3.36 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(c g+d g x)^4} \, dx\)

Optimal. Leaf size=183 \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}+\frac{b^2 B n}{3 d g^4 (c+d x) (b c-a d)^2}+\frac{b^3 B n \log (a+b x)}{3 d g^4 (b c-a d)^3}-\frac{b^3 B n \log (c+d x)}{3 d g^4 (b c-a d)^3}+\frac{b B n}{6 d g^4 (c+d x)^2 (b c-a d)}+\frac{B n}{9 d g^4 (c+d x)^3} \]

[Out]

(B*n)/(9*d*g^4*(c + d*x)^3) + (b*B*n)/(6*d*(b*c - a*d)*g^4*(c + d*x)^2) + (b^2*B*n)/(3*d*(b*c - a*d)^2*g^4*(c
+ d*x)) + (b^3*B*n*Log[a + b*x])/(3*d*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*d*g^4*(c
+ d*x)^3) - (b^3*B*n*Log[c + d*x])/(3*d*(b*c - a*d)^3*g^4)

________________________________________________________________________________________

Rubi [A]  time = 0.143203, antiderivative size = 183, normalized size of antiderivative = 1., 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} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}+\frac{b^2 B n}{3 d g^4 (c+d x) (b c-a d)^2}+\frac{b^3 B n \log (a+b x)}{3 d g^4 (b c-a d)^3}-\frac{b^3 B n \log (c+d x)}{3 d g^4 (b c-a d)^3}+\frac{b B n}{6 d g^4 (c+d x)^2 (b c-a d)}+\frac{B n}{9 d g^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^4,x]

[Out]

(B*n)/(9*d*g^4*(c + d*x)^3) + (b*B*n)/(6*d*(b*c - a*d)*g^4*(c + d*x)^2) + (b^2*B*n)/(3*d*(b*c - a*d)^2*g^4*(c
+ d*x)) + (b^3*B*n*Log[a + b*x])/(3*d*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*d*g^4*(c
+ d*x)^3) - (b^3*B*n*Log[c + d*x])/(3*d*(b*c - a*d)^3*g^4)

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]

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])

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)^4} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 d g^4 (c+d x)^3}+\frac{(B n) \int \frac{b c-a d}{g^3 (a+b x) (c+d x)^4} \, dx}{3 d g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 d g^4 (c+d x)^3}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x) (c+d x)^4} \, dx}{3 d g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 d g^4 (c+d x)^3}+\frac{(B (b c-a d) n) \int \left (\frac{b^4}{(b c-a d)^4 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^4}-\frac{b d}{(b c-a d)^2 (c+d x)^3}-\frac{b^2 d}{(b c-a d)^3 (c+d x)^2}-\frac{b^3 d}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 d g^4}\\ &=\frac{B n}{9 d g^4 (c+d x)^3}+\frac{b B n}{6 d (b c-a d) g^4 (c+d x)^2}+\frac{b^2 B n}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{b^3 B n \log (a+b x)}{3 d (b c-a d)^3 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 d g^4 (c+d x)^3}-\frac{b^3 B n \log (c+d x)}{3 d (b c-a d)^3 g^4}\\ \end{align*}

Mathematica [A]  time = 0.176667, size = 146, normalized size = 0.8 \[ \frac{\frac{B n \left ((b c-a d) \left (2 a^2 d^2-a b d (7 c+3 d x)+b^2 \left (11 c^2+15 c d x+6 d^2 x^2\right )\right )+6 b^3 (c+d x)^3 \log (a+b x)-6 b^3 (c+d x)^3 \log (c+d x)\right )}{(b c-a d)^3}-6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{18 d g^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^4,x]

[Out]

(-6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + (B*n*((b*c - a*d)*(2*a^2*d^2 - a*b*d*(7*c + 3*d*x) + b^2*(11*c^2
+ 15*c*d*x + 6*d^2*x^2)) + 6*b^3*(c + d*x)^3*Log[a + b*x] - 6*b^3*(c + d*x)^3*Log[c + d*x]))/(b*c - a*d)^3)/(1
8*d*g^4*(c + d*x)^3)

________________________________________________________________________________________

Maple [F]  time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{4}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^4,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^4,x)

________________________________________________________________________________________

Maxima [B]  time = 1.25888, size = 585, normalized size = 3.2 \begin{align*} \frac{1}{18} \, B n{\left (\frac{6 \, b^{2} d^{2} x^{2} + 11 \, b^{2} c^{2} - 7 \, a b c d + 2 \, a^{2} d^{2} + 3 \,{\left (5 \, b^{2} c d - a b d^{2}\right )} x}{{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} g^{4} x^{3} + 3 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} g^{4} x^{2} + 3 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} g^{4} x +{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} g^{4}} + \frac{6 \, b^{3} \log \left (b x + a\right )}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} g^{4}} - \frac{6 \, b^{3} \log \left (d x + c\right )}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} g^{4}}\right )} - \frac{B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{3 \,{\left (d^{4} g^{4} x^{3} + 3 \, c d^{3} g^{4} x^{2} + 3 \, c^{2} d^{2} g^{4} x + c^{3} d g^{4}\right )}} - \frac{A}{3 \,{\left (d^{4} g^{4} x^{3} + 3 \, c d^{3} g^{4} x^{2} + 3 \, c^{2} d^{2} g^{4} x + c^{3} d g^{4}\right )}} \end{align*}

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)^4,x, algorithm="maxima")

[Out]

1/18*B*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2 + 3*(5*b^2*c*d - a*b*d^2)*x)/((b^2*c^2*d^4 - 2*a
*b*c*d^5 + a^2*d^6)*g^4*x^3 + 3*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 + 3*(b^2*c^4*d^2 - 2*a*b*c^3
*d^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*g^4) + 6*b^3*log(b*x + a)/((b^3*c^3*d -
3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c
*d^3 - a^3*d^4)*g^4)) - 1/3*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^
2*g^4*x + c^3*d*g^4) - 1/3*A/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4)

________________________________________________________________________________________

Fricas [B]  time = 0.9341, size = 990, normalized size = 5.41 \begin{align*} -\frac{6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} - 6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \,{\left (5 \, B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} n x -{\left (11 \, B b^{3} c^{3} - 18 \, B a b^{2} c^{2} d + 9 \, B a^{2} b c d^{2} - 2 \, B a^{3} d^{3}\right )} n + 6 \,{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right ) - 6 \,{\left (B b^{3} d^{3} n x^{3} + 3 \, B b^{3} c d^{2} n x^{2} + 3 \, B b^{3} c^{2} d n x +{\left (3 \, B a b^{2} c^{2} d - 3 \, B a^{2} b c d^{2} + B a^{3} d^{3}\right )} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{18 \,{\left ({\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} g^{4} x^{3} + 3 \,{\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} g^{4} x^{2} + 3 \,{\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} g^{4} x +{\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} g^{4}\right )}} \end{align*}

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)^4,x, algorithm="fricas")

[Out]

-1/18*(6*A*b^3*c^3 - 18*A*a*b^2*c^2*d + 18*A*a^2*b*c*d^2 - 6*A*a^3*d^3 - 6*(B*b^3*c*d^2 - B*a*b^2*d^3)*n*x^2 -
3*(5*B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + B*a^2*b*d^3)*n*x - (11*B*b^3*c^3 - 18*B*a*b^2*c^2*d + 9*B*a^2*b*c*d^2 -
2*B*a^3*d^3)*n + 6*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2 - B*a^3*d^3)*log(e) - 6*(B*b^3*d^3*n*x^3 + 3
*B*b^3*c*d^2*n*x^2 + 3*B*b^3*c^2*d*n*x + (3*B*a*b^2*c^2*d - 3*B*a^2*b*c*d^2 + B*a^3*d^3)*n)*log((b*x + a)/(d*x
+ c)))/((b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*g^4*x^3 + 3*(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4
+ 3*a^2*b*c^2*d^5 - a^3*c*d^6)*g^4*x^2 + 3*(b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)*g^4
*x + (b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*g^4)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.45879, size = 655, normalized size = 3.58 \begin{align*} \frac{B b^{3} n \log \left (b x + a\right )}{3 \,{\left (b^{3} c^{3} d g^{4} - 3 \, a b^{2} c^{2} d^{2} g^{4} + 3 \, a^{2} b c d^{3} g^{4} - a^{3} d^{4} g^{4}\right )}} - \frac{B b^{3} n \log \left (d x + c\right )}{3 \,{\left (b^{3} c^{3} d g^{4} - 3 \, a b^{2} c^{2} d^{2} g^{4} + 3 \, a^{2} b c d^{3} g^{4} - a^{3} d^{4} g^{4}\right )}} - \frac{B n \log \left (\frac{b x + a}{d x + c}\right )}{3 \,{\left (d^{4} g^{4} x^{3} + 3 \, c d^{3} g^{4} x^{2} + 3 \, c^{2} d^{2} g^{4} x + c^{3} d g^{4}\right )}} + \frac{6 \, B b^{2} d^{2} n x^{2} + 15 \, B b^{2} c d n x - 3 \, B a b d^{2} n x + 11 \, B b^{2} c^{2} n - 7 \, B a b c d n + 2 \, B a^{2} d^{2} n - 6 \, A b^{2} c^{2} - 6 \, B b^{2} c^{2} + 12 \, A a b c d + 12 \, B a b c d - 6 \, A a^{2} d^{2} - 6 \, B a^{2} d^{2}}{18 \,{\left (b^{2} c^{2} d^{4} g^{4} x^{3} - 2 \, a b c d^{5} g^{4} x^{3} + a^{2} d^{6} g^{4} x^{3} + 3 \, b^{2} c^{3} d^{3} g^{4} x^{2} - 6 \, a b c^{2} d^{4} g^{4} x^{2} + 3 \, a^{2} c d^{5} g^{4} x^{2} + 3 \, b^{2} c^{4} d^{2} g^{4} x - 6 \, a b c^{3} d^{3} g^{4} x + 3 \, a^{2} c^{2} d^{4} g^{4} x + b^{2} c^{5} d g^{4} - 2 \, a b c^{4} d^{2} g^{4} + a^{2} c^{3} d^{3} g^{4}\right )}} \end{align*}

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)^4,x, algorithm="giac")

[Out]

1/3*B*b^3*n*log(b*x + a)/(b^3*c^3*d*g^4 - 3*a*b^2*c^2*d^2*g^4 + 3*a^2*b*c*d^3*g^4 - a^3*d^4*g^4) - 1/3*B*b^3*n
*log(d*x + c)/(b^3*c^3*d*g^4 - 3*a*b^2*c^2*d^2*g^4 + 3*a^2*b*c*d^3*g^4 - a^3*d^4*g^4) - 1/3*B*n*log((b*x + a)/
(d*x + c))/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4) + 1/18*(6*B*b^2*d^2*n*x^2 + 15*B*b^2*
c*d*n*x - 3*B*a*b*d^2*n*x + 11*B*b^2*c^2*n - 7*B*a*b*c*d*n + 2*B*a^2*d^2*n - 6*A*b^2*c^2 - 6*B*b^2*c^2 + 12*A*
a*b*c*d + 12*B*a*b*c*d - 6*A*a^2*d^2 - 6*B*a^2*d^2)/(b^2*c^2*d^4*g^4*x^3 - 2*a*b*c*d^5*g^4*x^3 + a^2*d^6*g^4*x
^3 + 3*b^2*c^3*d^3*g^4*x^2 - 6*a*b*c^2*d^4*g^4*x^2 + 3*a^2*c*d^5*g^4*x^2 + 3*b^2*c^4*d^2*g^4*x - 6*a*b*c^3*d^3
*g^4*x + 3*a^2*c^2*d^4*g^4*x + b^2*c^5*d*g^4 - 2*a*b*c^4*d^2*g^4 + a^2*c^3*d^3*g^4)