3.3.16 \(\int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{(A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\) [216]

Optimal. Leaf size=206 \[ \frac {e^{-\frac {A (1+m)}{B n}} (1+m) (a+b x) (g (a+b x))^m \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {1+m}{n}} (i (c+d x))^{-m} \text {Ei}\left (\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d) i^2 n^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B (b c-a d) i^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {2563, 2343, 2347, 2209} \begin {gather*} \frac {(m+1) (a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 i^2 n^2 (c+d x) (b c-a d)}-\frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

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

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]

Rubi steps

\begin {align*} \int \frac {(216 c+216 d x)^{-2-m} (a g+b g x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac {(216 c+216 d x)^{-2-m} (a g+b g x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]
time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m}}{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

g^m*(m + 1)*integrate(-(b*x + a)^m/(((-1)^(1/2*m)*B^2*d^2*n*x^2 + 2*(-1)^(1/2*m)*B^2*c*d*n*x + (-1)^(1/2*m)*B^
2*c^2*n)*(d*x + c)^m*log((b*x + a)^n) - ((-1)^(1/2*m)*B^2*d^2*n*x^2 + 2*(-1)^(1/2*m)*B^2*c*d*n*x + (-1)^(1/2*m
)*B^2*c^2*n)*(d*x + c)^m*log((d*x + c)^n) + ((-1)^(1/2*m)*A*B*c^2*n + (-1)^(1/2*m)*B^2*c^2*n + ((-1)^(1/2*m)*A
*B*d^2*n + (-1)^(1/2*m)*B^2*d^2*n)*x^2 + 2*((-1)^(1/2*m)*A*B*c*d*n + (-1)^(1/2*m)*B^2*c*d*n)*x)*(d*x + c)^m),
x) + (b*g^m*x + a*g^m)*(b*x + a)^m/((((-1)^(1/2*m)*b*c*d*n - (-1)^(1/2*m)*a*d^2*n)*B^2*x + ((-1)^(1/2*m)*b*c^2
*n - (-1)^(1/2*m)*a*c*d*n)*B^2)*(d*x + c)^m*log((b*x + a)^n) - (((-1)^(1/2*m)*b*c*d*n - (-1)^(1/2*m)*a*d^2*n)*
B^2*x + ((-1)^(1/2*m)*b*c^2*n - (-1)^(1/2*m)*a*c*d*n)*B^2)*(d*x + c)^m*log((d*x + c)^n) + (((-1)^(1/2*m)*b*c^2
*n - (-1)^(1/2*m)*a*c*d*n)*A*B + ((-1)^(1/2*m)*b*c^2*n - (-1)^(1/2*m)*a*c*d*n)*B^2 + (((-1)^(1/2*m)*b*c*d*n -
(-1)^(1/2*m)*a*d^2*n)*A*B + ((-1)^(1/2*m)*b*c*d*n - (-1)^(1/2*m)*a*d^2*n)*B^2)*x)*(d*x + c)^m)

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 240, normalized size = 1.17 \begin {gather*} -\frac {{\left (B b d n x^{2} + B a c n + {\left (B b c + B a d\right )} n x\right )} {\left (i \, d x + i \, c\right )}^{-m - 2} e^{\left (m \log \left (i \, d x + i \, c\right ) + m \log \left (-i \, g\right ) + m \log \left (\frac {b x + a}{d x + c}\right )\right )} + {\left ({\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A + B\right )} m + A + B\right )} {\rm Ei}\left (\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A + B\right )} m + A + B}{B n}\right ) e^{\left (\frac {B m n \log \left (-i \, g\right ) - {\left (A + B\right )} m - A - B}{B n}\right )}}{{\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (\frac {b x + a}{d x + c}\right ) + {\left ({\left (A B^{2} + B^{3}\right )} b c - {\left (A B^{2} + B^{3}\right )} a d\right )} n^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((B*b*d*n*x^2 + B*a*c*n + (B*b*c + B*a*d)*n*x)*(I*d*x + I*c)^(-m - 2)*e^(m*log(I*d*x + I*c) + m*log(-I*g) + m
*log((b*x + a)/(d*x + c))) + ((B*m + B)*n*log((b*x + a)/(d*x + c)) + (A + B)*m + A + B)*Ei(((B*m + B)*n*log((b
*x + a)/(d*x + c)) + (A + B)*m + A + B)/(B*n))*e^((B*m*n*log(-I*g) - (A + B)*m - A - B)/(B*n)))/((B^3*b*c - B^
3*a*d)*n^3*log((b*x + a)/(d*x + c)) + ((A*B^2 + B^3)*b*c - (A*B^2 + B^3)*a*d)*n^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6439 deep

________________________________________________________________________________________

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

[Out]

integrate((b*g*x + a*g)^m*(I*d*x + I*c)^(-m - 2)/(B*log(((b*x + a)/(d*x + c))^n*e) + A)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________