3.4.63 \(\int F^{a+b (c+d x)^n} \, dx\) [363]

Optimal. Leaf size=50 \[ -\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2239} \begin {gather*} -\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b \log (F) (c+d x)^n\right )}{d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n),x]

[Out]

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

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^n} \, dx &=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} -\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n),x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +b \left (d x +c \right )^{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^n*b + a), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="fricas")

[Out]

integral(F^((d*x + c)^n*b + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} F^{a + \frac {b}{c}} x & \text {for}\: d = 0 \wedge n = -1 \\F^{a + b c^{n}} x & \text {for}\: d = 0 \\\int F^{a + \frac {b}{c + d x}}\, dx & \text {for}\: n = -1 \\\frac {F^{a} F^{b \left (c + d x\right )^{n}} b c n \left (c + d x\right )^{n} \log {\left (F \right )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} b d n x \left (c + d x\right )^{n} \log {\left (F \right )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} c n}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} c}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d n x}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d x}{d n + d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**(a + b/c)*x, Eq(d, 0) & Eq(n, -1)), (F**(a + b*c**n)*x, Eq(d, 0)), (Integral(F**(a + b/(c + d*x)
), x), Eq(n, -1)), (F**a*F**(b*(c + d*x)**n)*b*c*n*(c + d*x)**n*log(F)/(d*n + d) - F**a*F**(b*(c + d*x)**n)*b*
d*n*x*(c + d*x)**n*log(F)/(d*n + d) - F**a*F**(b*(c + d*x)**n)*c*n/(d*n + d) + F**a*F**(b*(c + d*x)**n)*c/(d*n
+ d) + F**a*F**(b*(c + d*x)**n)*d*n*x/(d*n + d) + F**a*F**(b*(c + d*x)**n)*d*x/(d*n + d), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^n*b + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int F^{a+b\,{\left (c+d\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^n),x)

[Out]

int(F^(a + b*(c + d*x)^n), x)

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