Optimal. Leaf size=46 \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
[Out]
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Rubi [A] time = 0.0418323, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
Antiderivative was successfully verified.
[In] Int[f^(a + b*x^2)*x^m,x]
[Out]
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Rubi in Sympy [A] time = 3.31628, size = 46, normalized size = 1. \[ - \frac{f^{a} x^{m + 1} \left (- b x^{2} \log{\left (f \right )}\right )^{- \frac{m}{2} - \frac{1}{2}} \Gamma{\left (\frac{m}{2} + \frac{1}{2},- b x^{2} \log{\left (f \right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(b*x**2+a)*x**m,x)
[Out]
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Mathematica [A] time = 0.025781, size = 46, normalized size = 1. \[ -\frac{1}{2} f^a x^{m+1} \left (-b x^2 \log (f)\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b x^2 \log (f)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[f^(a + b*x^2)*x^m,x]
[Out]
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Maple [B] time = 0.046, size = 140, normalized size = 3. \[{\frac{{f}^{a}}{2} \left ( -b \right ) ^{-{\frac{m}{2}}-{\frac{1}{2}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{m}{2}}-{\frac{1}{2}}} \left ( 2\,{\frac{{x}^{1+m} \left ( -b \right ) ^{m/2+1/2} \left ( \ln \left ( f \right ) \right ) ^{m/2+1/2} \left ( m/2+1/2 \right ) \left ( -b{x}^{2}\ln \left ( f \right ) \right ) ^{-m/2-1/2}\Gamma \left ( m/2+1/2 \right ) }{1+m}}+2\,{\frac{{x}^{1+m} \left ( -b \right ) ^{m/2+1/2} \left ( \ln \left ( f \right ) \right ) ^{m/2+1/2} \left ( -m/2-1/2 \right ) \left ( -b{x}^{2}\ln \left ( f \right ) \right ) ^{-m/2-1/2}\Gamma \left ( m/2+1/2,-b{x}^{2}\ln \left ( f \right ) \right ) }{1+m}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(b*x^2+a)*x^m,x)
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Maxima [A] time = 0.845387, size = 51, normalized size = 1.11 \[ -\frac{1}{2} \, \left (-b x^{2} \log \left (f\right )\right )^{-\frac{1}{2} \, m - \frac{1}{2}} f^{a} x^{m + 1} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -b x^{2} \log \left (f\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^2 + a)*x^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256735, size = 54, normalized size = 1.17 \[ \frac{e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-b \log \left (f\right )\right ) + a \log \left (f\right )\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, b \log \left (f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^2 + a)*x^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + b x^{2}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(b*x**2+a)*x**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{b x^{2} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^2 + a)*x^m,x, algorithm="giac")
[Out]