate(x^4*(a+b*log(c*(e*x+d)^n))/(g*x^2+f),x, algorithm="fricas")
Output:
integral((b*x^4*log((e*x + d)^n*c) + a*x^4)/(g*x^2 + f), x)
Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\text {Timed out} \] Input:
integrate(x**4*(a+b*ln(c*(e*x+d)**n))/(g*x**2+f),x)
Output:
Timed out
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{g x^{2} + f} \,d x } \] Input:
integrate(x^4*(a+b*log(c*(e*x+d)^n))/(g*x^2+f),x, algorithm="maxima")
Output:
1/3*a*(3*f^2*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g^2) + (g*x^3 - 3*f*x)/g^2) + b*integrate((x^4*log((e*x + d)^n) + x^4*log(c))/(g*x^2 + f), x)
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{g x^{2} + f} \,d x } \] Input:
integrate(x^4*(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^4/(g*x^2 + f), x)
Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{g\,x^2+f} \,d x \] Input:
int((x^4*(a + b*log(c*(d + e*x)^n)))/(f + g*x^2),x)
Output:
int((x^4*(a + b*log(c*(d + e*x)^n)))/(f + g*x^2), x)
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx=\frac {18 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a \,e^{3} f +18 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b d \,e^{3} f^{2} g +18 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) x}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b \,e^{4} f^{2} g +6 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,d^{3} g^{2}-18 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b d \,e^{2} f g -18 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{3} f g x +6 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b \,e^{3} g^{2} x^{3}-18 a \,e^{3} f g x +6 a \,e^{3} g^{2} x^{3}-6 b \,d^{2} e \,g^{2} n x +3 b d \,e^{2} g^{2} n \,x^{2}+18 b \,e^{3} f g n x -2 b \,e^{3} g^{2} n \,x^{3}}{18 e^{3} g^{3}} \] Input:
int(x^4*(a+b*log(c*(e*x+d)^n))/(g*x^2+f),x)
Output:
(18*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*e**3*f + 18*int(log((d + e*x)**n*c)/(d*f + d*g*x**2 + e*f*x + e*g*x**3),x)*b*d*e**3*f**2*g + 18* int((log((d + e*x)**n*c)*x)/(d*f + d*g*x**2 + e*f*x + e*g*x**3),x)*b*e**4* f**2*g + 6*log((d + e*x)**n*c)*b*d**3*g**2 - 18*log((d + e*x)**n*c)*b*d*e* *2*f*g - 18*log((d + e*x)**n*c)*b*e**3*f*g*x + 6*log((d + e*x)**n*c)*b*e** 3*g**2*x**3 - 18*a*e**3*f*g*x + 6*a*e**3*g**2*x**3 - 6*b*d**2*e*g**2*n*x + 3*b*d*e**2*g**2*n*x**2 + 18*b*e**3*f*g*n*x - 2*b*e**3*g**2*n*x**3)/(18*e* *3*g**3)