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

Optimal. Leaf size=173 \[ -\frac{b i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)^2}+\frac{d i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac{b B i (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac{B d i (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.341942, antiderivative size = 225, normalized size of antiderivative = 1.3, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac{d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (a+b x)^3}+\frac{B d^2 i}{6 b^2 g^4 (a+b x) (b c-a d)}+\frac{B d^3 i \log (a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac{B d^3 i \log (c+d x)}{6 b^2 g^4 (b c-a d)^2}-\frac{B i (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac{B d i}{12 b^2 g^4 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)*i)/(9*b^2*g^4*(a + b*x)^3) - (B*d*i)/(12*b^2*g^4*(a + b*x)^2) + (B*d^2*i)/(6*b^2*(b*c - a*d)*g
^4*(a + b*x)) + (B*d^3*i*Log[a + b*x])/(6*b^2*(b*c - a*d)^2*g^4) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(3*b^2*g^4*(a + b*x)^3) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^4*(a + b*x)^2) - (B*d^
3*i*Log[c + d*x])/(6*b^2*(b*c - a*d)^2*g^4)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 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]

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{(8 c+8 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac{8 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^4}+\frac{8 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac{(8 d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^4}+\frac{(8 (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^4}\\ &=-\frac{8 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac{4 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac{(4 B d) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac{(8 B (b c-a d)) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac{8 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac{4 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac{(4 B d (b c-a d)) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac{\left (8 B (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac{8 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac{4 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac{(4 B d (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^4}+\frac{\left (8 B (b c-a d)^2\right ) \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^2 g^4}\\ &=-\frac{8 B (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac{2 B d}{3 b^2 g^4 (a+b x)^2}+\frac{4 B d^2}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac{8 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac{4 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}-\frac{4 B d^3 \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}\\ \end{align*}

Mathematica [A]  time = 0.397522, size = 187, normalized size = 1.08 \[ -\frac{i \left (\frac{12 A b c}{(a+b x)^3}+\frac{18 A d}{(a+b x)^2}-\frac{12 a A d}{(a+b x)^3}-\frac{6 B d^2}{(a+b x) (b c-a d)}-\frac{6 B d^3 \log (a+b x)}{(b c-a d)^2}+\frac{6 B d^3 \log (c+d x)}{(b c-a d)^2}+\frac{6 B (a d+2 b c+3 b d x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac{4 b B c}{(a+b x)^3}+\frac{3 B d}{(a+b x)^2}-\frac{4 a B d}{(a+b x)^3}\right )}{36 b^2 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(i*((12*A*b*c)/(a + b*x)^3 + (4*b*B*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 - (4*a*B*d)/(a + b*x)^3 + (18*A*d
)/(a + b*x)^2 + (3*B*d)/(a + b*x)^2 - (6*B*d^2)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*Log[a + b*x])/(b*c - a*d)^2
+ (6*B*(2*b*c + a*d + 3*b*d*x)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^3 + (6*B*d^3*Log[c + d*x])/(b*c - a*d)
^2))/(36*b^2*g^4)

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Maple [B]  time = 0.052, size = 804, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.00435, size = 1260, normalized size = 7.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*B*d*i*(6*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4
*x + a^3*b^2*g^4) + (5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 1
6*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d +
a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^
5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3
)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) -
1/18*B*c*i*((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) + 6*log(b*e*x/(d*x + c) + a*e/(d
*x + c))/(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)) - 1/6*(3*b*x + a)*A*d*i/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/
3*A*c*i/(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 = 1.05242, size = 747, normalized size = 4.32 \begin{align*} \frac{6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i x^{2} - 3 \,{\left ({\left (6 \, A + B\right )} b^{3} c^{2} d - 6 \,{\left (2 \, A + B\right )} a b^{2} c d^{2} +{\left (6 \, A + 5 \, B\right )} a^{2} b d^{3}\right )} i x -{\left (4 \,{\left (3 \, A + B\right )} b^{3} c^{3} - 9 \,{\left (2 \, A + B\right )} a b^{2} c^{2} d +{\left (6 \, A + 5 \, B\right )} a^{3} d^{3}\right )} i + 6 \,{\left (B b^{3} d^{3} i x^{3} + 3 \, B a b^{2} d^{3} i x^{2} - 3 \,{\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i x -{\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{36 \,{\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x +{\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*x^2 - 3*((6*A + B)*b^3*c^2*d - 6*(2*A + B)*a*b^2*c*d^2 + (6*A + 5*B)*a^2
*b*d^3)*i*x - (4*(3*A + B)*b^3*c^3 - 9*(2*A + B)*a*b^2*c^2*d + (6*A + 5*B)*a^3*d^3)*i + 6*(B*b^3*d^3*i*x^3 + 3
*B*a*b^2*d^3*i*x^2 - 3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2)*i*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i)*log((b*e*x + a
*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g
^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4
)

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Sympy [B]  time = 11.3072, size = 629, normalized size = 3.64 \begin{align*} - \frac{B d^{3} i \log{\left (x + \frac{- \frac{B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} + \frac{3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} - \frac{3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i + \frac{B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac{B d^{3} i \log{\left (x + \frac{\frac{B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} - \frac{3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} + \frac{3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i - \frac{B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac{\left (- B a d i - 2 B b c i - 3 B b d i x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{6 a^{3} b^{2} g^{4} + 18 a^{2} b^{3} g^{4} x + 18 a b^{4} g^{4} x^{2} + 6 b^{5} g^{4} x^{3}} - \frac{6 A a^{2} d^{2} i + 6 A a b c d i - 12 A b^{2} c^{2} i + 5 B a^{2} d^{2} i + 5 B a b c d i - 4 B b^{2} c^{2} i + 6 B b^{2} d^{2} i x^{2} + x \left (18 A a b d^{2} i - 18 A b^{2} c d i + 15 B a b d^{2} i - 3 B b^{2} c d i\right )}{36 a^{4} b^{2} d g^{4} - 36 a^{3} b^{3} c g^{4} + x^{3} \left (36 a b^{5} d g^{4} - 36 b^{6} c g^{4}\right ) + x^{2} \left (108 a^{2} b^{4} d g^{4} - 108 a b^{5} c g^{4}\right ) + x \left (108 a^{3} b^{3} d g^{4} - 108 a^{2} b^{4} c g^{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*d**3*i*log(x + (-B*a**3*d**6*i/(a*d - b*c)**2 + 3*B*a**2*b*c*d**5*i/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i
/(a*d - b*c)**2 + B*a*d**4*i + B*b**3*c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i))/(6*b**2*g**4*
(a*d - b*c)**2) + B*d**3*i*log(x + (B*a**3*d**6*i/(a*d - b*c)**2 - 3*B*a**2*b*c*d**5*i/(a*d - b*c)**2 + 3*B*a*
b**2*c**2*d**4*i/(a*d - b*c)**2 + B*a*d**4*i - B*b**3*c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i
))/(6*b**2*g**4*(a*d - b*c)**2) + (-B*a*d*i - 2*B*b*c*i - 3*B*b*d*i*x)*log(e*(a + b*x)/(c + d*x))/(6*a**3*b**2
*g**4 + 18*a**2*b**3*g**4*x + 18*a*b**4*g**4*x**2 + 6*b**5*g**4*x**3) - (6*A*a**2*d**2*i + 6*A*a*b*c*d*i - 12*
A*b**2*c**2*i + 5*B*a**2*d**2*i + 5*B*a*b*c*d*i - 4*B*b**2*c**2*i + 6*B*b**2*d**2*i*x**2 + x*(18*A*a*b*d**2*i
- 18*A*b**2*c*d*i + 15*B*a*b*d**2*i - 3*B*b**2*c*d*i))/(36*a**4*b**2*d*g**4 - 36*a**3*b**3*c*g**4 + x**3*(36*a
*b**5*d*g**4 - 36*b**6*c*g**4) + x**2*(108*a**2*b**4*d*g**4 - 108*a*b**5*c*g**4) + x*(108*a**3*b**3*d*g**4 - 1
08*a**2*b**4*c*g**4))

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Giac [B]  time = 1.39268, size = 544, normalized size = 3.14 \begin{align*} -\frac{B d^{3} \log \left (b x + a\right )}{6 \,{\left (b^{4} c^{2} g^{4} i - 2 \, a b^{3} c d g^{4} i + a^{2} b^{2} d^{2} g^{4} i\right )}} + \frac{B d^{3} \log \left (d x + c\right )}{6 \,{\left (b^{4} c^{2} g^{4} i - 2 \, a b^{3} c d g^{4} i + a^{2} b^{2} d^{2} g^{4} i\right )}} - \frac{{\left (3 \, B b d i x + 2 \, B b c i + B a d i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{6 \,{\left (b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}\right )}} + \frac{6 \, B b^{2} d^{2} i x^{2} - 18 \, A b^{2} c d i x - 21 \, B b^{2} c d i x + 18 \, A a b d^{2} i x + 33 \, B a b d^{2} i x - 12 \, A b^{2} c^{2} i - 16 \, B b^{2} c^{2} i + 6 \, A a b c d i + 11 \, B a b c d i + 6 \, A a^{2} d^{2} i + 11 \, B a^{2} d^{2} i}{36 \,{\left (b^{6} c g^{4} x^{3} - a b^{5} d g^{4} x^{3} + 3 \, a b^{5} c g^{4} x^{2} - 3 \, a^{2} b^{4} d g^{4} x^{2} + 3 \, a^{2} b^{4} c g^{4} x - 3 \, a^{3} b^{3} d g^{4} x + a^{3} b^{3} c g^{4} - a^{4} b^{2} d g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*B*d^3*log(b*x + a)/(b^4*c^2*g^4*i - 2*a*b^3*c*d*g^4*i + a^2*b^2*d^2*g^4*i) + 1/6*B*d^3*log(d*x + c)/(b^4*
c^2*g^4*i - 2*a*b^3*c*d*g^4*i + a^2*b^2*d^2*g^4*i) - 1/6*(3*B*b*d*i*x + 2*B*b*c*i + B*a*d*i)*log((b*x + a)/(d*
x + c))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) + 1/36*(6*B*b^2*d^2*i*x^2 - 18*A*b^2*c
*d*i*x - 21*B*b^2*c*d*i*x + 18*A*a*b*d^2*i*x + 33*B*a*b*d^2*i*x - 12*A*b^2*c^2*i - 16*B*b^2*c^2*i + 6*A*a*b*c*
d*i + 11*B*a*b*c*d*i + 6*A*a^2*d^2*i + 11*B*a^2*d^2*i)/(b^6*c*g^4*x^3 - a*b^5*d*g^4*x^3 + 3*a*b^5*c*g^4*x^2 -
3*a^2*b^4*d*g^4*x^2 + 3*a^2*b^4*c*g^4*x - 3*a^3*b^3*d*g^4*x + a^3*b^3*c*g^4 - a^4*b^2*d*g^4)