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

Optimal. Leaf size=63 \[ \frac {(c+d x)^n F^{a+b (c+d x)^n}}{b d n \log (F)}-\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2213, 2209} \[ \frac {(c+d x)^n F^{a+b (c+d x)^n}}{b d n \log (F)}-\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)} \]

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2213

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^Simplify[m
- n]*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && Lt
Q[0, Simplify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, -n]

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx &=\frac {F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}-\frac {\int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 32, normalized size = 0.51 \[ -\frac {F^a \Gamma \left (2,-b (c+d x)^n \log (F)\right )}{b^2 d n \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 44, normalized size = 0.70 \[ \frac {{\left ({\left (d x + c\right )}^{n} b \log \relax (F) - 1\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \relax (F) + a \log \relax (F)\right )}}{b^{2} d n \log \relax (F)^{2}} \]

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, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{2 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 74, normalized size = 1.17 \[ \frac {{\mathrm e}^{n \ln \left (d x +c \right )} {\mathrm e}^{\left (b \,{\mathrm e}^{n \ln \left (d x +c \right )}+a \right ) \ln \relax (F )}}{b d n \ln \relax (F )}-\frac {{\mathrm e}^{\left (b \,{\mathrm e}^{n \ln \left (d x +c \right )}+a \right ) \ln \relax (F )}}{b^{2} d n \ln \relax (F )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 45, normalized size = 0.71 \[ \frac {{\left ({\left (d x + c\right )}^{n} F^{a} b \log \relax (F) - F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{2} d n \log \relax (F)^{2}} \]

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, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{2\,n-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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