3.254 \(\int F^{a+b (c+d x)^2} (c+d x)^m \, dx\)

Optimal. Leaf size=61 \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]

[Out]

-(F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/2, -(b*(c + d*x)^2*Log[F])]*(-(b*(c + d*x)
^2*Log[F]))^((-1 - m)/2))/(2*d)

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Rubi [A]  time = 0.105459, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(c + d*x)^m,x]

[Out]

-(F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/2, -(b*(c + d*x)^2*Log[F])]*(-(b*(c + d*x)
^2*Log[F]))^((-1 - m)/2))/(2*d)

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Rubi in Sympy [A]  time = 5.48506, size = 58, normalized size = 0.95 \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{2} \log{\left (F \right )}\right )^{- \frac{m}{2} - \frac{1}{2}} \left (c + d x\right )^{m + 1} \Gamma{\left (\frac{m}{2} + \frac{1}{2},- b \left (c + d x\right )^{2} \log{\left (F \right )} \right )}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**m,x)

[Out]

-F**a*(-b*(c + d*x)**2*log(F))**(-m/2 - 1/2)*(c + d*x)**(m + 1)*Gamma(m/2 + 1/2,
-b*(c + d*x)**2*log(F))/(2*d)

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Mathematica [A]  time = 0.0626783, size = 61, normalized size = 1. \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^m,x]

[Out]

-(F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/2, -(b*(c + d*x)^2*Log[F])]*(-(b*(c + d*x)
^2*Log[F]))^((-1 - m)/2))/(2*d)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{F}^{a+b \left ( dx+c \right ) ^{2}} \left ( dx+c \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)*(d*x+c)^m,x)

[Out]

int(F^(a+b*(d*x+c)^2)*(d*x+c)^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^m*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*F^((d*x + c)^2*b + a), x)

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Fricas [A]  time = 0.261786, size = 80, normalized size = 1.31 \[ \frac{e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-b \log \left (F\right )\right ) + a \log \left (F\right )\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )}{2 \, b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^m*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

1/2*e^(-1/2*(m - 1)*log(-b*log(F)) + a*log(F))*gamma(1/2*m + 1/2, -(b*d^2*x^2 +
2*b*c*d*x + b*c^2)*log(F))/(b*d*log(F))

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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**(a+b*(d*x+c)**2)*(d*x+c)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^m*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

integrate((d*x + c)^m*F^((d*x + c)^2*b + a), x)