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

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

[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 = 0.13, 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 = {2247, 2246, 32} \[ \frac {\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]

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 2246

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 2247

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 ) \operatorname {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 = 0.32, size = 0, normalized size = 0.00 \[ \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}} \, dx \]

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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fricas [A]  time = 0.45, size = 88, normalized size = 1.10 \[ \frac {{\left (F^{d e n x + c e n} F^{\frac {{\left (d e f - c e g\right )} n}{g}} b + F^{\frac {{\left (d e f - c e g\right )} n}{g}} a\right )} {\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \relax (F)} \]

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

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giac [A]  time = 1.93, size = 156, normalized size = 1.95 \[ \frac {F^{\frac {d f n e}{g}} b e^{\left (2 \, d n x e \log \relax (F) + c n e \log \relax (F) + p \log \left (b e^{\left (d n x e \log \relax (F) + c n e \log \relax (F)\right )} + a\right )\right )} + F^{\frac {d f n e}{g}} a e^{\left (d n x e \log \relax (F) + p \log \left (b e^{\left (d n x e \log \relax (F) + c n e \log \relax (F)\right )} + a\right )\right )}}{b d n p e^{\left (d n x e \log \relax (F) + c n e \log \relax (F) + 1\right )} \log \relax (F) + b d n e^{\left (d n x e \log \relax (F) + c n e \log \relax (F) + 1\right )} \log \relax (F)} \]

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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maple [F]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (G^{\left (g x +f \right ) h}\right )^{\frac {d e n \ln \relax (F )}{g h \ln \relax (G )}} \left (b \left (F^{\left (d x +c \right ) e}\right )^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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

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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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((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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