3.375 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx\)

Optimal. Leaf size=56 \[ \frac{b F^a \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )}{d n}-\frac{(c+d x)^{-n} F^{a+b (c+d x)^n}}{d n} \]

[Out]

-(F^(a + b*(c + d*x)^n)/(d*n*(c + d*x)^n)) + (b*F^a*ExpIntegralEi[b*(c + d*x)^n*
Log[F]]*Log[F])/(d*n)

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Rubi [A]  time = 0.119054, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{b F^a \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )}{d n}-\frac{(c+d x)^{-n} F^{a+b (c+d x)^n}}{d n} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - n),x]

[Out]

-(F^(a + b*(c + d*x)^n)/(d*n*(c + d*x)^n)) + (b*F^a*ExpIntegralEi[b*(c + d*x)^n*
Log[F]]*Log[F])/(d*n)

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Rubi in Sympy [A]  time = 10.2107, size = 46, normalized size = 0.82 \[ \frac{F^{a} b \log{\left (F \right )} \operatorname{Ei}{\left (b \left (c + d x\right )^{n} \log{\left (F \right )} \right )}}{d n} - \frac{F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{- n}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1-n),x)

[Out]

F**a*b*log(F)*Ei(b*(c + d*x)**n*log(F))/(d*n) - F**(a + b*(c + d*x)**n)*(c + d*x
)**(-n)/(d*n)

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Mathematica [A]  time = 0.0639915, size = 50, normalized size = 0.89 \[ -\frac{F^a \left ((c+d x)^{-n} F^{b (c+d x)^n}-b \log (F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )\right )}{d n} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - n),x]

[Out]

-((F^a*(F^(b*(c + d*x)^n)/(c + d*x)^n - b*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Lo
g[F]))/(d*n))

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Maple [A]  time = 0.069, size = 60, normalized size = 1.1 \[ -{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{dn \left ( dx+c \right ) ^{n}}}-{\frac{b\ln \left ( F \right ){F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{dn}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-n),x)

[Out]

-F^(a+b*(d*x+c)^n)/d/n/((d*x+c)^n)-1/d/n*b*ln(F)*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{-n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-n - 1)*F^((d*x + c)^n*b + a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(-n - 1)*F^((d*x + c)^n*b + a), x)

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Fricas [A]  time = 0.253784, size = 84, normalized size = 1.5 \[ \frac{{\left (d x + c\right )}^{n} F^{a} b{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right ) - e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{{\left (d x + c\right )}^{n} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-n - 1)*F^((d*x + c)^n*b + a),x, algorithm="fricas")

[Out]

((d*x + c)^n*F^a*b*Ei((d*x + c)^n*b*log(F))*log(F) - e^((d*x + c)^n*b*log(F) + a
*log(F)))/((d*x + c)^n*d*n)

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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)**n)*(d*x+c)**(-1-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{-n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-n - 1)*F^((d*x + c)^n*b + a),x, algorithm="giac")

[Out]

integrate((d*x + c)^(-n - 1)*F^((d*x + c)^n*b + a), x)