3.3.23 \(\int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{(A+B \log (e (\frac {a+b x}{c+d x})^n))^3} \, dx\) [223]

Optimal. Leaf size=306 \[ \frac {e^{\frac {A (1+m)}{B n}} (1+m)^2 (a+b x) (g (a+b x))^{-2-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1+m}{n}} (i (c+d x))^{2+m} \text {Ei}\left (-\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 (b c-a d) i^2 n^3 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{2 B (b c-a d) i^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}+\frac {(1+m) (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{2 B^2 (b c-a d) i^2 n^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]

[Out]

1/2*exp(A*(1+m)/B/n)*(1+m)^2*(b*x+a)*(g*(b*x+a))^(-2-m)*(e*((b*x+a)/(d*x+c))^n)^((1+m)/n)*(i*(d*x+c))^(2+m)*Ei
(-(1+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/B/n)/B^3/(-a*d+b*c)/i^2/n^3/(d*x+c)-1/2*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*
(d*x+c))^(2+m)/B/(-a*d+b*c)/i^2/n/(d*x+c)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2+1/2*(1+m)*(b*x+a)*(g*(b*x+a))^(-2-
m)*(i*(d*x+c))^(2+m)/B^2/(-a*d+b*c)/i^2/n^2/(d*x+c)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))

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Rubi [A]
time = 0.26, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 5, 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)^2 (a+b x) e^{\frac {A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \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 )}{2 B^3 i^2 n^3 (c+d x) (b c-a d)}+\frac {(m+1) (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B^2 i^2 n^2 (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 )}-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 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 )^2} \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)/(c + d*x))^n])^3,x]

[Out]

(E^((A*(1 + m))/(B*n))*(1 + m)^2*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(
c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(2*B^3*(b*c - a*d)
*i^2*n^3*(c + d*x)) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(2*B*(b*c - a*d)*i^2*n*(c + d*x
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2) + ((1 + m)*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))
/(2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x)*(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 {(223 c+223 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx &=\int \frac {(223 c+223 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx\\ \end {align*}

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Mathematica [F]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, 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)/(c + d*x))^n])^3,x]

[Out]

Integrate[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3, x]

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{-2-m} \left (d i x +c i \right )^{m}}{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{3}}\, dx\]

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)/(d*x+c))^n))^3,x)

[Out]

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3,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)/(d*x+c))^n))^3,x, algorithm="maxima")

[Out]

-((-1)^(1/2*m)*m^2 + 2*(-1)^(1/2*m)*m + (-1)^(1/2*m))*integrate(-1/2*(d*x + c)^m/((B^3*b^2*g^(m + 2)*n^2*x^2 +
2*B^3*a*b*g^(m + 2)*n^2*x + B^3*a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((b*x + a)^n) - (B^3*b^2*g^(m + 2)*n^2*x^2
+ 2*B^3*a*b*g^(m + 2)*n^2*x + B^3*a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((d*x + c)^n) + (A*B^2*a^2*g^(m + 2)*n^2 +
B^3*a^2*g^(m + 2)*n^2 + (A*B^2*b^2*g^(m + 2)*n^2 + B^3*b^2*g^(m + 2)*n^2)*x^2 + 2*(A*B^2*a*b*g^(m + 2)*n^2 +
B^3*a*b*g^(m + 2)*n^2)*x)*(b*x + a)^m), x) + 1/2*((((-1)^(1/2*m)*m + (-1)^(1/2*m))*B*d*x + ((-1)^(1/2*m)*m + (
-1)^(1/2*m))*B*c)*(d*x + c)^m*log((b*x + a)^n) - (((-1)^(1/2*m)*m + (-1)^(1/2*m))*B*d*x + ((-1)^(1/2*m)*m + (-
1)^(1/2*m))*B*c)*(d*x + c)^m*log((d*x + c)^n) + (((-1)^(1/2*m)*m + (-1)^(1/2*m))*A*c + ((-1)^(1/2*m)*m - (-1)^
(1/2*m)*(n - 1))*B*c + (((-1)^(1/2*m)*m + (-1)^(1/2*m))*A*d + ((-1)^(1/2*m)*m - (-1)^(1/2*m)*(n - 1))*B*d)*x)*
(d*x + c)^m)/(((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)
*B^4)*(b*x + a)^m*log((b*x + a)^n)^2 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n
^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((d*x + c)^n)^2 + 2*((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)
*A*B^3 + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3
+ (b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4)*x)*(b*x + a)^m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a
^2*d*g^(m + 2)*n^2)*A^2*B^2 + 2*(a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b*c*g^(m + 2)*n^2 - a^2
*d*g^(m + 2)*n^2)*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A^2*B^2 + 2*(b^2*c*g^(m + 2)*n^2 - a*b*d*
g^(m + 2)*n^2)*A*B^3 + (b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4)*x)*(b*x + a)^m - 2*(((b^2*c*g^(m + 2)*
n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^
n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4 + ((
b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3 + (b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4)*x)*(b*x +
a)^m)*log((d*x + c)^n))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (297) = 594\).
time = 0.45, size = 612, normalized size = 2.00 \begin {gather*} -\frac {{\left (B^{2} a c n^{2} + {\left (B^{2} b d n^{2} - {\left ({\left (A B + B^{2}\right )} b d m + {\left (A B + B^{2}\right )} b d\right )} n\right )} x^{2} - {\left ({\left (A B + B^{2}\right )} a c m + {\left (A B + B^{2}\right )} a c\right )} n + {\left ({\left (B^{2} b c + B^{2} a d\right )} n^{2} - {\left ({\left (A B + B^{2}\right )} b c + {\left (A B + B^{2}\right )} a d + {\left ({\left (A B + B^{2}\right )} b c + {\left (A B + B^{2}\right )} a d\right )} m\right )} n\right )} x - {\left ({\left (B^{2} b d m + B^{2} b d\right )} n^{2} x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m\right )} n^{2} x + {\left (B^{2} a c m + B^{2} a c\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (i \, d x + i \, c\right )}^{m} e^{\left (-{\left (m + 2\right )} \log \left (i \, d x + i \, c\right ) - {\left (m + 2\right )} \log \left (-i \, g\right ) - {\left (m + 2\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} + {\left ({\left (B^{2} m^{2} + 2 \, B^{2} m + B^{2}\right )} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (A^{2} + 2 \, A B + B^{2}\right )} m^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} m^{2} + A B + B^{2} + 2 \, {\left (A B + B^{2}\right )} m\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A^{2} + 2 \, A B + B^{2} + 2 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} m\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 {{\left (B m + 2 \, B\right )} n \log \left (-i \, g\right ) - {\left (A + B\right )} m - A - B}{B n}\right )}}{2 \, {\left ({\left (B^{5} b c - B^{5} a d\right )} n^{5} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left ({\left (A B^{4} + B^{5}\right )} b c - {\left (A B^{4} + B^{5}\right )} a d\right )} n^{4} \log \left (\frac {b x + a}{d x + c}\right ) + {\left ({\left (A^{2} B^{3} + 2 \, A B^{4} + B^{5}\right )} b c - {\left (A^{2} B^{3} + 2 \, A B^{4} + B^{5}\right )} a d\right )} n^{3}\right )}} \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)/(d*x+c))^n))^3,x, algorithm="fricas")

[Out]

-1/2*((B^2*a*c*n^2 + (B^2*b*d*n^2 - ((A*B + B^2)*b*d*m + (A*B + B^2)*b*d)*n)*x^2 - ((A*B + B^2)*a*c*m + (A*B +
B^2)*a*c)*n + ((B^2*b*c + B^2*a*d)*n^2 - ((A*B + B^2)*b*c + (A*B + B^2)*a*d + ((A*B + B^2)*b*c + (A*B + B^2)*
a*d)*m)*n)*x - ((B^2*b*d*m + B^2*b*d)*n^2*x^2 + (B^2*b*c + B^2*a*d + (B^2*b*c + B^2*a*d)*m)*n^2*x + (B^2*a*c*m
+ B^2*a*c)*n^2)*log((b*x + a)/(d*x + c)))*(I*d*x + I*c)^m*e^(-(m + 2)*log(I*d*x + I*c) - (m + 2)*log(-I*g) -
(m + 2)*log((b*x + a)/(d*x + c))) + ((B^2*m^2 + 2*B^2*m + B^2)*n^2*log((b*x + a)/(d*x + c))^2 + (A^2 + 2*A*B +
B^2)*m^2 + 2*((A*B + B^2)*m^2 + A*B + B^2 + 2*(A*B + B^2)*m)*n*log((b*x + a)/(d*x + c)) + A^2 + 2*A*B + B^2 +
2*(A^2 + 2*A*B + B^2)*m)*Ei(-((B*m + B)*n*log((b*x + a)/(d*x + c)) + (A + B)*m + A + B)/(B*n))*e^(-((B*m + 2*
B)*n*log(-I*g) - (A + B)*m - A - B)/(B*n)))/((B^5*b*c - B^5*a*d)*n^5*log((b*x + a)/(d*x + c))^2 + 2*((A*B^4 +
B^5)*b*c - (A*B^4 + B^5)*a*d)*n^4*log((b*x + a)/(d*x + c)) + ((A^2*B^3 + 2*A*B^4 + B^5)*b*c - (A^2*B^3 + 2*A*B
^4 + B^5)*a*d)*n^3)

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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)/(d*x+c))**n))**3,x)

[Out]

Timed out

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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)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^m}{{\left (a\,g+b\,g\,x\right )}^{m+2}\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*i + d*i*x)^m/((a*g + b*g*x)^(m + 2)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^3),x)

[Out]

int((c*i + d*i*x)^m/((a*g + b*g*x)^(m + 2)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^3), x)

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