3.22 \(\int (a g+b g x) (c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=271 \[ -\frac{g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac{b g i^3 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^2}+\frac{B g i^3 (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g i^3 (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac{B g i^3 (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac{B g i^3 x (b c-a d)^4}{20 b^3 d}+\frac{B g i^3 (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac{B g i^3 (c+d x)^4 (b c-a d)}{20 d^2} \]

[Out]

(B*(b*c - a*d)^4*g*i^3*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2*g*i
^3*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4)/(20*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[(a + b*x)
/(c + d*x)])/(20*b^4*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2) + (b*
g*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[c + d*x])/(20*b^4
*d^2)

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Rubi [A]  time = 0.340664, antiderivative size = 232, normalized size of antiderivative = 0.86, 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, 43} \[ -\frac{g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac{b g i^3 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^2}+\frac{B g i^3 (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g i^3 (b c-a d)^5 \log (a+b x)}{20 b^4 d^2}+\frac{B g i^3 x (b c-a d)^4}{20 b^3 d}+\frac{B g i^3 (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac{B g i^3 (c+d x)^4 (b c-a d)}{20 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)*(c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(B*(b*c - a*d)^4*g*i^3*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2*g*i
^3*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4)/(20*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[a + b*x])
/(20*b^4*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2) + (b*g*i^3*(c + d
*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^2)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (22 c+22 d x)^3 (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{(-b c+a d) g (22 c+22 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{b g (22 c+22 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{22 d}\right ) \, dx\\ &=\frac{(b g) \int (22 c+22 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{22 d}+\frac{((-b c+a d) g) \int (22 c+22 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d}\\ &=-\frac{2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{10648 b g (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac{(b B g) \int \frac{5153632 (b c-a d) (c+d x)^4}{a+b x} \, dx}{2420 d^2}+\frac{(B (b c-a d) g) \int \frac{234256 (b c-a d) (c+d x)^3}{a+b x} \, dx}{88 d^2}\\ &=-\frac{2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{10648 b g (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac{(10648 b B (b c-a d) g) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^2}+\frac{\left (2662 B (b c-a d)^2 g\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^2}\\ &=-\frac{2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{10648 b g (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac{(10648 b B (b c-a d) g) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^2}+\frac{\left (2662 B (b c-a d)^2 g\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{d^2}\\ &=\frac{2662 B (b c-a d)^4 g x}{5 b^3 d}+\frac{1331 B (b c-a d)^3 g (c+d x)^2}{5 b^2 d^2}+\frac{2662 B (b c-a d)^2 g (c+d x)^3}{15 b d^2}-\frac{2662 B (b c-a d) g (c+d x)^4}{5 d^2}+\frac{2662 B (b c-a d)^5 g \log (a+b x)}{5 b^4 d^2}-\frac{2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{10648 b g (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^2}\\ \end{align*}

Mathematica [A]  time = 0.192793, size = 261, normalized size = 0.96 \[ \frac{g i^3 \left (24 b (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-30 (c+d x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{5 B (b c-a d)^2 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^4}-\frac{2 B (b c-a d) \left (6 b^2 (c+d x)^2 (b c-a d)^2+4 b^3 (c+d x)^3 (b c-a d)+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{b^4}\right )}{120 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)*(c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(g*i^3*((5*B*(b*c - a*d)^2*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c
- a*d)^3*Log[a + b*x]))/b^4 - (2*B*(b*c - a*d)*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 + 4*
b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]))/b^4 - 30*(b*c - a*d)*(c + d*
x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*b*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(120*d^2)

________________________________________________________________________________________

Maple [B]  time = 0.191, size = 4481, normalized size = 16.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

-1/4*e^2*d*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3*c^2-1/20*e/d^2*B*g*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c
)*c^5-1/12*e^3*d^2*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^4*c-1/60*e^3/d^2*B*g*i^3*b^4/(d*e/(d*x+c)*a-e/(d*
x+c)*b*c)^3*c^5+1/40*e^2/d^2*B*g*i^3*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^5-1/20*e^4/d^2*B*g*i^3/(d*e/(d*x+c)
*a-e/(d*x+c)*b*c)^4*b^5*c^5+1/20*e*d^3*B*g*i^3/b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^5-1/5*e^5/d^2*A*g*i^3*b^6/(
d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^5-1/40*e^2*d^3*B*g*i^3/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^5+1/5*e^5*d^3*A*
g*i^3*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^5+1/60*e^3*d^3*B*g*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^5+1/4*e^2
*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*c^3*b-2*e^5*A*g*i^3*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^2*c^3-1
/2*e^4*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*b^3*c^3-5/2*e^4*A*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2
*b^3*c^3-1/6*e^3*B*g*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c^3-1/4*e^4/d^2*A*g*i^3/(d*e/(d*x+c)*a-e/(d*x
+c)*b*c)^4*b^5*c^5+1/4*e^4*d^3*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^5-1/4
*d^2*B*g*i^3/b^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^4*c+1/2*d*B*g*i^3/b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*
x+c))-b*e)*a^3*c^2-1/2*e*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c^3-e^5*d^2*B*g*i^3*b^2*ln(b*e/d+(a*d-b*c)*
e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^4*c+1/5*e^5*d^8*B*g*i^3/b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*
e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^10/(d*x+c)^5+5/4*e^4/d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e
/(d*x+c)*b*c)^4*c^4*b^4*a-5/4*e^4*d^2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*
a^4*b*c-1/4*e^4*d^7*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^9/(d*x+c)^4+
5/2*e^4*d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*c^2+2*e^5*d*B*g*i^3*
ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^3*b^3*c^2+42*e^5*d^4*B*g*i^3*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^4/(d*x+c)^5*a^6-1/20/d^2*B*g*i^3*ln(d*(b*e/d+(a*d-b*c)*e/d/
(d*x+c))-b*e)*c^5*b+1/20*d^3*B*g*i^3/b^4*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^5-1/2*B*g*i^3/b*ln(d*(b*e/d
+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2*c^3+1/20*e^4*d^3*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^5+1/4*e^4*d^3*A*g*
i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^5+1/4/d*B*g*i^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^4*a+1/4*e^4/d^
2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^9/(d*x+c)^4*b^5-63/2*e^4*d^3*B*g*i
^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^5*c^4/(d*x+c)^4+1/5*e^5/d^2*B*g*i^3*ln(b*
e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^10/(d*x+c)^5*b^6+9*e^5*B*g*i^3*ln(b*e/d+(a*d-b*c)
*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^8/(d*x+c)^5*a^2*b^4+9*e^4*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*c^7/(d*x+c)^4+e^5/d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(
d*x+c)*a-e/(d*x+c)*b*c)^5*c^4*b^5*a-2*e^5*d^7*B*g*i^3/b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*
x+c)*b*c)^5*a^9/(d*x+c)^5*c+9/4*e^4*d^6*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b
*c)^4*a^8/(d*x+c)^4*c-252/5*e^5*d^3*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^
5/(d*x+c)^5*a^5*b+42*e^5*d^2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^6/(d*x+
c)^5*a^4*b^2-24*e^5*d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^7/(d*x+c)^5*a^
3*b^3-9/4*e^4/d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^8/(d*x+c)^4*b^4*a-2*
e^5/d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^9/(d*x+c)^5*b^5*a+9*e^5*d^6*B*
g*i^3/b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^8*c^2/(d*x+c)^5-24*e^5*d^5*B*g*i^3
/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^7*c^3/(d*x+c)^5-9*e^4*d^5*B*g*i^3*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^7*c^2/(d*x+c)^4+21*e^4*d^4*B*g*i^3*ln(b*e/d+(a*
d-b*c)*e/d/(d*x+c))/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^6*c^3/(d*x+c)^4-21*e^4*d*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/
d/(d*x+c))*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*c^6/(d*x+c)^4+63/2*e^4*d^2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(
d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4*c^5/(d*x+c)^4*b+1/2*e*d*B*g*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^
3*c^2+1/4*e/d*B*g*i^3*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^4*a-1/4*e^4*d^2*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^
4*a^4*b*c-5/4*e^4*d^2*A*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4*b*c+1/12*e^3/d*B*g*i^3*b^3/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^3*c^4*a-1/8*e^2/d*B*g*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^4*a+1/4*e^4/d*B*g*i^3/(d*e/(d*x+c
)*a-e/(d*x+c)*b*c)^4*b^4*c^4*a-1/4*e^4/d^2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*
c)^4*c^5*b^5-1/5*e^5/d^2*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*c^5*b^6-5/2*e
^4*B*g*i^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^2*c^3-2*e^5*B*g*i^3*ln(b*e/d+
(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^2*b^4*c^3+1/5*e^5*d^3*B*g*i^3*b*ln(b*e/d+(a*d-b*c)*e/
d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^5+1/8*e^2*d^2*B*g*i^3/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^4*c+1/6
*e^3*d*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3*c^2*b+2*e^5*d*A*g*i^3*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a
^3*c^2+1/2*e^4*d*B*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*b^2*c^2+5/2*e^4*d*A*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)^4*a^3*b^2*c^2-e^5*d^2*A*g*i^3*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^5*a^4*c+e^5/d*A*g*i^3*b^5/(d*e/(d*x+c)*
a-e/(d*x+c)*b*c)^5*c^4*a+5/4*e^4/d*A*g*i^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*b^4*c^4*a-1/4*e*d^2*B*g*i^3/b^2/(d*
e/(d*x+c)*a-e/(d*x+c)*b*c)*a^4*c

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Maxima [B]  time = 1.36154, size = 1380, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/5*A*b*d^3*g*i^3*x^5 + 3/4*A*b*c*d^2*g*i^3*x^4 + 1/4*A*a*d^3*g*i^3*x^4 + A*b*c^2*d*g*i^3*x^3 + A*a*c*d^2*g*i^
3*x^3 + 1/2*A*b*c^3*g*i^3*x^2 + 3/2*A*a*c^2*d*g*i^3*x^2 + (x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x
+ a)/b - c*log(d*x + c)/d)*B*a*c^3*g*i^3 + 1/2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^
2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*b*c^3*g*i^3 + 3/2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a*c^2*d*g*i^3 + 1/2*(2*x^3*log(b*e*x/(d
*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2
*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*b*c^2*d*g*i^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*
x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*c*d^2
*g*i^3 + 1/8*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (
2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b*c*d^2*
g*i^3 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (
2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a*d^3*g*
i^3 + 1/60*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 -
(3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c
^4 - a^4*d^4)*x)/(b^4*d^4))*B*b*d^3*g*i^3 + A*a*c^3*g*i^3*x

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Fricas [A]  time = 1.21094, size = 1057, normalized size = 3.9 \begin{align*} \frac{24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \,{\left ({\left (15 \, A - B\right )} b^{5} c d^{4} +{\left (5 \, A + B\right )} a b^{4} d^{5}\right )} g i^{3} x^{4} + 2 \,{\left ({\left (60 \, A - 11 \, B\right )} b^{5} c^{2} d^{3} + 10 \,{\left (6 \, A + B\right )} a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g i^{3} x^{3} + 3 \,{\left ({\left (20 \, A - 9 \, B\right )} b^{5} c^{3} d^{2} + 5 \,{\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g i^{3} x^{2} - 6 \,{\left (B b^{5} c^{4} d - 5 \,{\left (4 \, A - B\right )} a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} x + 6 \,{\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} \log \left (b x + a\right ) + 6 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} \log \left (d x + c\right ) + 6 \,{\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \,{\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \,{\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \,{\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{120 \, b^{4} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/120*(24*A*b^5*d^5*g*i^3*x^5 + 6*((15*A - B)*b^5*c*d^4 + (5*A + B)*a*b^4*d^5)*g*i^3*x^4 + 2*((60*A - 11*B)*b^
5*c^2*d^3 + 10*(6*A + B)*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g*i^3*x^3 + 3*((20*A - 9*B)*b^5*c^3*d^2 + 5*(12*A + B)*a
*b^4*c^2*d^3 + 5*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g*i^3*x^2 - 6*(B*b^5*c^4*d - 5*(4*A - B)*a*b^4*c^3*d^2 - 10*
B*a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g*i^3*x + 6*(10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3
+ 5*B*a^4*b*c*d^4 - B*a^5*d^5)*g*i^3*log(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c^4*d)*g*i^3*log(d*x + c) + 6*(4*
B*b^5*d^5*g*i^3*x^5 + 20*B*a*b^4*c^3*d^2*g*i^3*x + 5*(3*B*b^5*c*d^4 + B*a*b^4*d^5)*g*i^3*x^4 + 20*(B*b^5*c^2*d
^3 + B*a*b^4*c*d^4)*g*i^3*x^3 + 10*(B*b^5*c^3*d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*x^2)*log((b*e*x + a*e)/(d*x + c))
)/(b^4*d^2)

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Sympy [B]  time = 9.48581, size = 1187, normalized size = 4.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)*(d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b*d**3*g*i**3*x**5/5 - B*a**2*g*i**3*(a**3*d**3 - 5*a**2*b*c*d**2 + 10*a*b**2*c**2*d - 10*b**3*c**3)*log(x +
(B*a**5*c*d**4*g*i**3 - 5*B*a**4*b*c**2*d**3*g*i**3 + 10*B*a**3*b**2*c**3*d**2*g*i**3 + B*a**3*d**2*g*i**3*(a
**3*d**3 - 5*a**2*b*c*d**2 + 10*a*b**2*c**2*d - 10*b**3*c**3)/b - 15*B*a**2*b**3*c**4*d*g*i**3 - B*a**2*c*d*g*
i**3*(a**3*d**3 - 5*a**2*b*c*d**2 + 10*a*b**2*c**2*d - 10*b**3*c**3) + B*a*b**4*c**5*g*i**3)/(B*a**5*d**5*g*i*
*3 - 5*B*a**4*b*c*d**4*g*i**3 + 10*B*a**3*b**2*c**2*d**3*g*i**3 - 10*B*a**2*b**3*c**3*d**2*g*i**3 - 5*B*a*b**4
*c**4*d*g*i**3 + B*b**5*c**5*g*i**3))/(20*b**4) - B*c**4*g*i**3*(5*a*d - b*c)*log(x + (B*a**5*c*d**4*g*i**3 -
5*B*a**4*b*c**2*d**3*g*i**3 + 10*B*a**3*b**2*c**3*d**2*g*i**3 - 15*B*a**2*b**3*c**4*d*g*i**3 + B*a*b**4*c**5*g
*i**3 + B*a*b**3*c**4*g*i**3*(5*a*d - b*c) - B*b**4*c**5*g*i**3*(5*a*d - b*c)/d)/(B*a**5*d**5*g*i**3 - 5*B*a**
4*b*c*d**4*g*i**3 + 10*B*a**3*b**2*c**2*d**3*g*i**3 - 10*B*a**2*b**3*c**3*d**2*g*i**3 - 5*B*a*b**4*c**4*d*g*i*
*3 + B*b**5*c**5*g*i**3))/(20*d**2) + x**4*(A*a*d**3*g*i**3/4 + 3*A*b*c*d**2*g*i**3/4 + B*a*d**3*g*i**3/20 - B
*b*c*d**2*g*i**3/20) + (B*a*c**3*g*i**3*x + 3*B*a*c**2*d*g*i**3*x**2/2 + B*a*c*d**2*g*i**3*x**3 + B*a*d**3*g*i
**3*x**4/4 + B*b*c**3*g*i**3*x**2/2 + B*b*c**2*d*g*i**3*x**3 + 3*B*b*c*d**2*g*i**3*x**4/4 + B*b*d**3*g*i**3*x*
*5/5)*log(e*(a + b*x)/(c + d*x)) + x**3*(60*A*a*b*c*d**2*g*i**3 + 60*A*b**2*c**2*d*g*i**3 + B*a**2*d**3*g*i**3
+ 10*B*a*b*c*d**2*g*i**3 - 11*B*b**2*c**2*d*g*i**3)/(60*b) - x**2*(-60*A*a*b**2*c**2*d*g*i**3 - 20*A*b**3*c**
3*g*i**3 + B*a**3*d**3*g*i**3 - 5*B*a**2*b*c*d**2*g*i**3 - 5*B*a*b**2*c**2*d*g*i**3 + 9*B*b**3*c**3*g*i**3)/(4
0*b**2) + x*(20*A*a*b**3*c**3*d*g*i**3 + B*a**4*d**4*g*i**3 - 5*B*a**3*b*c*d**3*g*i**3 + 10*B*a**2*b**2*c**2*d
**2*g*i**3 - 5*B*a*b**3*c**3*d*g*i**3 - B*b**4*c**4*g*i**3)/(20*b**3*d)

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Giac [B]  time = 21.1528, size = 690, normalized size = 2.55 \begin{align*} -\frac{1}{5} \,{\left (A b d^{3} g i + B b d^{3} g i\right )} x^{5} - \frac{1}{20} \,{\left (15 \, A b c d^{2} g i + 14 \, B b c d^{2} g i + 5 \, A a d^{3} g i + 6 \, B a d^{3} g i\right )} x^{4} - \frac{{\left (60 \, A b^{2} c^{2} d g i + 49 \, B b^{2} c^{2} d g i + 60 \, A a b c d^{2} g i + 70 \, B a b c d^{2} g i + B a^{2} d^{3} g i\right )} x^{3}}{60 \, b} - \frac{1}{20} \,{\left (4 \, B b d^{3} g i x^{5} + 20 \, B a c^{3} g i x + 5 \,{\left (3 \, B b c d^{2} g i + B a d^{3} g i\right )} x^{4} + 20 \,{\left (B b c^{2} d g i + B a c d^{2} g i\right )} x^{3} + 10 \,{\left (B b c^{3} g i + 3 \, B a c^{2} d g i\right )} x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (20 \, A b^{3} c^{3} g i + 11 \, B b^{3} c^{3} g i + 60 \, A a b^{2} c^{2} d g i + 65 \, B a b^{2} c^{2} d g i + 5 \, B a^{2} b c d^{2} g i - B a^{3} d^{3} g i\right )} x^{2}}{40 \, b^{2}} - \frac{{\left (B b c^{5} g i - 5 \, B a c^{4} d g i\right )} \log \left (-d i x - c i\right )}{20 \, d^{2}} + \frac{{\left (B b^{4} c^{4} g i - 20 \, A a b^{3} c^{3} d g i - 15 \, B a b^{3} c^{3} d g i - 10 \, B a^{2} b^{2} c^{2} d^{2} g i + 5 \, B a^{3} b c d^{3} g i - B a^{4} d^{4} g i\right )} x}{20 \, b^{3} d} - \frac{{\left (10 \, B a^{2} b^{3} c^{3} g i - 10 \, B a^{3} b^{2} c^{2} d g i + 5 \, B a^{4} b c d^{2} g i - B a^{5} d^{3} g i\right )} \log \left (b x + a\right )}{20 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

-1/5*(A*b*d^3*g*i + B*b*d^3*g*i)*x^5 - 1/20*(15*A*b*c*d^2*g*i + 14*B*b*c*d^2*g*i + 5*A*a*d^3*g*i + 6*B*a*d^3*g
*i)*x^4 - 1/60*(60*A*b^2*c^2*d*g*i + 49*B*b^2*c^2*d*g*i + 60*A*a*b*c*d^2*g*i + 70*B*a*b*c*d^2*g*i + B*a^2*d^3*
g*i)*x^3/b - 1/20*(4*B*b*d^3*g*i*x^5 + 20*B*a*c^3*g*i*x + 5*(3*B*b*c*d^2*g*i + B*a*d^3*g*i)*x^4 + 20*(B*b*c^2*
d*g*i + B*a*c*d^2*g*i)*x^3 + 10*(B*b*c^3*g*i + 3*B*a*c^2*d*g*i)*x^2)*log((b*x + a)/(d*x + c)) - 1/40*(20*A*b^3
*c^3*g*i + 11*B*b^3*c^3*g*i + 60*A*a*b^2*c^2*d*g*i + 65*B*a*b^2*c^2*d*g*i + 5*B*a^2*b*c*d^2*g*i - B*a^3*d^3*g*
i)*x^2/b^2 - 1/20*(B*b*c^5*g*i - 5*B*a*c^4*d*g*i)*log(-d*i*x - c*i)/d^2 + 1/20*(B*b^4*c^4*g*i - 20*A*a*b^3*c^3
*d*g*i - 15*B*a*b^3*c^3*d*g*i - 10*B*a^2*b^2*c^2*d^2*g*i + 5*B*a^3*b*c*d^3*g*i - B*a^4*d^4*g*i)*x/(b^3*d) - 1/
20*(10*B*a^2*b^3*c^3*g*i - 10*B*a^3*b^2*c^2*d*g*i + 5*B*a^4*b*c*d^2*g*i - B*a^5*d^3*g*i)*log(b*x + a)/b^4