3.1.17 \(\int (a+b (F^{e (c+d x)})^n)^p (G^{h (f+g x)})^{\frac {d e n \log (F)}{g h \log (G)}} \, dx\) [17]

Optimal. Leaf size=80 \[ \frac {\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}}{b d e n (1+p) \log (F)} \]

[Out]

(a+b*(F^(e*(d*x+c)))^n)^(1+p)*(G^(h*(g*x+f)))^(d*e*n*ln(F)/g/h/ln(G))/b/d/e/((F^(e*(d*x+c)))^n)/n/(1+p)/ln(F)

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Rubi [A]
time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {2279, 2278, 32} \begin {gather*} \frac {\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*(F^(e*(c + d*x)))^n)^p*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])),x]

[Out]

((a + b*(F^(e*(c + d*x)))^n)^(1 + p)*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])))/(b*d*e*(F^(e*(c + d*x)))
^n*n*(1 + p)*Log[F])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2278

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 2279

Int[((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.)*((G_)^((h_.)*((f_.) + (g_.)*(x_))))^(m_.),
x_Symbol] :> Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)
^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]

Rubi steps

\begin {align*} \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}} \, dx &=\left (\left (F^{e (c+d x)}\right )^{-n} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}\right ) \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx\\ &=\frac {\left (\left (F^{e (c+d x)}\right )^{-n} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}\right ) \text {Subst}\left (\int (a+b x)^p \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=\frac {\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}}{b d e n (1+p) \log (F)}\\ \end {align*}

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Mathematica [F]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*(F^(e*(c + d*x)))^n)^p*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])),x]

[Out]

Integrate[(a + b*(F^(e*(c + d*x)))^n)^p*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])), x]

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )^{p} \left (G^{h \left (g x +f \right )}\right )^{\frac {d e n \ln \left (F \right )}{g h \ln \left (G \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(F^(e*(d*x+c)))^n)^p*(G^(h*(g*x+f)))^(d*e*n*ln(F)/g/h/ln(G)),x)

[Out]

int((a+b*(F^(e*(d*x+c)))^n)^p*(G^(h*(g*x+f)))^(d*e*n*ln(F)/g/h/ln(G)),x)

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Maxima [A]
time = 0.31, size = 86, normalized size = 1.08 \begin {gather*} \frac {{\left (F^{d e n x} F^{c e n + \frac {d e f n}{g}} b + F^{\frac {d e f n}{g}} a\right )} {\left (F^{d e n x} F^{c e n} b + a\right )}^{p}}{F^{c e n} b d e n {\left (p + 1\right )} \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(e*(d*x+c)))^n)^p*(G^(h*(g*x+f)))^(d*e*n*log(F)/g/h/log(G)),x, algorithm="maxima")

[Out]

(F^(d*e*n*x)*F^(c*e*n + d*e*f*n/g)*b + F^(d*e*f*n/g)*a)*(F^(d*e*n*x)*F^(c*e*n)*b + a)^p/(F^(c*e*n)*b*d*e*n*(p
+ 1)*log(F))

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Fricas [A]
time = 0.41, size = 90, normalized size = 1.12 \begin {gather*} \frac {{\left (F^{{\left (d n x + c n\right )} e} F^{\frac {{\left (d f - c g\right )} n e}{g}} b + F^{\frac {{\left (d f - c g\right )} n e}{g}} a\right )} {\left (F^{{\left (d n x + c n\right )} e} b + a\right )}^{p} e^{\left (-1\right )}}{{\left (b d n p + b d n\right )} \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(e*(d*x+c)))^n)^p*(G^(h*(g*x+f)))^(d*e*n*log(F)/g/h/log(G)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F**(e*(d*x+c)))**n)**p*(G**(h*(g*x+f)))**(d*e*n*ln(F)/g/h/ln(G)),x)

[Out]

Timed out

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Giac [A]
time = 2.95, size = 156, normalized size = 1.95 \begin {gather*} \frac {F^{\frac {d f n e}{g}} b e^{\left (2 \, d n x e \log \left (F\right ) + c n e \log \left (F\right ) + p \log \left (b e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right )\right )} + a\right )\right )} + F^{\frac {d f n e}{g}} a e^{\left (d n x e \log \left (F\right ) + p \log \left (b e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right )\right )} + a\right )\right )}}{b d n p e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right ) + 1\right )} \log \left (F\right ) + b d n e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right ) + 1\right )} \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(F^(e*(d*x+c)))^n)^p*(G^(h*(g*x+f)))^(d*e*n*log(F)/g/h/log(G)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (G^{h\,\left (f+g\,x\right )}\right )}^{\frac {d\,e\,n\,\ln \left (F\right )}{g\,h\,\ln \left (G\right )}}\,{\left (a+b\,{\left (F^{e\,\left (c+d\,x\right )}\right )}^n\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((G^(h*(f + g*x)))^((d*e*n*log(F))/(g*h*log(G)))*(a + b*(F^(e*(c + d*x)))^n)^p,x)

[Out]

int((G^(h*(f + g*x)))^((d*e*n*log(F))/(g*h*log(G)))*(a + b*(F^(e*(c + d*x)))^n)^p, x)

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