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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]