3.3.27 \(\int (a g+b g x)^{-2-m} (c i+d i x)^m (A+B \log (e (a+b x)^n (c+d x)^{-n}))^p \, dx\) [227]

Optimal. Leaf size=194 \[ -\frac {e^{\frac {A (1+m)}{B n}} (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{\frac {1+m}{n}} \Gamma \left (1+p,\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \left (\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )^{-p}}{(b c-a d) i^2 (1+m) (c+d x)} \]

[Out]

-exp(A*(1+m)/B/n)*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(e*(b*x+a)^n/((d*x+c)^n))^((1+m)/n)*GAMMA(1+p,(
1+m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/n)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p/(-a*d+b*c)/i^2/(1+m)/(d*x+c)/(
((1+m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/n)^p)

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Rubi [A]
time = 0.30, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2573, 2563, 2347, 2212} \begin {gather*} -\frac {(a+b x) e^{\frac {A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e (a+b x)^n (c+d x)^{-n}\right )^{\frac {m+1}{n}} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p \left (\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )^{-p} \text {Gamma}\left (p+1,\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p,x]

[Out]

-((E^((A*(1 + m))/(B*n))*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*((e*(a + b*x)^n)/(c + d*x)^n)^
((1 + m)/n)*Gamma[1 + p, ((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n)]*(A + B*Log[(e*(a + b*x)^n)/
(c + d*x)^n])^p)/((b*c - a*d)*i^2*(1 + m)*(c + d*x)*(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n))
^p))

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2347

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

Rule 2563

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[d^2*((g*((a + b*x)/b))^m/(i^2*(b*c - a*d)*(i*((c + d*x)/d))^
m*((a + b*x)/(c + d*x))^m)), Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a,
b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] &
& EqQ[m + q + 2, 0]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

Rubi steps

\begin {align*} \int (227 c+227 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx &=\int (227 c+227 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx\\ \end {align*}

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Mathematica [F]
time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p,x]

[Out]

Integrate[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p, x]

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Maple [F]
time = 1.44, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{-2-m} \left (d i x +c i \right )^{m} \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,x)

[Out]

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,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((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="maxima")

[Out]

integrate((b*g*x + a*g)^(-m - 2)*(I*d*x + I*c)^m*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p, 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((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="fricas")

[Out]

integral((I*d*x + I*c)^m*(cosh(-p*log(1/2*I*pi*B*n + B*n*log(b*x + a) - B*n*log(I*d*x + I*c) + A + B)) - sinh(
-p*log(1/2*I*pi*B*n + B*n*log(b*x + a) - B*n*log(I*d*x + I*c) + A + B)))*e^(-(m + 2)*log(b*x + a) - (m + 2)*lo
g(g)), x)

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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((b*g*x+a*g)**(-2-m)*(d*i*x+c*i)**m*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**p,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Simplification as
suming sageVARc near 0Simplification assuming sageVARc near 0Simplification assuming t_nostep near 0Simplifica
tion assuming

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(a*g + b*g*x)^(m + 2),x)

[Out]

int(((c*i + d*i*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(a*g + b*g*x)^(m + 2), x)

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