3.96 \(\int \frac{A+B \log (\frac{e (a+b x)}{c+d x})}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=206 \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{16 b g^5 (a+b x)^4} \]

[Out]

-B/(16*b*g^5*(a + b*x)^4) + (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x
)^2) + (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(
e*(a + b*x))/(c + d*x)])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5)

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Rubi [A]  time = 0.156583, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{16 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^5,x]

[Out]

-B/(16*b*g^5*(a + b*x)^4) + (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x
)^2) + (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(
e*(a + b*x))/(c + d*x)])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5)

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 (\frac{e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{b c-a d}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b g^5}\\ &=-\frac{B}{16 b g^5 (a+b x)^4}+\frac{B d}{12 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}-\frac{B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}\\ \end{align*}

Mathematica [A]  time = 0.228634, size = 158, normalized size = 0.77 \[ \frac{\frac{B \left (\frac{12 d^3 (b c-a d)}{a+b x}-\frac{6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac{4 d (b c-a d)^3}{(a+b x)^3}-\frac{3 (b c-a d)^4}{(a+b x)^4}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{(a+b x)^4}}{4 b g^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^5,x]

[Out]

(-((A + B*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^4) + (B*((-3*(b*c - a*d)^4)/(a + b*x)^4 + (4*d*(b*c - a*d)^3
)/(a + b*x)^3 - (6*d^2*(b*c - a*d)^2)/(a + b*x)^2 + (12*d^3*(b*c - a*d))/(a + b*x) + 12*d^4*Log[a + b*x] - 12*
d^4*Log[c + d*x]))/(12*(b*c - a*d)^4))/(4*b*g^5)

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Maple [B]  time = 0.051, size = 1607, normalized size = 7.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x)

[Out]

e*d^4/(a*d-b*c)^5/g^5*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a-e*d^3/(a*d-b*c)^5/g^5*A/(b*e/d+e/(d*x+c)*a-e/d/(
d*x+c)*b*c)*b*c-3/2*e^2*d^3/(a*d-b*c)^5/g^5*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+3/2*e^2*d^2/(a*d-b*c)^
5/g^5*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+e^3*d^2/(a*d-b*c)^5/g^5*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^3*a-e^3*d/(a*d-b*c)^5/g^5*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-1/4*e^4*d/(a*d-b*c)^5/g^5*A*b^
3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/4*e^4/(a*d-b*c)^5/g^5*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*
c+e*d^4/(a*d-b*c)^5/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-e*d^3/(a*d-b*c
)^5/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+e*d^4/(a*d-b*c)^5/g^5*B/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a-e*d^3/(a*d-b*c)^5/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*b*c-3/2*e^2*d^3/
(a*d-b*c)^5/g^5*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3/2*e^2*d^2/(a*d-b
*c)^5/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-3/4*e^2*d^3/(a*d-b*c)^
5/g^5*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+3/4*e^2*d^2/(a*d-b*c)^5/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^2*c+e^3*d^2/(a*d-b*c)^5/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))*a-e^3*d/(a*d-b*c)^5/g^5*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+1/3*
e^3*d^2/(a*d-b*c)^5/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/3*e^3*d/(a*d-b*c)^5/g^5*B*b^3/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-1/4*e^4*d/(a*d-b*c)^5/g^5*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*a+1/4*e^4/(a*d-b*c)^5/g^5*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*c-1/16*e^4*d/(a*d-b*c)^5/g^5*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/16*e^4/(a*d-b*c)^
5/g^5*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c

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Maxima [B]  time = 1.39642, size = 873, normalized size = 4.24 \begin{align*} \frac{1}{48} \, B{\left (\frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x +{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac{12 \, \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac{12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac{12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac{A}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\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)))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/48*B*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^
3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5
*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a
^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a
^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*e*x/(d*x + c
) + a*e/(d*x + c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*
log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*
x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*A/(b^5*g^5*x^4
+ 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 1.07202, size = 1284, normalized size = 6.23 \begin{align*} -\frac{3 \,{\left (4 \, A + B\right )} b^{4} c^{4} - 16 \,{\left (3 \, A + B\right )} a b^{3} c^{3} d + 36 \,{\left (2 \, A + B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \,{\left (A + B\right )} a^{3} b c d^{3} +{\left (12 \, A + 25 \, B\right )} a^{4} d^{4} - 12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \,{\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{48 \,{\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x +{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\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)))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/48*(3*(4*A + B)*b^4*c^4 - 16*(3*A + B)*a*b^3*c^3*d + 36*(2*A + B)*a^2*b^2*c^2*d^2 - 48*(A + B)*a^3*b*c*d^3
+ (12*A + 25*B)*a^4*d^4 - 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 + 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^
2*d^4)*x^2 - 4*(B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*x - 12*(B*b^4*d^4*x^4 +
4*B*a*b^3*d^4*x^3 + 6*B*a^2*b^2*d^4*x^2 + 4*B*a^3*b*d^4*x - B*b^4*c^4 + 4*B*a*b^3*c^3*d - 6*B*a^2*b^2*c^2*d^2
+ 4*B*a^3*b*c*d^3)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*
d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^
4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4
*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*
a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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Sympy [B]  time = 11.107, size = 944, normalized size = 4.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(a + b*x)/(c + d*x))/(4*a**4*b*g**5 + 16*a**3*b**2*g**5*x + 24*a**2*b**3*g**5*x**2 + 16*a*b**4*g**5*x
**3 + 4*b**5*g**5*x**4) - B*d**4*log(x + (-B*a**5*d**9/(a*d - b*c)**4 + 5*B*a**4*b*c*d**8/(a*d - b*c)**4 - 10*
B*a**3*b**2*c**2*d**7/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 - 5*B*a*b**4*c**4*d**5/(a*d - b
*c)**4 + B*a*d**5 + B*b**5*c**5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) + B*
d**4*log(x + (B*a**5*d**9/(a*d - b*c)**4 - 5*B*a**4*b*c*d**8/(a*d - b*c)**4 + 10*B*a**3*b**2*c**2*d**7/(a*d -
b*c)**4 - 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 + 5*B*a*b**4*c**4*d**5/(a*d - b*c)**4 + B*a*d**5 - B*b**5*c*
*5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) - (12*A*a**3*d**3 - 36*A*a**2*b*c
*d**2 + 36*A*a*b**2*c**2*d - 12*A*b**3*c**3 + 25*B*a**3*d**3 - 23*B*a**2*b*c*d**2 + 13*B*a*b**2*c**2*d - 3*B*b
**3*c**3 + 12*B*b**3*d**3*x**3 + x**2*(42*B*a*b**2*d**3 - 6*B*b**3*c*d**2) + x*(52*B*a**2*b*d**3 - 20*B*a*b**2
*c*d**2 + 4*B*b**3*c**2*d))/(48*a**7*b*d**3*g**5 - 144*a**6*b**2*c*d**2*g**5 + 144*a**5*b**3*c**2*d*g**5 - 48*
a**4*b**4*c**3*g**5 + x**4*(48*a**3*b**5*d**3*g**5 - 144*a**2*b**6*c*d**2*g**5 + 144*a*b**7*c**2*d*g**5 - 48*b
**8*c**3*g**5) + x**3*(192*a**4*b**4*d**3*g**5 - 576*a**3*b**5*c*d**2*g**5 + 576*a**2*b**6*c**2*d*g**5 - 192*a
*b**7*c**3*g**5) + x**2*(288*a**5*b**3*d**3*g**5 - 864*a**4*b**4*c*d**2*g**5 + 864*a**3*b**5*c**2*d*g**5 - 288
*a**2*b**6*c**3*g**5) + x*(192*a**6*b**2*d**3*g**5 - 576*a**5*b**3*c*d**2*g**5 + 576*a**4*b**4*c**2*d*g**5 - 1
92*a**3*b**5*c**3*g**5))

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Giac [B]  time = 1.37798, size = 967, normalized size = 4.69 \begin{align*} \frac{B d^{4} \log \left (b x + a\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac{B d^{4} \log \left (d x + c\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac{B \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} + \frac{12 \, B b^{3} d^{3} x^{3} - 6 \, B b^{3} c d^{2} x^{2} + 42 \, B a b^{2} d^{3} x^{2} + 4 \, B b^{3} c^{2} d x - 20 \, B a b^{2} c d^{2} x + 52 \, B a^{2} b d^{3} x - 12 \, A b^{3} c^{3} - 15 \, B b^{3} c^{3} + 36 \, A a b^{2} c^{2} d + 49 \, B a b^{2} c^{2} d - 36 \, A a^{2} b c d^{2} - 59 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 37 \, B a^{3} d^{3}}{48 \,{\left (b^{8} c^{3} g^{5} x^{4} - 3 \, a b^{7} c^{2} d g^{5} x^{4} + 3 \, a^{2} b^{6} c d^{2} g^{5} x^{4} - a^{3} b^{5} d^{3} g^{5} x^{4} + 4 \, a b^{7} c^{3} g^{5} x^{3} - 12 \, a^{2} b^{6} c^{2} d g^{5} x^{3} + 12 \, a^{3} b^{5} c d^{2} g^{5} x^{3} - 4 \, a^{4} b^{4} d^{3} g^{5} x^{3} + 6 \, a^{2} b^{6} c^{3} g^{5} x^{2} - 18 \, a^{3} b^{5} c^{2} d g^{5} x^{2} + 18 \, a^{4} b^{4} c d^{2} g^{5} x^{2} - 6 \, a^{5} b^{3} d^{3} g^{5} x^{2} + 4 \, a^{3} b^{5} c^{3} g^{5} x - 12 \, a^{4} b^{4} c^{2} d g^{5} x + 12 \, a^{5} b^{3} c d^{2} g^{5} x - 4 \, a^{6} b^{2} d^{3} g^{5} x + a^{4} b^{4} c^{3} g^{5} - 3 \, a^{5} b^{3} c^{2} d g^{5} + 3 \, a^{6} b^{2} c d^{2} g^{5} - a^{7} b d^{3} g^{5}\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)))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

1/4*B*d^4*log(b*x + a)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*
d^4*g^5) - 1/4*B*d^4*log(d*x + c)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g
^5 + a^4*b*d^4*g^5) - 1/4*B*log((b*x + a)/(d*x + c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^
3*b^2*g^5*x + a^4*b*g^5) + 1/48*(12*B*b^3*d^3*x^3 - 6*B*b^3*c*d^2*x^2 + 42*B*a*b^2*d^3*x^2 + 4*B*b^3*c^2*d*x -
20*B*a*b^2*c*d^2*x + 52*B*a^2*b*d^3*x - 12*A*b^3*c^3 - 15*B*b^3*c^3 + 36*A*a*b^2*c^2*d + 49*B*a*b^2*c^2*d - 3
6*A*a^2*b*c*d^2 - 59*B*a^2*b*c*d^2 + 12*A*a^3*d^3 + 37*B*a^3*d^3)/(b^8*c^3*g^5*x^4 - 3*a*b^7*c^2*d*g^5*x^4 + 3
*a^2*b^6*c*d^2*g^5*x^4 - a^3*b^5*d^3*g^5*x^4 + 4*a*b^7*c^3*g^5*x^3 - 12*a^2*b^6*c^2*d*g^5*x^3 + 12*a^3*b^5*c*d
^2*g^5*x^3 - 4*a^4*b^4*d^3*g^5*x^3 + 6*a^2*b^6*c^3*g^5*x^2 - 18*a^3*b^5*c^2*d*g^5*x^2 + 18*a^4*b^4*c*d^2*g^5*x
^2 - 6*a^5*b^3*d^3*g^5*x^2 + 4*a^3*b^5*c^3*g^5*x - 12*a^4*b^4*c^2*d*g^5*x + 12*a^5*b^3*c*d^2*g^5*x - 4*a^6*b^2
*d^3*g^5*x + a^4*b^4*c^3*g^5 - 3*a^5*b^3*c^2*d*g^5 + 3*a^6*b^2*c*d^2*g^5 - a^7*b*d^3*g^5)