3.34 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(c g+d g x)^2} \, dx\)

Optimal. Leaf size=102 \[ \frac {A (a+b x)}{g^2 (c+d x) (b c-a d)}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g^2 (c+d x) (b c-a d)}-\frac {B n (a+b x)}{g^2 (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{d g^2 (c+d x)}+\frac {b B n \log (a+b x)}{d g^2 (b c-a d)}-\frac {b B n \log (c+d x)}{d g^2 (b c-a d)}+\frac {B n}{d g^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 114, normalized size = 1.12 \[ \frac {B n (b c-a d) \left (\frac {1}{(c+d x) (b c-a d)}+\frac {b \log (a+b x)}{(b c-a d)^2}-\frac {b \log (c+d x)}{(b c-a d)^2}\right )}{d g^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{d g (c g+d g x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 105, normalized size = 1.03 \[ -\frac {A b c - A a d - {\left (B b c - B a d\right )} n + {\left (B b c - B a d\right )} \log \relax (e) - {\left (B b d n x + B a d n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} g^{2} x + {\left (b c^{2} d - a c d^{2}\right )} g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 3.77, size = 89, normalized size = 0.87 \[ {\left (\frac {{\left (b x + a\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (d x + c\right )} g^{2}} - \frac {{\left (B n - A - B\right )} {\left (b x + a\right )}}{{\left (d x + c\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (d g x +c g \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.13, size = 136, normalized size = 1.33 \[ B n {\left (\frac {1}{d^{2} g^{2} x + c d g^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} g^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} g^{2}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{2} g^{2} x + c d g^{2}} - \frac {A}{d^{2} g^{2} x + c d g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.02, size = 113, normalized size = 1.11 \[ -\frac {A-B\,n}{x\,d^2\,g^2+c\,d\,g^2}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{d\,\left (c\,g^2+d\,g^2\,x\right )}+\frac {B\,b\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,g^2\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError

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