3.208 \(\int \frac{A+B \log (\frac{e (c+d x)^2}{(a+b x)^2})}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=177 \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{3 b g^4 (a+b x)^3}+\frac{2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac{2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac{B d}{3 b g^4 (a+b x)^2 (b c-a d)}+\frac{2 B}{9 b g^4 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.120882, antiderivative size = 177, 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} \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{3 b g^4 (a+b x)^3}+\frac{2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac{2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac{B d}{3 b g^4 (a+b x)^2 (b c-a d)}+\frac{2 B}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.120337, size = 140, normalized size = 0.79 \[ \frac{\frac{B \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}-3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^4,x]

[Out]

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

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Maple [B]  time = 0.059, size = 427, normalized size = 2.4 \begin{align*} -{\frac{A}{3\,b \left ( bx+a \right ) ^{3}{g}^{4}}}-{\frac{B}{3\,b \left ( bx+a \right ) ^{3}{g}^{4}}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{2\,B{a}^{3}{d}^{3}}{9\,b{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{3}}}-{\frac{2\,B{a}^{2}{d}^{2}c}{3\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{3}}}+{\frac{2\,bBad{c}^{2}}{3\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{3}}}+{\frac{B{a}^{2}{d}^{3}}{3\,b{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{2\,Ba{d}^{2}c}{3\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{2\,Ba{d}^{3}}{3\,b{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}+{\frac{2\,Ba{d}^{4}}{3\,b{g}^{4} \left ( ad-bc \right ) ^{4}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }-{\frac{2\,{b}^{2}B{c}^{3}}{9\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{3}}}+{\frac{bB{c}^{2}d}{3\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{2\,Bc{d}^{2}}{3\,{g}^{4} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-{\frac{2\,Bc{d}^{3}}{3\,{g}^{4} \left ( ad-bc \right ) ^{4}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^4,x)

[Out]

-1/3/b/(b*x+a)^3/g^4*A-1/3/b/g^4*B/(b*x+a)^3*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/b^2)+2/9/b/g^4*B*a^3*d^3/(a*
d-b*c)^3/(b*x+a)^3-2/3/g^4*B*a^2*d^2/(a*d-b*c)^3/(b*x+a)^3*c+2/3*b/g^4*B*a*d/(a*d-b*c)^3/(b*x+a)^3*c^2+1/3/b/g
^4*B*a^2*d^3/(a*d-b*c)^3/(b*x+a)^2-2/3/g^4*B*a*d^2/(a*d-b*c)^3/(b*x+a)^2*c+2/3/b/g^4*B*a*d^3/(a*d-b*c)^3/(b*x+
a)+2/3/b/g^4*B*a*d^4/(a*d-b*c)^4*ln(1/(b*x+a)*a*d-b*c/(b*x+a)-d)-2/9*b^2/g^4*B*c^3/(a*d-b*c)^3/(b*x+a)^3+1/3*b
/g^4*B*c^2/(a*d-b*c)^3/(b*x+a)^2*d-2/3/g^4*B*c/(a*d-b*c)^3/(b*x+a)*d^2-2/3/g^4*B*c*d^3/(a*d-b*c)^4*ln(1/(b*x+a
)*a*d-b*c/(b*x+a)-d)

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Maxima [B]  time = 1.31059, size = 648, normalized size = 3.66 \begin{align*} \frac{1}{9} \, B{\left (\frac{6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x +{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} - \frac{3 \, \log \left (\frac{d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac{6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac{6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac{A}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.07831, size = 869, normalized size = 4.91 \begin{align*} -\frac{{\left (3 \, A - 2 \, B\right )} b^{3} c^{3} - 9 \,{\left (A - B\right )} a b^{2} c^{2} d + 9 \,{\left (A - 2 \, B\right )} a^{2} b c d^{2} -{\left (3 \, A - 11 \, B\right )} a^{3} d^{3} - 6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} + 3 \,{\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 3 \,{\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{9 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x +{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 5.36037, size = 677, normalized size = 3.82 \begin{align*} - \frac{B \log{\left (\frac{e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} + \frac{2 B d^{3} \log{\left (x + \frac{- \frac{2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac{8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac{12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} - \frac{2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} - \frac{2 B d^{3} \log{\left (x + \frac{\frac{2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac{8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac{12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} + \frac{2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac{- 3 A a^{2} d^{2} + 6 A a b c d - 3 A b^{2} c^{2} + 11 B a^{2} d^{2} - 7 B a b c d + 2 B b^{2} c^{2} + 6 B b^{2} d^{2} x^{2} + x \left (15 B a b d^{2} - 3 B b^{2} c d\right )}{9 a^{5} b d^{2} g^{4} - 18 a^{4} b^{2} c d g^{4} + 9 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (9 a^{2} b^{4} d^{2} g^{4} - 18 a b^{5} c d g^{4} + 9 b^{6} c^{2} g^{4}\right ) + x^{2} \left (27 a^{3} b^{3} d^{2} g^{4} - 54 a^{2} b^{4} c d g^{4} + 27 a b^{5} c^{2} g^{4}\right ) + x \left (27 a^{4} b^{2} d^{2} g^{4} - 54 a^{3} b^{3} c d g^{4} + 27 a^{2} b^{4} c^{2} g^{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(d*x+c)**2/(b*x+a)**2))/(b*g*x+a*g)**4,x)

[Out]

-B*log(e*(c + d*x)**2/(a + b*x)**2)/(3*a**3*b*g**4 + 9*a**2*b**2*g**4*x + 9*a*b**3*g**4*x**2 + 3*b**4*g**4*x**
3) + 2*B*d**3*log(x + (-2*B*a**4*d**7/(a*d - b*c)**3 + 8*B*a**3*b*c*d**6/(a*d - b*c)**3 - 12*B*a**2*b**2*c**2*
d**5/(a*d - b*c)**3 + 8*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + 2*B*a*d**4 - 2*B*b**4*c**4*d**3/(a*d - b*c)**3 + 2
*B*b*c*d**3)/(4*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) - 2*B*d**3*log(x + (2*B*a**4*d**7/(a*d - b*c)**3 - 8*B*a*
*3*b*c*d**6/(a*d - b*c)**3 + 12*B*a**2*b**2*c**2*d**5/(a*d - b*c)**3 - 8*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + 2
*B*a*d**4 + 2*B*b**4*c**4*d**3/(a*d - b*c)**3 + 2*B*b*c*d**3)/(4*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + (-3*A*
a**2*d**2 + 6*A*a*b*c*d - 3*A*b**2*c**2 + 11*B*a**2*d**2 - 7*B*a*b*c*d + 2*B*b**2*c**2 + 6*B*b**2*d**2*x**2 +
x*(15*B*a*b*d**2 - 3*B*b**2*c*d))/(9*a**5*b*d**2*g**4 - 18*a**4*b**2*c*d*g**4 + 9*a**3*b**3*c**2*g**4 + x**3*(
9*a**2*b**4*d**2*g**4 - 18*a*b**5*c*d*g**4 + 9*b**6*c**2*g**4) + x**2*(27*a**3*b**3*d**2*g**4 - 54*a**2*b**4*c
*d*g**4 + 27*a*b**5*c**2*g**4) + x*(27*a**4*b**2*d**2*g**4 - 54*a**3*b**3*c*d*g**4 + 27*a**2*b**4*c**2*g**4))

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Giac [B]  time = 1.38949, size = 639, normalized size = 3.61 \begin{align*} \frac{2 \, B d^{3} \log \left (b x + a\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac{2 \, B d^{3} \log \left (d x + c\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac{B \log \left (\frac{d^{2} x^{2} + 2 \, c d x + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} + \frac{6 \, B b^{2} d^{2} x^{2} - 3 \, B b^{2} c d x + 15 \, B a b d^{2} x - 3 \, A b^{2} c^{2} - B b^{2} c^{2} + 6 \, A a b c d - B a b c d - 3 \, A a^{2} d^{2} + 8 \, B a^{2} d^{2}}{9 \,{\left (b^{6} c^{2} g^{4} x^{3} - 2 \, a b^{5} c d g^{4} x^{3} + a^{2} b^{4} d^{2} g^{4} x^{3} + 3 \, a b^{5} c^{2} g^{4} x^{2} - 6 \, a^{2} b^{4} c d g^{4} x^{2} + 3 \, a^{3} b^{3} d^{2} g^{4} x^{2} + 3 \, a^{2} b^{4} c^{2} g^{4} x - 6 \, a^{3} b^{3} c d g^{4} x + 3 \, a^{4} b^{2} d^{2} g^{4} x + a^{3} b^{3} c^{2} g^{4} - 2 \, a^{4} b^{2} c d g^{4} + a^{5} b d^{2} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

2/3*B*d^3*log(b*x + a)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) - 2/3*B*d^3*log
(d*x + c)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) - 1/3*B*log((d^2*x^2 + 2*c*d
*x + c^2)/(b^2*x^2 + 2*a*b*x + a^2))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 1/9*(6*B*
b^2*d^2*x^2 - 3*B*b^2*c*d*x + 15*B*a*b*d^2*x - 3*A*b^2*c^2 - B*b^2*c^2 + 6*A*a*b*c*d - B*a*b*c*d - 3*A*a^2*d^2
+ 8*B*a^2*d^2)/(b^6*c^2*g^4*x^3 - 2*a*b^5*c*d*g^4*x^3 + a^2*b^4*d^2*g^4*x^3 + 3*a*b^5*c^2*g^4*x^2 - 6*a^2*b^4
*c*d*g^4*x^2 + 3*a^3*b^3*d^2*g^4*x^2 + 3*a^2*b^4*c^2*g^4*x - 6*a^3*b^3*c*d*g^4*x + 3*a^4*b^2*d^2*g^4*x + a^3*b
^3*c^2*g^4 - 2*a^4*b^2*c*d*g^4 + a^5*b*d^2*g^4)