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

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

[Out]

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

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Rubi [A]  time = 0.0751403, 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 \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

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

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Rubi in Sympy [A]  time = 6.0649, size = 56, normalized size = 0.92 \[ \frac{F^{a} \left (- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}}\right )^{\frac{m}{2} + \frac{1}{2}} \left (c + d x\right )^{m + 1} \Gamma{\left (- \frac{m}{2} - \frac{1}{2},- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}} \right )}}{2 d} \]

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

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Mathematica [A]  time = 0.0623221, size = 61, normalized size = 1. \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{2}}}} \left ( dx+c \right ) ^{m}\, dx \]

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. \[ \int{\left (d x + c\right )}^{m} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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. \[ \int{\left (d x + c\right )}^{m} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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