3.2.39 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x) (c i+d i x)} \, dx\) [139]

Optimal. Leaf size=50 \[ \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 B (b c-a d) g i n} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2561, 2338} \begin {gather*} \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 B g i n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(139 c+139 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac {(B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(B n) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}+\frac {(B n) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(b B n) \int \frac {\log (a+b x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac {(b B n) \int \frac {\log (c+d x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac {(B d n) \int \frac {\log (a+b x)}{c+d x} \, dx}{139 (b c-a d) g}-\frac {(B d n) \int \frac {\log (c+d x)}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac {(B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{139 (b c-a d) g}-\frac {(B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{139 (b c-a d) g}-\frac {(b B n) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac {(B d n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=-\frac {B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac {(B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{139 (b c-a d) g}-\frac {(B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{139 (b c-a d) g}\\ &=-\frac {B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac {B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac {B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.08, size = 219, normalized size = 4.38 \begin {gather*} \frac {2 A \log (a+b x)-B n \log ^2(a+b x)+2 B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 A \log (c+d x)+2 B n \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)-B n \log ^2(c+d x)+2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 B n \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )+2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d) g i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right ) \left (d i x +c i \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (46) = 92\).
time = 0.28, size = 172, normalized size = 3.44 \begin {gather*} -B {\left (\frac {\log \left (b x + a\right )}{{\left (-i \, b c + i \, a d\right )} g} - \frac {\log \left (d x + c\right )}{{\left (-i \, b c + i \, a d\right )} g}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {{\left (i \, \log \left (b x + a\right )^{2} - 2 i \, \log \left (b x + a\right ) \log \left (d x + c\right ) + i \, \log \left (d x + c\right )^{2}\right )} B n}{2 \, {\left (b c g - a d g\right )}} - A {\left (\frac {\log \left (b x + a\right )}{{\left (-i \, b c + i \, a d\right )} g} - \frac {\log \left (d x + c\right )}{{\left (-i \, b c + i \, a d\right )} g}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 59, normalized size = 1.18 \begin {gather*} -\frac {i \, B n \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (-i \, A - i \, B\right )} \log \left (\frac {b x + a}{d x + c}\right )}{2 \, {\left (b c - a d\right )} g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.58, size = 88, normalized size = 1.76 \begin {gather*} -\frac {{\left (i \, B n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 i \, A \log \left (\frac {b x + a}{d x + c}\right ) + 2 i \, B \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.72, size = 76, normalized size = 1.52 \begin {gather*} -\frac {B\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2-A\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{2\,g\,i\,n\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*log(e*((a + b*x)/(c + d*x))^n)^2 - A*n*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*4i)/(2*g*i*n*(a*d - b*c)
)

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