3.52 \(\int \frac{f^x x}{\left (a+b f^{2 x}\right )^3} \, dx\)

Optimal. Leaf size=223 \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac{3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

[Out]

-f^x/(8*a^2*(a + b*f^(2*x))*Log[f]^2) - ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(5/2)
*Sqrt[b]*Log[f]^2) + (f^x*x)/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x*x)/(8*a^2*(
a + b*f^(2*x))*Log[f]) + (3*x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*
Log[f]) - (((3*I)/16)*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*L
og[f]^2) + (((3*I)/16)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log
[f]^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.380101, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b} \log ^2(f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b} \log ^2(f)}+\frac{3 x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}-\frac{f^x}{8 a^2 \log ^2(f) \left (a+b f^{2 x}\right )}+\frac{3 x f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{x f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(f^x*x)/(a + b*f^(2*x))^3,x]

[Out]

-f^x/(8*a^2*(a + b*f^(2*x))*Log[f]^2) - ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(5/2)
*Sqrt[b]*Log[f]^2) + (f^x*x)/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x*x)/(8*a^2*(
a + b*f^(2*x))*Log[f]) + (3*x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*
Log[f]) - (((3*I)/16)*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*L
og[f]^2) + (((3*I)/16)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log
[f]^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 65.1531, size = 209, normalized size = 0.94 \[ \frac{f^{x} x}{4 a \left (a + b f^{2 x}\right )^{2} \log{\left (f \right )}} + \frac{3 f^{x} x}{8 a^{2} \left (a + b f^{2 x}\right ) \log{\left (f \right )}} - \frac{f^{x}}{8 a^{2} \left (a + b f^{2 x}\right ) \log{\left (f \right )}^{2}} + \frac{3 x \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}^{2}} - \frac{3 i \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{16 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{3 i \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{16 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**x*x/(a+b*f**(2*x))**3,x)

[Out]

f**x*x/(4*a*(a + b*f**(2*x))**2*log(f)) + 3*f**x*x/(8*a**2*(a + b*f**(2*x))*log(
f)) - f**x/(8*a**2*(a + b*f**(2*x))*log(f)**2) + 3*x*atan(sqrt(b)*f**x/sqrt(a))/
(8*a**(5/2)*sqrt(b)*log(f)) - atan(sqrt(b)*f**x/sqrt(a))/(2*a**(5/2)*sqrt(b)*log
(f)**2) - 3*I*polylog(2, -I*sqrt(b)*f**x/sqrt(a))/(16*a**(5/2)*sqrt(b)*log(f)**2
) + 3*I*polylog(2, I*sqrt(b)*f**x/sqrt(a))/(16*a**(5/2)*sqrt(b)*log(f)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.433942, size = 184, normalized size = 0.83 \[ \frac{\frac{6 i \left (-\text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x \log (f) \left (\log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-\log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )\right )}{\sqrt{a} \sqrt{b}}-\frac{16 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}+\frac{8 a x f^x \log (f)}{\left (a+b f^{2 x}\right )^2}+\frac{4 f^x (3 x \log (f)-1)}{a+b f^{2 x}}}{32 a^2 \log ^2(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[(f^x*x)/(a + b*f^(2*x))^3,x]

[Out]

((-16*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (8*a*f^x*x*Log[f])/(a +
b*f^(2*x))^2 + (4*f^x*(-1 + 3*x*Log[f]))/(a + b*f^(2*x)) + ((6*I)*(x*Log[f]*(Lo
g[1 - (I*Sqrt[b]*f^x)/Sqrt[a]] - Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]]) - PolyLog[2,
((-I)*Sqrt[b]*f^x)/Sqrt[a]] + PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]]))/(Sqrt[a]*Sqr
t[b]))/(32*a^2*Log[f]^2)

_______________________________________________________________________________________

Maple [C]  time = 0.058, size = 223, normalized size = 1. \[{\frac{ \left ( 3\,\ln \left ( f \right ) bx \left ({f}^{x} \right ) ^{2}+5\,\ln \left ( f \right ) ax-b \left ({f}^{x} \right ) ^{2}-a \right ){f}^{x}}{8\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2} \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) ^{2}}}-{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}\arctan \left ({b{f}^{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,x}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({1 \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{3\,x}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({1 \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{16\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}{\it dilog} \left ({1 \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{3}{16\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}}{\it dilog} \left ({1 \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x*x/(a+b*f^(2*x))^3,x)

[Out]

1/8*f^x*(3*ln(f)*b*x*(f^x)^2+5*ln(f)*a*x-b*(f^x)^2-a)/ln(f)^2/a^2/(a+b*(f^x)^2)^
2-1/2/ln(f)^2/a^2/(a*b)^(1/2)*arctan(b*f^x/(a*b)^(1/2))+3/16/ln(f)/a^2*x/(-a*b)^
(1/2)*ln((-b*f^x+(-a*b)^(1/2))/(-a*b)^(1/2))-3/16/ln(f)/a^2*x/(-a*b)^(1/2)*ln((b
*f^x+(-a*b)^(1/2))/(-a*b)^(1/2))+3/16/ln(f)^2/a^2/(-a*b)^(1/2)*dilog((-b*f^x+(-a
*b)^(1/2))/(-a*b)^(1/2))-3/16/ln(f)^2/a^2/(-a*b)^(1/2)*dilog((b*f^x+(-a*b)^(1/2)
)/(-a*b)^(1/2))

_______________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^x*x/(b*f^(2*x) + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.262802, size = 756, normalized size = 3.39 \[ \frac{2 \,{\left (3 \, b^{2} x \log \left (f\right ) - b^{2}\right )} f^{3 \, x} + 2 \,{\left (5 \, a b x \log \left (f\right ) - a b\right )} f^{x} + 3 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (-\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) - 3 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) - 4 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt{-\frac{b}{a}}\right ) + 4 \,{\left (b^{2} f^{4 \, x} \sqrt{-\frac{b}{a}} + 2 \, a b f^{2 \, x} \sqrt{-\frac{b}{a}} + a^{2} \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt{-\frac{b}{a}}\right ) + 3 \,{\left (b^{2} f^{4 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a^{2} x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) - 3 \,{\left (b^{2} f^{4 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + 2 \, a b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a^{2} x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (-\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right )}{16 \,{\left (a^{2} b^{3} f^{4 \, x} \log \left (f\right )^{2} + 2 \, a^{3} b^{2} f^{2 \, x} \log \left (f\right )^{2} + a^{4} b \log \left (f\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^x*x/(b*f^(2*x) + a)^3,x, algorithm="fricas")

[Out]

1/16*(2*(3*b^2*x*log(f) - b^2)*f^(3*x) + 2*(5*a*b*x*log(f) - a*b)*f^x + 3*(b^2*f
^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*dilog(-(b*f^x + a
*sqrt(-b/a))/(a*sqrt(-b/a)) + 1) - 3*(b^2*f^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqr
t(-b/a) + a^2*sqrt(-b/a))*dilog((b*f^x - a*sqrt(-b/a))/(a*sqrt(-b/a)) + 1) - 4*(
b^2*f^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*sqrt(-b/a))*log(2*b*f^x
+ 2*a*sqrt(-b/a)) + 4*(b^2*f^(4*x)*sqrt(-b/a) + 2*a*b*f^(2*x)*sqrt(-b/a) + a^2*s
qrt(-b/a))*log(2*b*f^x - 2*a*sqrt(-b/a)) + 3*(b^2*f^(4*x)*x*sqrt(-b/a)*log(f) +
2*a*b*f^(2*x)*x*sqrt(-b/a)*log(f) + a^2*x*sqrt(-b/a)*log(f))*log((b*f^x + a*sqrt
(-b/a))/(a*sqrt(-b/a))) - 3*(b^2*f^(4*x)*x*sqrt(-b/a)*log(f) + 2*a*b*f^(2*x)*x*s
qrt(-b/a)*log(f) + a^2*x*sqrt(-b/a)*log(f))*log(-(b*f^x - a*sqrt(-b/a))/(a*sqrt(
-b/a))))/(a^2*b^3*f^(4*x)*log(f)^2 + 2*a^3*b^2*f^(2*x)*log(f)^2 + a^4*b*log(f)^2
)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{f^{3 x} \left (3 b x \log{\left (f \right )} - b\right ) + f^{x} \left (5 a x \log{\left (f \right )} - a\right )}{8 a^{4} \log{\left (f \right )}^{2} + 16 a^{3} b f^{2 x} \log{\left (f \right )}^{2} + 8 a^{2} b^{2} f^{4 x} \log{\left (f \right )}^{2}} + \frac{\int \left (- \frac{4 f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac{3 f^{x} x \log{\left (f \right )}}{a + b f^{2 x}}\, dx}{8 a^{2} \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**x*x/(a+b*f**(2*x))**3,x)

[Out]

(f**(3*x)*(3*b*x*log(f) - b) + f**x*(5*a*x*log(f) - a))/(8*a**4*log(f)**2 + 16*a
**3*b*f**(2*x)*log(f)**2 + 8*a**2*b**2*f**(4*x)*log(f)**2) + (Integral(-4*f**x/(
a + b*f**(2*x)), x) + Integral(3*f**x*x*log(f)/(a + b*f**(2*x)), x))/(8*a**2*log
(f))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^x*x/(b*f^(2*x) + a)^3,x, algorithm="giac")

[Out]

integrate(f^x*x/(b*f^(2*x) + a)^3, x)