Integrand size = 18, antiderivative size = 501 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {3 x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i x \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)} \]
1/2*f^x*x^3/a/(a+b*f^(2*x))/ln(f)-3/2*x^2*arctan(f^x*b^(1/2)/a^(1/2))/a^(3 /2)/ln(f)^2/b^(1/2)+1/2*x^3*arctan(f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)/b^(1 /2)+3/2*I*x*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^3/b^(1/2)-3/4* I*x^2*polylog(2,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^2/b^(1/2)-3/2*I*x*po lylog(2,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^3/b^(1/2)+3/4*I*x^2*polylog(2 ,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^2/b^(1/2)-3/2*I*polylog(3,-I*f^x*b^( 1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)+3/2*I*x*polylog(3,-I*f^x*b^(1/2)/a^( 1/2))/a^(3/2)/ln(f)^3/b^(1/2)+3/2*I*polylog(3,I*f^x*b^(1/2)/a^(1/2))/a^(3/ 2)/ln(f)^4/b^(1/2)-3/2*I*x*polylog(3,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^ 3/b^(1/2)-3/2*I*polylog(4,-I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)+ 3/2*I*polylog(4,I*f^x*b^(1/2)/a^(1/2))/a^(3/2)/ln(f)^4/b^(1/2)
Time = 0.21 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.87 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {\frac {2 \sqrt {a} f^x x^3 \log ^3(f)}{a+b f^{2 x}}-\frac {3 i x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {i x^3 \log ^3(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {i x^3 \log ^3(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {3 i x \log (f) (-2+x \log (f)) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x \log (f) (-2+x \log (f)) \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i x \log (f) \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i x \log (f) \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}}{4 a^{3/2} \log ^4(f)} \]
((2*Sqrt[a]*f^x*x^3*Log[f]^3)/(a + b*f^(2*x)) - ((3*I)*x^2*Log[f]^2*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + (I*x^3*Log[f]^3*Log[1 - (I*Sqrt[b]*f ^x)/Sqrt[a]])/Sqrt[b] + ((3*I)*x^2*Log[f]^2*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a ]])/Sqrt[b] - (I*x^3*Log[f]^3*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((3*I)*x*Log[f]*(-2 + x*Log[f])*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sq rt[b] + ((3*I)*x*Log[f]*(-2 + x*Log[f])*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a] ])/Sqrt[b] - ((6*I)*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6* I)*x*Log[f]*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*PolyL og[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*x*Log[f]*PolyLog[3, (I*Sq rt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a ]])/Sqrt[b] + ((6*I)*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b])/(4*a^(3 /2)*Log[f]^4)
Time = 0.75 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2675, 27, 2010, 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.
| \(\displaystyle \int \frac {x^3 f^x}{\left (a+b f^{2 x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 2675 |
| \(\displaystyle -3 \int \frac {1}{2} x^2 \left (\frac {f^x}{a \left (b f^{2 x}+a\right ) \log (f)}+\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log (f)}\right )dx+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} \int x^2 \left (\frac {f^x}{a \left (b f^{2 x}+a\right ) \log (f)}+\frac {\arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log (f)}\right )dx+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {x^2 f^x}{a \left (b f^{2 x}+a\right ) \log (f)}+\frac {x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log (f)}\right )dx+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i x^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^4(f)}-\frac {i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^4(f)}+\frac {i \operatorname {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^4(f)}-\frac {i \operatorname {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^4(f)}-\frac {i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i x \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i x \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^3(f)}\right )+\frac {x^3 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}\) |
(f^x*x^3)/(2*a*(a + b*f^(2*x))*Log[f]) + (x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a] ])/(2*a^(3/2)*Sqrt[b]*Log[f]) - (3*((x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(a ^(3/2)*Sqrt[b]*Log[f]^2) - (I*x*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a ^(3/2)*Sqrt[b]*Log[f]^3) + ((I/2)*x^2*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a ]])/(a^(3/2)*Sqrt[b]*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]]) /(a^(3/2)*Sqrt[b]*Log[f]^3) - ((I/2)*x^2*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a ]])/(a^(3/2)*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] )/(a^(3/2)*Sqrt[b]*Log[f]^4) - (I*x*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] )/(a^(3/2)*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^ (3/2)*Sqrt[b]*Log[f]^4) + (I*x*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/ 2)*Sqrt[b]*Log[f]^3) + (I*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2) *Sqrt[b]*Log[f]^4) - (I*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt [b]*Log[f]^4)))/2
3.1.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[(F_)^((e_.)*((c_.) + (d_.)*(x_)))*((a_.) + (b_.)*(F_)^(v_))^(p_)*(x_)^( m_.), x_Symbol] :> With[{u = IntHide[F^(e*(c + d*x))*(a + b*F^v)^p, x]}, Si mp[x^m u, x] - Simp[m Int[x^(m - 1)*u, x], x]] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[v, 2*e*(c + d*x)] && GtQ[m, 0] && ILtQ[p, 0]
\[\int \frac {f^{x} x^{3}}{\left (a +b \,f^{2 x}\right )^{2}}d x\]
Time = 0.27 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.10 \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {2 \, b f^{x} x^{3} \log \left (f\right )^{3} + 3 \, {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (b x^{3} \log \left (f\right )^{3} - 3 \, b x^{2} \log \left (f\right )^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \left (f\right )^{3} - 3 \, a x^{2} \log \left (f\right )^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, -f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right )}{4 \, {\left (a b^{2} f^{2 \, x} \log \left (f\right )^{4} + a^{2} b \log \left (f\right )^{4}\right )}} \]
1/4*(2*b*f^x*x^3*log(f)^3 + 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqr t(-b/a) + (a*x^2*log(f)^2 - 2*a*x*log(f))*sqrt(-b/a))*dilog(f^x*sqrt(-b/a) ) - 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (a*x^2*log(f)^ 2 - 2*a*x*log(f))*sqrt(-b/a))*dilog(-f^x*sqrt(-b/a)) - ((b*x^3*log(f)^3 - 3*b*x^2*log(f)^2)*f^(2*x)*sqrt(-b/a) + (a*x^3*log(f)^3 - 3*a*x^2*log(f)^2) *sqrt(-b/a))*log(f^x*sqrt(-b/a) + 1) + ((b*x^3*log(f)^3 - 3*b*x^2*log(f)^2 )*f^(2*x)*sqrt(-b/a) + (a*x^3*log(f)^3 - 3*a*x^2*log(f)^2)*sqrt(-b/a))*log (-f^x*sqrt(-b/a) + 1) + 6*(b*f^(2*x)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4, f^x*sqrt(-b/a)) - 6*(b*f^(2*x)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4, -f^x *sqrt(-b/a)) - 6*((b*x*log(f) - b)*f^(2*x)*sqrt(-b/a) + (a*x*log(f) - a)*s qrt(-b/a))*polylog(3, f^x*sqrt(-b/a)) + 6*((b*x*log(f) - b)*f^(2*x)*sqrt(- b/a) + (a*x*log(f) - a)*sqrt(-b/a))*polylog(3, -f^x*sqrt(-b/a)))/(a*b^2*f^ (2*x)*log(f)^4 + a^2*b*log(f)^4)
\[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {f^{x} x^{3}}{2 a^{2} \log {\left (f \right )} + 2 a b f^{2 x} \log {\left (f \right )}} + \frac {\int \left (- \frac {3 f^{x} x^{2}}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x^{3} \log {\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log {\left (f \right )}} \]
f**x*x**3/(2*a**2*log(f) + 2*a*b*f**(2*x)*log(f)) + (Integral(-3*f**x*x**2 /(a + b*f**(2*x)), x) + Integral(f**x*x**3*log(f)/(a + b*f**(2*x)), x))/(2 *a*log(f))
\[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x^{3}}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
1/2*f^x*x^3/(a*b*f^(2*x)*log(f) + a^2*log(f)) + integrate(1/2*(x^3*log(f) - 3*x^2)*f^x/(a*b*f^(2*x)*log(f) + a^2*log(f)), x)
\[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x^{3}}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx=\int \frac {f^x\,x^3}{{\left (a+b\,f^{2\,x}\right )}^2} \,d x \]