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

Optimal. Leaf size=239 \[ -\frac{g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d^2}+\frac{b g i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac{B g i^2 (b c-a d)^4 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^3 d^2}+\frac{B g i^2 (b c-a d)^4 \log (c+d x)}{12 b^3 d^2}+\frac{B g i^2 x (b c-a d)^3}{12 b^2 d}+\frac{B g i^2 (c+d x)^2 (b c-a d)^2}{24 b d^2}-\frac{B g i^2 (c+d x)^3 (b c-a d)}{12 d^2} \]

[Out]

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

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Rubi [A]  time = 0.341305, antiderivative size = 200, normalized size of antiderivative = 0.84, 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} \[ -\frac{g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d^2}+\frac{b g i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac{B g i^2 (b c-a d)^4 \log (a+b x)}{12 b^3 d^2}+\frac{B g i^2 x (b c-a d)^3}{12 b^2 d}+\frac{B g i^2 (c+d x)^2 (b c-a d)^2}{24 b d^2}-\frac{B g i^2 (c+d x)^3 (b c-a d)}{12 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^3*g*i^2*x)/(12*b^2*d) + (B*(b*c - a*d)^2*g*i^2*(c + d*x)^2)/(24*b*d^2) - (B*(b*c - a*d)*g*i^2*(
c + d*x)^3)/(12*d^2) + (B*(b*c - a*d)^4*g*i^2*Log[a + b*x])/(12*b^3*d^2) - ((b*c - a*d)*g*i^2*(c + d*x)^3*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^2) + (b*g*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*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 (12 c+12 d x)^2 (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 (12 c+12 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{b g (12 c+12 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{12 d}\right ) \, dx\\ &=\frac{(b g) \int (12 c+12 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{12 d}+\frac{((-b c+a d) g) \int (12 c+12 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d}\\ &=-\frac{48 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{36 b g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac{(b B g) \int \frac{20736 (b c-a d) (c+d x)^3}{a+b x} \, dx}{576 d^2}+\frac{(B (b c-a d) g) \int \frac{1728 (b c-a d) (c+d x)^2}{a+b x} \, dx}{36 d^2}\\ &=-\frac{48 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{36 b g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac{(36 b B (b c-a d) g) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^2}+\frac{\left (48 B (b c-a d)^2 g\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{d^2}\\ &=-\frac{48 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{36 b g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac{(36 b B (b c-a d) g) \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{\left (48 B (b c-a d)^2 g\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{d^2}\\ &=\frac{12 B (b c-a d)^3 g x}{b^2 d}+\frac{6 B (b c-a d)^2 g (c+d x)^2}{b d^2}-\frac{12 B (b c-a d) g (c+d x)^3}{d^2}+\frac{12 B (b c-a d)^4 g \log (a+b x)}{b^3 d^2}-\frac{48 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac{36 b g (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.182512, size = 216, normalized size = 0.9 \[ \frac{g i^2 \left (6 b (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-8 (c+d x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{4 B (b c-a d)^2 \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )}{b^3}-\frac{B (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^3}\right )}{24 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^2*((4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]))/b^3 - (B*(b
*c - a*d)*(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^3 - 8*(b*c - a*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*b*(c + d*x)^4*(A + B*Log[(e*(
a + b*x))/(c + d*x)])))/(24*d^2)

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Maple [B]  time = 0.173, size = 3439, normalized size = 14.4 \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)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

1/24*e^2/d^2*B*g*i^2*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^4-1/12*e*d^2*B*g*i^2/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b
*c)*a^4+1/4*e^4*d^2*A*g*i^2*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4-1/6*e^2*d*B*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b
*c)^2*a^3*c+1/3*e^3*d^2*B*g*i^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)^3*a^4+1/3*d*B*g*
i^2/b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3*c+1/24*e^2*d^2*B*g*i^2/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a
^4+1/12*e^3/d^2*B*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*b^4*c^4+1/3*e^3/d^2*A*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b
*c)^3*b^4*c^4+1/4*e^4/d^2*A*g*i^2*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^4-1/12*e/d^2*B*g*i^2*b^2/(d*e/(d*x+c)*
a-e/(d*x+c)*b*c)*c^4+2*e^3*A*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*b^2*c^2+3/2*e^4*A*g*i^2*b^3/(d*e/(d*x+c
)*a-e/(d*x+c)*b*c)^4*a^2*c^2+1/2*e^3*B*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*b^2*c^2+1/4*e^2*B*g*i^2/(d*e/
(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*c^2*b-1/4*e^4*d^6*B*g*i^2/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)^4*a^8/(d*x+c)^4+14*e^4*d^3*B*g*i^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)^
4*c^3/(d*x+c)^4*a^5-1/2*B*g*i^2/b*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2*c^2+1/12*e^3*d^2*B*g*i^2/(d*e/(d
*x+c)*a-e/(d*x+c)*b*c)^3*a^4+1/3/d*B*g*i^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^3*a-1/12*d^2*B*g*i^2/b^3*
ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^4-1/12/d^2*B*g*i^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^4*b-1/2
*e*B*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c^2+1/3*e^3*d^2*A*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^4+35/3*
e^3*d*B*g*i^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)^3*a^3*c^4/(d*x+c)^3*b+14*e^4*d*B*g
*i^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)^4*c^5/(d*x+c)^4*a^3*b^2+7/3*e^3/d*B*g*i^2*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^6/(d*x+c)^3*b^3*a+2*e^4*d^5*B*g*i^2/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)^4*a^7/(d*x+c)^4*c+2*e^4/d*B*g*i^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)^4*c^7/(d*x+c)^4*b^4*a-35/2*e^4*d^2*B*g*i^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)^4*c^4/(d*x+c)^4*a^4*b-7*e^4*d^4*B*g*i^2/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)^4*a^6/(d*x+c)^4*c^2-7/3*e^3*d^4*B*g*i^2*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)^3*a^6*c/(d*x+c)^3+7*e^3*d^3*B*g*i^2*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)^3*a^5*c^2/(d*x+c)^3+1/3*e*d*B*g*i^2/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3*c+1/3*e/d*B*g*i^2*b/
(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^3*a-e^4/d*A*g*i^2*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^3*a-1/3*e^3/d^2*B*g*i^
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)^3*c^7/(d*x+c)^3*b^4-1/4*e^4/d^2*B*g*i^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)^4*c^8/(d*x+c)^4*b^5-7*e^3*B*g*i^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)^3*c^5/(d*x+c)^3*a^2*b^2-7*e^4*B*g*i^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)^4*c^6/(d*x+c)^4*a^2*b^3-e^4*d*B*g*i^2*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)^4*a^3*c-e^4/d*B*g*i^2*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)^4*c^3*a-4/3*e^3*d*B*g*i^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)^3*a^3*b*c+1/3*e^3*
d^5*B*g*i^2*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)^3*a^7/(d*x+c)^3-35/3*e^3*d^2*B*g
*i^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)^3*a^4*c^3/(d*x+c)^3-4/3*e^3/d*B*g*i^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)^3*a*c^3*b^3-1/3*e^3*d*B*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)^3*a^3*b*c-4/3*e^3*d*A*g*i^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3*b*c-1/3*e^3/d*B*g*i^2/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^3*b^3*c^3*a-1/6*e^2/d*B*g*i^2*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^3*a-4/3*e^3/d*A*g*i^2/(d*e/(d
*x+c)*a-e/(d*x+c)*b*c)^3*b^3*c^3*a-e^4*d*A*g*i^2*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*c+3/2*e^4*B*g*i^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)^4*a^2*b^3*c^2+2*e^3*B*g*i^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)^3*a^2*c^2*b^2+1/4*e^4/d^2*B*g*i^2*b^5*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+1/3*e^3/d^2*B*g*i^2*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)^3*c^4+1/4*e^4*d^2*B*g*i^2*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)^4*a^4

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Maxima [B]  time = 1.53032, size = 906, normalized size = 3.79 \begin{align*} \frac{1}{4} \, A b d^{2} g i^{2} x^{4} + \frac{2}{3} \, A b c d g i^{2} x^{3} + \frac{1}{3} \, A a d^{2} g i^{2} x^{3} + \frac{1}{2} \, A b c^{2} g i^{2} x^{2} + A a c d g i^{2} x^{2} +{\left (x \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} B a c^{2} g i^{2} + \frac{1}{2} \,{\left (x^{2} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{a^{2} \log \left (b x + a\right )}{b^{2}} + \frac{c^{2} \log \left (d x + c\right )}{d^{2}} - \frac{{\left (b c - a d\right )} x}{b d}\right )} B b c^{2} g i^{2} +{\left (x^{2} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{a^{2} \log \left (b x + a\right )}{b^{2}} + \frac{c^{2} \log \left (d x + c\right )}{d^{2}} - \frac{{\left (b c - a d\right )} x}{b d}\right )} B a c d g i^{2} + \frac{1}{3} \,{\left (2 \, x^{3} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b c d g i^{2} + \frac{1}{6} \,{\left (2 \, x^{3} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a d^{2} g i^{2} + \frac{1}{24} \,{\left (6 \, x^{4} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac{6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac{2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b d^{2} g i^{2} + A a c^{2} g i^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*A*b*d^2*g*i^2*x^4 + 2/3*A*b*c*d*g*i^2*x^3 + 1/3*A*a*d^2*g*i^2*x^3 + 1/2*A*b*c^2*g*i^2*x^2 + A*a*c*d*g*i^2*
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^2*g*i^2 + 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^2*g*i^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*d*g*i^2 + 1/3*(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*d*g*i^2 + 1/6*(2*x^3*l
og(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*d^2*g*i^2 + 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*b*d^2*g*i^2 + A*a*c^2*g*i^2*x

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Fricas [A]  time = 1.54713, size = 774, normalized size = 3.24 \begin{align*} \frac{6 \, A b^{4} d^{4} g i^{2} x^{4} + 2 \,{\left ({\left (8 \, A - B\right )} b^{4} c d^{3} +{\left (4 \, A + B\right )} a b^{3} d^{4}\right )} g i^{2} x^{3} +{\left ({\left (12 \, A - 5 \, B\right )} b^{4} c^{2} d^{2} + 4 \,{\left (6 \, A + B\right )} a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g i^{2} x^{2} - 2 \,{\left (B b^{4} c^{3} d - 2 \,{\left (6 \, A - B\right )} a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g i^{2} x + 2 \,{\left (6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} g i^{2} \log \left (b x + a\right ) + 2 \,{\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} g i^{2} \log \left (d x + c\right ) + 2 \,{\left (3 \, B b^{4} d^{4} g i^{2} x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} x + 4 \,{\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} x^{3} + 6 \,{\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{24 \, b^{3} 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)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 6.60334, size = 870, normalized size = 3.64 \begin{align*} \frac{A b d^{2} g i^{2} x^{4}}{4} + \frac{B a^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) \log{\left (x + \frac{B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + \frac{B a^{3} d^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{b} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a^{2} c d g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) - B a b^{3} c^{4} g i^{2}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 b^{3}} - \frac{B c^{3} g i^{2} \left (4 a d - b c\right ) \log{\left (x + \frac{B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a b^{3} c^{4} g i^{2} - B a b^{2} c^{3} g i^{2} \left (4 a d - b c\right ) + \frac{B b^{3} c^{4} g i^{2} \left (4 a d - b c\right )}{d}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 d^{2}} + x^{3} \left (\frac{A a d^{2} g i^{2}}{3} + \frac{2 A b c d g i^{2}}{3} + \frac{B a d^{2} g i^{2}}{12} - \frac{B b c d g i^{2}}{12}\right ) + \left (B a c^{2} g i^{2} x + B a c d g i^{2} x^{2} + \frac{B a d^{2} g i^{2} x^{3}}{3} + \frac{B b c^{2} g i^{2} x^{2}}{2} + \frac{2 B b c d g i^{2} x^{3}}{3} + \frac{B b d^{2} g i^{2} x^{4}}{4}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )} + \frac{x^{2} \left (24 A a b c d g i^{2} + 12 A b^{2} c^{2} g i^{2} + B a^{2} d^{2} g i^{2} + 4 B a b c d g i^{2} - 5 B b^{2} c^{2} g i^{2}\right )}{24 b} - \frac{x \left (- 12 A a b^{2} c^{2} d g i^{2} + B a^{3} d^{3} g i^{2} - 4 B a^{2} b c d^{2} g i^{2} + 2 B a b^{2} c^{2} d g i^{2} + B b^{3} c^{3} g i^{2}\right )}{12 b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*d**2*g*i**2*x**4/4 + B*a**2*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c**2)*log(x + (B*a**4*c*d**3*g*i**2 - 4
*B*a**3*b*c**2*d**2*g*i**2 + B*a**3*d**2*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c**2)/b + 10*B*a**2*b**2*c**3*
d*g*i**2 - B*a**2*c*d*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c**2) - B*a*b**3*c**4*g*i**2)/(B*a**4*d**4*g*i**2
- 4*B*a**3*b*c*d**3*g*i**2 + 6*B*a**2*b**2*c**2*d**2*g*i**2 + 4*B*a*b**3*c**3*d*g*i**2 - B*b**4*c**4*g*i**2))
/(12*b**3) - B*c**3*g*i**2*(4*a*d - b*c)*log(x + (B*a**4*c*d**3*g*i**2 - 4*B*a**3*b*c**2*d**2*g*i**2 + 10*B*a*
*2*b**2*c**3*d*g*i**2 - B*a*b**3*c**4*g*i**2 - B*a*b**2*c**3*g*i**2*(4*a*d - b*c) + B*b**3*c**4*g*i**2*(4*a*d
- b*c)/d)/(B*a**4*d**4*g*i**2 - 4*B*a**3*b*c*d**3*g*i**2 + 6*B*a**2*b**2*c**2*d**2*g*i**2 + 4*B*a*b**3*c**3*d*
g*i**2 - B*b**4*c**4*g*i**2))/(12*d**2) + x**3*(A*a*d**2*g*i**2/3 + 2*A*b*c*d*g*i**2/3 + B*a*d**2*g*i**2/12 -
B*b*c*d*g*i**2/12) + (B*a*c**2*g*i**2*x + B*a*c*d*g*i**2*x**2 + B*a*d**2*g*i**2*x**3/3 + B*b*c**2*g*i**2*x**2/
2 + 2*B*b*c*d*g*i**2*x**3/3 + B*b*d**2*g*i**2*x**4/4)*log(e*(a + b*x)/(c + d*x)) + x**2*(24*A*a*b*c*d*g*i**2 +
12*A*b**2*c**2*g*i**2 + B*a**2*d**2*g*i**2 + 4*B*a*b*c*d*g*i**2 - 5*B*b**2*c**2*g*i**2)/(24*b) - x*(-12*A*a*b
**2*c**2*d*g*i**2 + B*a**3*d**3*g*i**2 - 4*B*a**2*b*c*d**2*g*i**2 + 2*B*a*b**2*c**2*d*g*i**2 + B*b**3*c**3*g*i
**2)/(12*b**2*d)

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Giac [A]  time = 14.4937, size = 441, normalized size = 1.85 \begin{align*} -\frac{1}{4} \,{\left (A b d^{2} g + B b d^{2} g\right )} x^{4} - \frac{1}{12} \,{\left (8 \, A b c d g + 7 \, B b c d g + 4 \, A a d^{2} g + 5 \, B a d^{2} g\right )} x^{3} - \frac{{\left (12 \, A b^{2} c^{2} g + 7 \, B b^{2} c^{2} g + 24 \, A a b c d g + 28 \, B a b c d g + B a^{2} d^{2} g\right )} x^{2}}{24 \, b} - \frac{1}{12} \,{\left (3 \, B b d^{2} g x^{4} + 12 \, B a c^{2} g x + 4 \,{\left (2 \, B b c d g + B a d^{2} g\right )} x^{3} + 6 \,{\left (B b c^{2} g + 2 \, B a c d g\right )} x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b c^{4} g - 4 \, B a c^{3} d g\right )} \log \left (-d x - c\right )}{12 \, d^{2}} + \frac{{\left (B b^{3} c^{3} g - 12 \, A a b^{2} c^{2} d g - 10 \, B a b^{2} c^{2} d g - 4 \, B a^{2} b c d^{2} g + B a^{3} d^{3} g\right )} x}{12 \, b^{2} d} - \frac{{\left (6 \, B a^{2} b^{2} c^{2} g - 4 \, B a^{3} b c d g + B a^{4} d^{2} g\right )} \log \left (b x + a\right )}{12 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(A*b*d^2*g + B*b*d^2*g)*x^4 - 1/12*(8*A*b*c*d*g + 7*B*b*c*d*g + 4*A*a*d^2*g + 5*B*a*d^2*g)*x^3 - 1/24*(12
*A*b^2*c^2*g + 7*B*b^2*c^2*g + 24*A*a*b*c*d*g + 28*B*a*b*c*d*g + B*a^2*d^2*g)*x^2/b - 1/12*(3*B*b*d^2*g*x^4 +
12*B*a*c^2*g*x + 4*(2*B*b*c*d*g + B*a*d^2*g)*x^3 + 6*(B*b*c^2*g + 2*B*a*c*d*g)*x^2)*log((b*x + a)/(d*x + c)) -
1/12*(B*b*c^4*g - 4*B*a*c^3*d*g)*log(-d*x - c)/d^2 + 1/12*(B*b^3*c^3*g - 12*A*a*b^2*c^2*d*g - 10*B*a*b^2*c^2*
d*g - 4*B*a^2*b*c*d^2*g + B*a^3*d^3*g)*x/(b^2*d) - 1/12*(6*B*a^2*b^2*c^2*g - 4*B*a^3*b*c*d*g + B*a^4*d^2*g)*lo
g(b*x + a)/b^3