Optimal. Leaf size=71 \[ \frac{\left ((d+e x)^2\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-2 m} \text{Gamma}\left (2 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
[Out]
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Rubi [A] time = 0.097998, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\left ((d+e x)^2\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-2 m} \text{Gamma}\left (2 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))*(d^2 + 2*d*e*x + e^2*x^2)^m,x]
[Out]
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Rubi in Sympy [A] time = 18.4941, size = 65, normalized size = 0.92 \[ \frac{F^{\frac{c \left (a e - b d\right )}{e}} \left (\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e}\right )^{- 2 m} \left (\left (d + e x\right )^{2}\right )^{m} \Gamma{\left (2 m + 1,\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b c \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))*(e**2*x**2+2*d*e*x+d**2)**m,x)
[Out]
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Mathematica [A] time = 0.04669, size = 73, normalized size = 1.03 \[ -\frac{(d+e x) \left ((d+e x)^2\right )^m F^{a c-\frac{b c d}{e}} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-2 m-1} \text{Gamma}\left (2 m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))*(d^2 + 2*d*e*x + e^2*x^2)^m,x]
[Out]
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Maple [F] time = 0.138, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))*(e^2*x^2+2*d*e*x+d^2)^m,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2 + 2*d*e*x + d^2)^m*F^((b*x + a)*c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{b c x + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2 + 2*d*e*x + d^2)^m*F^((b*x + a)*c),x, algorithm="fricas")
[Out]
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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))*(e**2*x**2+2*d*e*x+d**2)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2 + 2*d*e*x + d^2)^m*F^((b*x + a)*c),x, algorithm="giac")
[Out]