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

Optimal. Leaf size=111 \[ \frac{\sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^2}-\frac{c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{2 b^2}+\frac{(a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{2 b^2}-\frac{a (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.198994, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{\sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^2}-\frac{c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{2 b^2}+\frac{(a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{2 b^2}-\frac{a (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x)^2)*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.6343, size = 105, normalized size = 0.95 \[ \frac{\sqrt{\pi } a \sqrt{c} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{b^{2}} - \frac{a f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )}{b^{2}} - \frac{c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{\left (a + b x\right )^{2}} \right )}}{2 b^{2}} + \frac{f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{2}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.049486, size = 89, normalized size = 0.8 \[ \frac{\left (b^2 x^2-a^2\right ) f^{\frac{c}{(a+b x)^2}}+2 \sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )-c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 93, normalized size = 0.8 \[{\frac{{x}^{2}}{2}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{{a}^{2}}{2\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{c\ln \left ( f \right ) }{2\,{b}^{2}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+{\frac{ac\ln \left ( f \right ) \sqrt{\pi }}{{b}^{2}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a)^2)*x,x)

[Out]

1/2*f^(c/(b*x+a)^2)*x^2-1/2/b^2*f^(c/(b*x+a)^2)*a^2+1/2/b^2*c*ln(f)*Ei(1,-c*ln(f
)/(b*x+a)^2)+1/b^2*a*c*ln(f)*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. \[ b c \int \frac{f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x^{2}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\,{d x} \log \left (f\right ) + \frac{1}{2} \, f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.32739, size = 167, normalized size = 1.5 \[ \frac{2 \, \sqrt{\pi } a c \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) \log \left (f\right ) -{\left (b c{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) \log \left (f\right ) -{\left (b^{3} x^{2} - a^{2} b\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}\right )} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{2 \, b^{3} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(c/(b*x + a)^2)*x, x)