3.224 \(\int f^{\frac{c}{(a+b x)^2}} x^4 \, dx\)

Optimal. Leaf size=415 \[ -\frac{\sqrt{\pi } a^4 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{a^4 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 a^3 c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}-\frac{2 a^3 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } a^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{4 a^2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } c^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{15 b^5}+\frac{a c^2 \log ^2(f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{4 c^2 \log ^2(f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{15 b^5}+\frac{(a+b x)^5 f^{\frac{c}{(a+b x)^2}}}{5 b^5}-\frac{a (a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 c \log (f) (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{15 b^5}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5} \]

[Out]

(a^4*f^(c/(a + b*x)^2)*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^
5 + (2*a^2*f^(c/(a + b*x)^2)*(a + b*x)^3)/b^5 - (a*f^(c/(a + b*x)^2)*(a + b*x)^4
)/b^5 + (f^(c/(a + b*x)^2)*(a + b*x)^5)/(5*b^5) - (a^4*Sqrt[c]*Sqrt[Pi]*Erfi[(Sq
rt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^5 + (4*a^2*c*f^(c/(a + b*x)^2)*(a
+ b*x)*Log[f])/b^5 - (a*c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/b^5 + (2*c*f^(c
/(a + b*x)^2)*(a + b*x)^3*Log[f])/(15*b^5) + (2*a^3*c*ExpIntegralEi[(c*Log[f])/(
a + b*x)^2]*Log[f])/b^5 - (4*a^2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a
+ b*x)]*Log[f]^(3/2))/b^5 + (4*c^2*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]^2)/(15*b^
5) + (a*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/b^5 - (4*c^(5/2)*Sqr
t[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(5/2))/(15*b^5)

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Rubi [A]  time = 0.791376, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt{\pi } a^4 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{a^4 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 a^3 c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}-\frac{2 a^3 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } a^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{4 a^2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } c^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{15 b^5}+\frac{a c^2 \log ^2(f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{4 c^2 \log ^2(f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{15 b^5}+\frac{(a+b x)^5 f^{\frac{c}{(a+b x)^2}}}{5 b^5}-\frac{a (a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 c \log (f) (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{15 b^5}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x)^2)*x^4,x]

[Out]

(a^4*f^(c/(a + b*x)^2)*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^
5 + (2*a^2*f^(c/(a + b*x)^2)*(a + b*x)^3)/b^5 - (a*f^(c/(a + b*x)^2)*(a + b*x)^4
)/b^5 + (f^(c/(a + b*x)^2)*(a + b*x)^5)/(5*b^5) - (a^4*Sqrt[c]*Sqrt[Pi]*Erfi[(Sq
rt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^5 + (4*a^2*c*f^(c/(a + b*x)^2)*(a
+ b*x)*Log[f])/b^5 - (a*c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/b^5 + (2*c*f^(c
/(a + b*x)^2)*(a + b*x)^3*Log[f])/(15*b^5) + (2*a^3*c*ExpIntegralEi[(c*Log[f])/(
a + b*x)^2]*Log[f])/b^5 - (4*a^2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a
+ b*x)]*Log[f]^(3/2))/b^5 + (4*c^2*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]^2)/(15*b^
5) + (a*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/b^5 - (4*c^(5/2)*Sqr
t[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(5/2))/(15*b^5)

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Rubi in Sympy [A]  time = 68.0169, size = 415, normalized size = 1. \[ - \frac{\sqrt{\pi } a^{4} \sqrt{c} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{b^{5}} + \frac{a^{4} f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )}{b^{5}} + \frac{2 a^{3} c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{\left (a + b x\right )^{2}} \right )}}{b^{5}} - \frac{2 a^{3} f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{2}}{b^{5}} - \frac{4 \sqrt{\pi } a^{2} c^{\frac{3}{2}} \log{\left (f \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{b^{5}} + \frac{4 a^{2} c f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right ) \log{\left (f \right )}}{b^{5}} + \frac{2 a^{2} f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{3}}{b^{5}} + \frac{a c^{2} \log{\left (f \right )}^{2} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{\left (a + b x\right )^{2}} \right )}}{b^{5}} - \frac{a c f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{2} \log{\left (f \right )}}{b^{5}} - \frac{a f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{4}}{b^{5}} - \frac{4 \sqrt{\pi } c^{\frac{5}{2}} \log{\left (f \right )}^{\frac{5}{2}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{15 b^{5}} + \frac{4 c^{2} f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right ) \log{\left (f \right )}^{2}}{15 b^{5}} + \frac{2 c f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{3} \log{\left (f \right )}}{15 b^{5}} + \frac{f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{5}}{5 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c/(b*x+a)**2)*x**4,x)

[Out]

-sqrt(pi)*a**4*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/(a + b*x))/b**5 +
a**4*f**(c/(a + b*x)**2)*(a + b*x)/b**5 + 2*a**3*c*log(f)*Ei(c*log(f)/(a + b*x)*
*2)/b**5 - 2*a**3*f**(c/(a + b*x)**2)*(a + b*x)**2/b**5 - 4*sqrt(pi)*a**2*c**(3/
2)*log(f)**(3/2)*erfi(sqrt(c)*sqrt(log(f))/(a + b*x))/b**5 + 4*a**2*c*f**(c/(a +
b*x)**2)*(a + b*x)*log(f)/b**5 + 2*a**2*f**(c/(a + b*x)**2)*(a + b*x)**3/b**5 +
a*c**2*log(f)**2*Ei(c*log(f)/(a + b*x)**2)/b**5 - a*c*f**(c/(a + b*x)**2)*(a +
b*x)**2*log(f)/b**5 - a*f**(c/(a + b*x)**2)*(a + b*x)**4/b**5 - 4*sqrt(pi)*c**(5
/2)*log(f)**(5/2)*erfi(sqrt(c)*sqrt(log(f))/(a + b*x))/(15*b**5) + 4*c**2*f**(c/
(a + b*x)**2)*(a + b*x)*log(f)**2/(15*b**5) + 2*c*f**(c/(a + b*x)**2)*(a + b*x)*
*3*log(f)/(15*b**5) + f**(c/(a + b*x)**2)*(a + b*x)**5/(5*b**5)

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Mathematica [A]  time = 0.242088, size = 195, normalized size = 0.47 \[ \frac{a \left (3 a^4+47 a^2 c \log (f)+4 c^2 \log ^2(f)\right ) f^{\frac{c}{(a+b x)^2}}}{15 b^5}+\frac{b x f^{\frac{c}{(a+b x)^2}} \left (c \log (f) \left (36 a^2-9 a b x+2 b^2 x^2\right )+3 b^4 x^4+4 c^2 \log ^2(f)\right )+15 a c \log (f) \left (2 a^2+c \log (f)\right ) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )-\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \left (15 a^4+60 a^2 c \log (f)+4 c^2 \log ^2(f)\right ) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{15 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x)^2)*x^4,x]

[Out]

(a*f^(c/(a + b*x)^2)*(3*a^4 + 47*a^2*c*Log[f] + 4*c^2*Log[f]^2))/(15*b^5) + (15*
a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]*(2*a^2 + c*Log[f]) - Sqrt[c]*Sq
rt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]]*(15*a^4 + 60*a^2*c*Lo
g[f] + 4*c^2*Log[f]^2) + b*f^(c/(a + b*x)^2)*x*(3*b^4*x^4 + c*(36*a^2 - 9*a*b*x
+ 2*b^2*x^2)*Log[f] + 4*c^2*Log[f]^2))/(15*b^5)

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Maple [A]  time = 0.097, size = 343, normalized size = 0.8 \[{\frac{47\,c{a}^{3}\ln \left ( f \right ) }{15\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{4\, \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}}{15\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{{a}^{4}c\ln \left ( f \right ) \sqrt{\pi }}{{b}^{5}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-4\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}{c}^{2}\sqrt{\pi }}{{b}^{5}\sqrt{-c\ln \left ( f \right ) }}{\it Erf} \left ({\frac{\sqrt{-c\ln \left ( f \right ) }}{bx+a}} \right ) }-{\frac{3\,ac\ln \left ( f \right ){x}^{2}}{5\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{12\,{a}^{2}c\ln \left ( f \right ) x}{5\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{x}^{5}}{5}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{a}^{5}}{5\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-2\,{\frac{c{a}^{3}\ln \left ( f \right ) }{{b}^{5}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}}{{b}^{5}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+{\frac{2\,c\ln \left ( f \right ){x}^{3}}{15\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{4\,{c}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}x}{15\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{4\, \left ( \ln \left ( f \right ) \right ) ^{3}{c}^{3}\sqrt{\pi }}{15\,{b}^{5}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a)^2)*x^4,x)

[Out]

47/15/b^5*c*ln(f)*f^(c/(b*x+a)^2)*a^3+4/15/b^5*c^2*ln(f)^2*f^(c/(b*x+a)^2)*a-1/b
^5*a^4*c*ln(f)*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-4/b^5*a^2
*c^2*ln(f)^2*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-3/5/b^3*c*l
n(f)*f^(c/(b*x+a)^2)*a*x^2+12/5/b^4*c*ln(f)*f^(c/(b*x+a)^2)*a^2*x+1/5*f^(c/(b*x+
a)^2)*x^5+1/5/b^5*a^5*f^(c/(b*x+a)^2)-2/b^5*a^3*c*ln(f)*Ei(1,-c*ln(f)/(b*x+a)^2)
-1/b^5*a*c^2*ln(f)^2*Ei(1,-c*ln(f)/(b*x+a)^2)+2/15/b^2*c*ln(f)*f^(c/(b*x+a)^2)*x
^3+4/15/b^4*c^2*ln(f)^2*f^(c/(b*x+a)^2)*x-4/15/b^5*c^3*ln(f)^3*Pi^(1/2)/(-c*ln(f
))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (3 \, b^{4} x^{5} + 2 \, b^{2} c x^{3} \log \left (f\right ) - 9 \, a b c x^{2} \log \left (f\right ) + 4 \,{\left (9 \, a^{2} c \log \left (f\right ) + c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \, b^{4}} - \int \frac{2 \,{\left (18 \, a^{5} c \log \left (f\right ) + 2 \, a^{3} c^{2} \log \left (f\right )^{2} + 15 \,{\left (2 \, a^{3} b^{2} c \log \left (f\right ) + a b^{2} c^{2} \log \left (f\right )^{2}\right )} x^{2} +{\left (45 \, a^{4} b c \log \left (f\right ) - 30 \, a^{2} b c^{2} \log \left (f\right )^{2} - 4 \, b c^{3} \log \left (f\right )^{3}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a)^2)*x^4,x, algorithm="maxima")

[Out]

1/15*(3*b^4*x^5 + 2*b^2*c*x^3*log(f) - 9*a*b*c*x^2*log(f) + 4*(9*a^2*c*log(f) +
c^2*log(f)^2)*x)*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^4 - integrate(2/15*(18*a^5*c*
log(f) + 2*a^3*c^2*log(f)^2 + 15*(2*a^3*b^2*c*log(f) + a*b^2*c^2*log(f)^2)*x^2 +
(45*a^4*b*c*log(f) - 30*a^2*b*c^2*log(f)^2 - 4*b*c^3*log(f)^3)*x)*f^(c/(b^2*x^2
+ 2*a*b*x + a^2))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4), x)

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Fricas [A]  time = 0.274548, size = 306, normalized size = 0.74 \[ -\frac{\sqrt{\pi }{\left (15 \, a^{4} c \log \left (f\right ) + 60 \, a^{2} c^{2} \log \left (f\right )^{2} + 4 \, c^{3} \log \left (f\right )^{3}\right )} \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) -{\left ({\left (3 \, b^{6} x^{5} + 3 \, a^{5} b + 4 \,{\left (b^{2} c^{2} x + a b c^{2}\right )} \log \left (f\right )^{2} +{\left (2 \, b^{4} c x^{3} - 9 \, a b^{3} c x^{2} + 36 \, a^{2} b^{2} c x + 47 \, a^{3} b c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 15 \,{\left (2 \, a^{3} b c \log \left (f\right ) + a b c^{2} \log \left (f\right )^{2}\right )}{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )\right )} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{15 \, b^{6} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a)^2)*x^4,x, algorithm="fricas")

[Out]

-1/15*(sqrt(pi)*(15*a^4*c*log(f) + 60*a^2*c^2*log(f)^2 + 4*c^3*log(f)^3)*erf(b*s
qrt(-c*log(f)/b^2)/(b*x + a)) - ((3*b^6*x^5 + 3*a^5*b + 4*(b^2*c^2*x + a*b*c^2)*
log(f)^2 + (2*b^4*c*x^3 - 9*a*b^3*c*x^2 + 36*a^2*b^2*c*x + 47*a^3*b*c)*log(f))*f
^(c/(b^2*x^2 + 2*a*b*x + a^2)) + 15*(2*a^3*b*c*log(f) + a*b*c^2*log(f)^2)*Ei(c*l
og(f)/(b^2*x^2 + 2*a*b*x + a^2)))*sqrt(-c*log(f)/b^2))/(b^6*sqrt(-c*log(f)/b^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c/(b*x+a)**2)*x**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{\frac{c}{{\left (b x + a\right )}^{2}}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a)^2)*x^4,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a)^2)*x^4, x)