3.376 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1-2 n} \, dx\)

Optimal. Leaf size=100 \[ \frac{b^2 F^a \log ^2(F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )}{2 d n}-\frac{(c+d x)^{-2 n} F^{a+b (c+d x)^n}}{2 d n}-\frac{b \log (F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{2 d n} \]

[Out]

-F^(a + b*(c + d*x)^n)/(2*d*n*(c + d*x)^(2*n)) - (b*F^(a + b*(c + d*x)^n)*Log[F]
)/(2*d*n*(c + d*x)^n) + (b^2*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^2)/(
2*d*n)

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Rubi [A]  time = 0.18833, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{b^2 F^a \log ^2(F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )}{2 d n}-\frac{(c+d x)^{-2 n} F^{a+b (c+d x)^n}}{2 d n}-\frac{b \log (F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{2 d n} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 2*n),x]

[Out]

-F^(a + b*(c + d*x)^n)/(2*d*n*(c + d*x)^(2*n)) - (b*F^(a + b*(c + d*x)^n)*Log[F]
)/(2*d*n*(c + d*x)^n) + (b^2*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^2)/(
2*d*n)

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Rubi in Sympy [A]  time = 16.2958, size = 83, normalized size = 0.83 \[ \frac{F^{a} b^{2} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (b \left (c + d x\right )^{n} \log{\left (F \right )} \right )}}{2 d n} - \frac{F^{a + b \left (c + d x\right )^{n}} b \left (c + d x\right )^{- n} \log{\left (F \right )}}{2 d n} - \frac{F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{- 2 n}}{2 d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1-2*n),x)

[Out]

F**a*b**2*log(F)**2*Ei(b*(c + d*x)**n*log(F))/(2*d*n) - F**(a + b*(c + d*x)**n)*
b*(c + d*x)**(-n)*log(F)/(2*d*n) - F**(a + b*(c + d*x)**n)*(c + d*x)**(-2*n)/(2*
d*n)

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Mathematica [A]  time = 0.0716282, size = 78, normalized size = 0.78 \[ \frac{F^a (c+d x)^{-2 n} \left (b^2 \log ^2(F) (c+d x)^{2 n} \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )-F^{b (c+d x)^n} \left (b \log (F) (c+d x)^n+1\right )\right )}{2 d n} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 2*n),x]

[Out]

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

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Maple [A]  time = 0.1, size = 97, normalized size = 1. \[ -{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{2\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}}}-{\frac{b{F}^{a+b \left ( dx+c \right ) ^{n}}\ln \left ( F \right ) }{2\,dn \left ( dx+c \right ) ^{n}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{2\,dn}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-2*n),x)

[Out]

-1/2/d/n*F^(a+b*(d*x+c)^n)/((d*x+c)^n)^2-1/2*b*F^(a+b*(d*x+c)^n)*ln(F)/d/n/((d*x
+c)^n)-1/2/d/n*b^2*ln(F)^2*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{-2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-2*n - 1)*F^((d*x + c)^n*b + a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(-2*n - 1)*F^((d*x + c)^n*b + a), x)

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Fricas [A]  time = 0.262139, size = 113, normalized size = 1.13 \[ \frac{{\left (d x + c\right )}^{2 \, n} F^{a} b^{2}{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{2} -{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + 1\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \,{\left (d x + c\right )}^{2 \, n} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-2*n - 1)*F^((d*x + c)^n*b + a),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{-2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-2*n - 1)*F^((d*x + c)^n*b + a),x, algorithm="giac")

[Out]

integrate((d*x + c)^(-2*n - 1)*F^((d*x + c)^n*b + a), x)