3.127 \(\int \frac{A+B \log (\frac{e (a+b x)^2}{(c+d x)^2})}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=208 \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}+\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{8 b g^5 (a+b x)^4} \]

[Out]

-B/(8*b*g^5*(a + b*x)^4) + (B*d)/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^
2) + (B*d^3)/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*
(a + b*x)^2)/(c + d*x)^2])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5)

________________________________________________________________________________________

Rubi [A]  time = 0.144397, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}+\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac{B}{8 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^5,x]

[Out]

-B/(8*b*g^5*(a + b*x)^4) + (B*d)/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^
2) + (B*d^3)/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*
(a + b*x)^2)/(c + d*x)^2])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(2*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)^2}{(c+d x)^2}\right )}{(a g+b g x)^5} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{2 (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)^2}{(c+d x)^2}\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}{2 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\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}{2 b g^5}\\ &=-\frac{B}{8 b g^5 (a+b x)^4}+\frac{B d}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \log (a+b x)}{2 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac{B d^4 \log (c+d x)}{2 b (b c-a d)^4 g^5}\\ \end{align*}

Mathematica [A]  time = 0.2014, size = 162, normalized size = 0.78 \[ -\frac{6 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac{B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )}{(b c-a d)^4}}{24 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^5,x]

[Out]

-(6*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (B*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b
*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*
x)^4*Log[c + d*x]))/(b*c - a*d)^4)/(24*b*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B]  time = 0.207, size = 833, normalized size = 4. \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)^2/(d*x+c)^2))/(b*g*x+a*g)^5,x)

[Out]

-3/2*d^4/g^5*A*b/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2+d^4/g^5*A/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x+c)+
b)+d^4/g^5*A*b^2/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^3-1/4*d^4/g^5*A*b^3/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/
(d*x+c)+b)^4-25/24*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B/b/(d*x+c)^4+1/4*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c
)+b)^4*b^3*B/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b
)^2/d^2)-1/2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+
c)-7/4*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2-13/6*d^4/g^5/(1/(d*x+
c)*a*d-b*c/(d*x+c)+b)^4*B/(a*d-b*c)/(d*x+c)^3+d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B/(a*d-b*c)/(d*x+c)^3*ln
(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+3/2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*b*B/(a^2*d^2-2*a*b*c*d+b^2
*c^2)/(d*x+c)^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B*b^2/(a^3*d
^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)

________________________________________________________________________________________

Maxima [B]  time = 1.3979, size = 944, normalized size = 4.54 \begin{align*} \frac{1}{24} \, 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{6 \, \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\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)^2/(d*x+c)^2))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/24*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) - 6*log(b^2*e*x^2/(d^2*
x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(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)

________________________________________________________________________________________

Fricas [B]  time = 1.08479, size = 1328, normalized size = 6.38 \begin{align*} -\frac{3 \,{\left (2 \, A + B\right )} b^{4} c^{4} - 8 \,{\left (3 \, A + 2 \, B\right )} a b^{3} c^{3} d + 36 \,{\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 24 \,{\left (A + 2 \, B\right )} a^{3} b c d^{3} +{\left (6 \, 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 - 6 \,{\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^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{24 \,{\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)^2/(d*x+c)^2))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/24*(3*(2*A + B)*b^4*c^4 - 8*(3*A + 2*B)*a*b^3*c^3*d + 36*(A + B)*a^2*b^2*c^2*d^2 - 24*(A + 2*B)*a^3*b*c*d^3
+ (6*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 - 6*(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^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((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)

________________________________________________________________________________________

Sympy [B]  time = 8.26937, size = 947, normalized size = 4.55 \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)**2/(d*x+c)**2))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(a + b*x)**2/(c + d*x)**2)/(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))/(2*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))/(2*b*g**5*(a*d - b*c)**4) - (6*A*a**3*d**3 - 18*A*a**
2*b*c*d**2 + 18*A*a*b**2*c**2*d - 6*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))/(24*a**7*b*d**3*g**5 - 72*a**6*b**2*c*d**2*g**5 + 72*a**5*b**3*c**2*d*g**5 - 2
4*a**4*b**4*c**3*g**5 + x**4*(24*a**3*b**5*d**3*g**5 - 72*a**2*b**6*c*d**2*g**5 + 72*a*b**7*c**2*d*g**5 - 24*b
**8*c**3*g**5) + x**3*(96*a**4*b**4*d**3*g**5 - 288*a**3*b**5*c*d**2*g**5 + 288*a**2*b**6*c**2*d*g**5 - 96*a*b
**7*c**3*g**5) + x**2*(144*a**5*b**3*d**3*g**5 - 432*a**4*b**4*c*d**2*g**5 + 432*a**3*b**5*c**2*d*g**5 - 144*a
**2*b**6*c**3*g**5) + x*(96*a**6*b**2*d**3*g**5 - 288*a**5*b**3*c*d**2*g**5 + 288*a**4*b**4*c**2*d*g**5 - 96*a
**3*b**5*c**3*g**5))

________________________________________________________________________________________

Giac [B]  time = 1.44465, size = 566, normalized size = 2.72 \begin{align*} -\frac{B d^{4} \log \left (-\frac{b c g}{b g x + a g} + \frac{a d g}{b g x + a g} - d\right )}{2 \,{\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^{3}}{2 \,{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )}{\left (b g x + a g\right )} b g} - \frac{B d^{2}}{4 \,{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )}{\left (b g x + a g\right )}^{2} b g^{2}} - \frac{B \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{4 \,{\left (b g x + a g\right )}^{4} b g} + \frac{B d}{6 \,{\left (b g x + a g\right )}^{3}{\left (b c - a d\right )} b g^{2}} - \frac{2 \, A b^{3} g^{3} + 3 \, B b^{3} g^{3}}{8 \,{\left (b g x + a g\right )}^{4} b^{4} g^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

-1/2*B*d^4*log(-b*c*g/(b*g*x + a*g) + a*d*g/(b*g*x + a*g) - d)/(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) + 1/2*B*d^3/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2
*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 1/4*B*d^2/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^
2) - 1/4*B*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2
+ 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))/((b*g*x + a*g)^4*b*g) + 1/6*B*d/((b*g*x + a*g)^3*(
b*c - a*d)*b*g^2) - 1/8*(2*A*b^3*g^3 + 3*B*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)