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

Optimal. Leaf size=62 \[ \frac{(a+b x) f^{\frac{c}{(a+b x)^2}}}{b}-\frac{\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0745087, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{(a+b x) f^{\frac{c}{(a+b x)^2}}}{b}-\frac{\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.50836, size = 53, normalized size = 0.85 \[ - \frac{\sqrt{\pi } \sqrt{c} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\frac{\sqrt{c} \sqrt{\log{\left (f \right )}}}{a + b x} \right )}}{b} + \frac{f^{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0227393, size = 62, normalized size = 1. \[ \frac{(a+b x) f^{\frac{c}{(a+b x)^2}}}{b}-\frac{\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 65, normalized size = 1.1 \[{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}x+{\frac{a}{b}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{c\ln \left ( f \right ) \sqrt{\pi }}{b}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a)^2),x)

[Out]

f^(c/(b*x+a)^2)*x+1/b*f^(c/(b*x+a)^2)*a-1/b*c*ln(f)*Pi^(1/2)/(-c*ln(f))^(1/2)*er
f((-c*ln(f))^(1/2)/(b*x+a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.310337, size = 116, normalized size = 1.87 \[ -\frac{\sqrt{\pi } c \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) \log \left (f\right ) -{\left (b^{2} x + a b\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b^{2} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c/(b*x+a)**2),x)

[Out]

Integral(f**(c/(a + b*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(c/(b*x + a)^2), x)