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

Optimal. Leaf size=206 \[ -\frac{\sqrt{\pi } a^2 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^3}+\frac{a^2 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^3}-\frac{2 \sqrt{\pi } c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{3 b^3}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac{(a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{3 b^3}-\frac{a (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^3}+\frac{2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.372813, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt{\pi } a^2 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^3}+\frac{a^2 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^3}-\frac{2 \sqrt{\pi } c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{3 b^3}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^3}+\frac{(a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{3 b^3}-\frac{a (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^3}+\frac{2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x)^2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 35.2048, size = 201, normalized size = 0.98 \[ - \frac{\sqrt{\pi } a^{2} \sqrt{c} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{b^{3}} + \frac{a^{2} f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )}{b^{3}} + \frac{a c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{\left (a + b x\right )^{2}} \right )}}{b^{3}} - \frac{a f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{2}}{b^{3}} - \frac{2 \sqrt{\pi } c^{\frac{3}{2}} \log{\left (f \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{3 b^{3}} + \frac{2 c f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right ) \log{\left (f \right )}}{3 b^{3}} + \frac{f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{3}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121642, size = 131, normalized size = 0.64 \[ \frac{a \left (a^2+2 c \log (f)\right ) f^{\frac{c}{(a+b x)^2}}}{3 b^3}+\frac{-\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \left (3 a^2+2 c \log (f)\right ) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )+b x f^{\frac{c}{(a+b x)^2}} \left (b^2 x^2+2 c \log (f)\right )+3 a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{3 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(a*f^(c/(a + b*x)^2)*(a^2 + 2*c*Log[f]))/(3*b^3) + (3*a*c*ExpIntegralEi[(c*Log[f
])/(a + b*x)^2]*Log[f] - Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]
*Sqrt[Log[f]]*(3*a^2 + 2*c*Log[f]) + b*f^(c/(a + b*x)^2)*x*(b^2*x^2 + 2*c*Log[f]
))/(3*b^3)

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Maple [A]  time = 0.039, size = 175, normalized size = 0.9 \[{\frac{{x}^{3}}{3}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{a}^{3}}{3\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,c\ln \left ( f \right ) x}{3\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,ac\ln \left ( f \right ) }{3\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{2\,{c}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}\sqrt{\pi }}{3\,{b}^{3}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{a}^{2}c\ln \left ( f \right ) \sqrt{\pi }}{{b}^{3}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{ac\ln \left ( f \right ) }{{b}^{3}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a)^2)*x^2,x)

[Out]

1/3*f^(c/(b*x+a)^2)*x^3+1/3/b^3*a^3*f^(c/(b*x+a)^2)+2/3/b^2*c*ln(f)*f^(c/(b*x+a)
^2)*x+2/3/b^3*c*ln(f)*f^(c/(b*x+a)^2)*a-2/3/b^3*c^2*ln(f)^2*Pi^(1/2)/(-c*ln(f))^
(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a^2*c*ln(f)*Pi^(1/2)/(-c*ln(f))^(1/2)*
erf((-c*ln(f))^(1/2)/(b*x+a))-1/b^3*a*c*ln(f)*Ei(1,-c*ln(f)/(b*x+a)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^2*x^3 + 2*c*x*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^2 - integrate(2/3
*(3*a*b^2*c*x^2*log(f) + a^3*c*log(f) + (3*a^2*b*c*log(f) - 2*b*c^2*log(f)^2)*x)
*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)
, x)

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Fricas [A]  time = 0.252103, size = 205, normalized size = 1. \[ -\frac{\sqrt{\pi }{\left (3 \, a^{2} c \log \left (f\right ) + 2 \, c^{2} \log \left (f\right )^{2}\right )} \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) -{\left (3 \, a b c{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \left (f\right ) +{\left (b^{4} x^{3} + a^{3} b + 2 \,{\left (b^{2} c x + a b c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}\right )} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{3 \, b^{4} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(sqrt(pi)*(3*a^2*c*log(f) + 2*c^2*log(f)^2)*erf(b*sqrt(-c*log(f)/b^2)/(b*x
+ a)) - (3*a*b*c*Ei(c*log(f)/(b^2*x^2 + 2*a*b*x + a^2))*log(f) + (b^4*x^3 + a^3*
b + 2*(b^2*c*x + a*b*c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2)))*sqrt(-c*log(f)/
b^2))/(b^4*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**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(c/(b*x + a)^2)*x^2, x)