3.2 \(\int (a g+b g x)^2 (c i+d i x) (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)
Optimal. Leaf size=180 \[ \frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A-B\right )}{12 b^2}+\frac{g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b}-\frac{B g^2 i (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac{B g^2 i (a+b x)^2 (b c-a d)^2}{24 b^2 d}+\frac{B g^2 i x (b c-a d)^3}{12 b d^2} \]
[Out]
(B*(b*c - a*d)^3*g^2*i*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*(a + b*x)^2)/(24*b^2*d) + (g^2*i*(a + b*x)^3*(c
+ d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A - B + B*Log[(e*(a + b*x
))/(c + d*x)]))/(12*b^2) - (B*(b*c - a*d)^4*g^2*i*Log[c + d*x])/(12*b^2*d^3)
________________________________________________________________________________________
Rubi [A] time = 0.293748, antiderivative size = 200, normalized size of antiderivative =
1.11, 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} \[ \frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2}+\frac{d g^2 i (a+b x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^2}-\frac{B g^2 i (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac{B g^2 i (a+b x)^2 (b c-a d)^2}{24 b^2 d}-\frac{B g^2 i (a+b x)^3 (b c-a d)}{12 b^2}+\frac{B g^2 i x (b c-a d)^3}{12 b d^2} \]
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
(B*(b*c - a*d)^3*g^2*i*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*(a + b*x)^2)/(24*b^2*d) - (B*(b*c - a*d)*g^2*i*(
a + b*x)^3)/(12*b^2) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^2) + (d*g^2*i
*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2) - (B*(b*c - a*d)^4*g^2*i*Log[c + d*x])/(12*b^2*d^3)
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 (2 c+2 d x) (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{2 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}+\frac{2 d (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g}\right ) \, dx\\ &=\frac{(2 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b}+\frac{(2 d) \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}\\ &=\frac{2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac{d g^2 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac{(B d) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{2 b^2 g^2}-\frac{(2 B (b c-a d)) \int \frac{(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b^2 g}\\ &=\frac{2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac{d g^2 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac{\left (B d (b c-a d) g^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{2 b^2}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b^2}\\ &=\frac{2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac{d g^2 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac{\left (B d (b c-a d) g^2\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{2 b^2}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac{B (b c-a d)^3 g^2 x}{6 b d^2}-\frac{B (b c-a d)^2 g^2 (a+b x)^2}{12 b^2 d}-\frac{B (b c-a d) g^2 (a+b x)^3}{6 b^2}+\frac{2 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac{d g^2 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2}-\frac{B (b c-a d)^4 g^2 \log (c+d x)}{6 b^2 d^3}\\ \end{align*}
Mathematica [A] time = 0.157149, size = 217, normalized size = 1.21 \[ \frac{g^2 i \left (6 d (a+b x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+8 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{4 B (b c-a d)^2 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )}{d^3}-\frac{B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{d^3}\right )}{24 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*d*(a + b*x)^4*(A + B*Log[(e*(a + b*
x))/(c + d*x)]) + (4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]))/d
^3 - (B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a
*d)^3*Log[c + d*x]))/d^3))/(24*b^2)
________________________________________________________________________________________
Maple [B] time = 0.202, size = 4593, normalized size = 25.5 \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)^2*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
-1/3*e/d^2*B*g^2*i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^3*a+1/2*e^3/d*B*g^2*i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)
^3*a^2*c^2+5/4*e^2/d*B*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*b^2*c^2+1/4*e^4*d*B*g^2*i*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^4+2/3*e^3*d*B*g^2*i*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)^3*a^4+1/4*e^4/d^3*B*g^2*i*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*b^6*c^4-8/3*e^3*B*g^2*i*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)^3*a^3*c-2*e^2*B
*g^2*i*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)^2*a^3*b*c-e^4*B*g^2*i*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^3*c-1/3*e^3/d^2*B*g^2*i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3
*c^3*a-5/6*e^2/d^2*B*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^3*c^3*a+3/2*e^4/d*A*g^2*i*b^4/(d*e/(d*x+c)*a-e/(d
*x+c)*b*c)^4*a^2*c^2+4*e^3/d*A*g^2*i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c^2+1/12/d^3*B*g^2*i*b^2*ln(d*(b*
e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^4-1/3*B*g^2*i/b*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3*c+5/24*e^2*d*B*g
^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^4+1/2*e^2*d*A*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^4+1/2/d*B*g^2*i*l
n(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2*c^2+1/12*d*B*g^2*i/b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^4-
1/3*e*B*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3*c-1/4*e^4*d^5*B*g^2*i/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^8/(d*x+c)^4-1/4*e^4/d^3*B*g^2*i*b^6*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)^4-1/2*e^2*d^3*B*g^2*i*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)^2*a^6/(d*x+c)^2-1/2*e^2/d^3*B*g^2*i*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
)^2*c^6/(d*x+c)^2-7*e^4*d^3*B*g^2*i*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+14*e^3*d^2*B*g^2*i*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^5/(d*x+c)^3*c^2-8
/3*e^3/d^2*B*g^2*i*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)^3*c^3*a+10*e^2*B*g^2*i*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)^2*a^3/(d*x+c)^2*c^3*b+14*e^4*B*g^2*i*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^3/(d*x+c)^4*c^5-e^4/d^2*A*g^2*i*b^5/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^4*c^3*a-8/3*e^3/d^2*A*g^2*i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3*a+5/24*e^2/d^3*B*g^2*i/(d*e/(
d*x+c)*a-e/(d*x+c)*b*c)^2*b^4*c^4+1/4*e^4/d^3*A*g^2*i*b^6/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*c^4+1/4*e^4*d*A*g^2*
i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^4+1/12*e^3/d^3*B*g^2*i*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^4-1/3*e^3
*B*g^2*i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3*c-5/6*e^2*B*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3*b*c-1/3
/d^2*B*g^2*i*b*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^3*a+1/2*e^2*d*B*g^2*i*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)^2*a^4+1/12*e/d^3*B*g^2*i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^4+1/12*e*d*B*g^2*i
/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^4+1/12*e^3*d*B*g^2*i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^4+1/2*e^2/d^3*A*g^
2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^4*c^4-8/3*e^3*A*g^2*i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3*c-e^4*A*g^
2*i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^4*a^3*c-2*e^2*A*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3*b*c+2/3*e^3*d*
A*g^2*i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^4+2/3*e^3/d^3*A*g^2*i*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^4+3*e^
2/d*A*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*b^2*c^2-2*e^2/d^2*A*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^3*
c^3*a+1/2*e/d*B*g^2*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c^2*b+1/2*e^2/d^3*B*g^2*i*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)^2*c^4+2/3*e^3/d^3*B*g^2*i*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)^3*c^4+14/3*e^3/d^2*B*g^2*i*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)^3*c^6/(d*x+c)^3*a+2*e^4*d^4*B*g^2*i/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^7/(
d*x+c)^4*c-14/3*e^3*d^3*B*g^2*i/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)^3*a^6/(d*x+c)^
3*c-35/2*e^4*d*B*g^2*i*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^4/(d*x+c)^4*c^4+3
*e^2*d^2*B*g^2*i*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)^2*a^5/(d*x+c)^2*c-14*e^3/d*B*
g^2*i*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)^3*c^5/(d*x+c)^3*a^2-15/2*e^2/d*B*g^2*i
*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)^2*c^4/(d*x+c)^2*a^2+2*e^4/d^2*B*g^2*i*b^5*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)^4*c^7/(d*x+c)^4*a+3*e^2/d^2*B*g^2*i*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)^2*c^5/(d*x+c)^2*a-70/3*e^3*d*B*g^2*i*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/(d*x+c)^3*c^3*b+14*e^4*d^2*B*g^2*i*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/(d*x+c)^4*c^3*b-7*e^4/d*B*g^2*i*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^6/(d*x+c)^4*a^2-15/2*e^2*d*B*g^2*i*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)^2*a^4/(d*x+c)^2*c^2-2*e^2/d^2*B*g^2*i*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)^2*c^3*a-e^4/d^2*B*g^2*i*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*b^5*c^3*
a+3/2*e^4/d*B*g^2*i*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^4*c^2+4*e^3/d*B*g^2*
i*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*b^3*c^2+2/3*e^3*d^4*B*g^2*i/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)^3*a^7/(d*x+c)^3+3*e^2/d*B*g^2*i*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)^2*a^2*c^2-2/3*e^3/d^3*B*g^2*i*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)^3*c^7/(d*x+c)^3+70/3*e^3*B*g^2*i*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)^3*c^4/(d*x+c)^3*a^3
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Maxima [B] time = 3.18149, size = 906, normalized size = 5.03 \begin{align*} \frac{1}{4} \, A b^{2} d g^{2} i x^{4} + \frac{1}{3} \, A b^{2} c g^{2} i x^{3} + \frac{2}{3} \, A a b d g^{2} i x^{3} + A a b c g^{2} i x^{2} + \frac{1}{2} \, A a^{2} d g^{2} i 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^{2} c g^{2} i +{\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 b c g^{2} i + \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 b^{2} c g^{2} i + \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 a^{2} d g^{2} i + \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 a b d g^{2} i + \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^{2} d g^{2} i + A a^{2} c g^{2} i x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")
[Out]
1/4*A*b^2*d*g^2*i*x^4 + 1/3*A*b^2*c*g^2*i*x^3 + 2/3*A*a*b*d*g^2*i*x^3 + A*a*b*c*g^2*i*x^2 + 1/2*A*a^2*d*g^2*i*
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^2*c*g^2*i + (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*b*c*
g^2*i + 1/6*(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^2*c*g^2*i + 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*a^2*d*g^2*i + 1/3*(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*b*d*g^2*i + 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^2*d*g^2*i + A*a^2*c*g^2*i*x
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Fricas [B] time = 1.20818, size = 774, normalized size = 4.3 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \,{\left ({\left (4 \, A - B\right )} b^{4} c d^{3} +{\left (8 \, A + B\right )} a b^{3} d^{4}\right )} g^{2} i x^{3} -{\left (B b^{4} c^{2} d^{2} - 4 \,{\left (6 \, A - B\right )} a b^{3} c d^{3} -{\left (12 \, A + 5 \, B\right )} a^{2} b^{2} d^{4}\right )} g^{2} i x^{2} + 2 \,{\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \,{\left (6 \, A + B\right )} a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i x + 2 \,{\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i \log \left (b x + a\right ) - 2 \,{\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i \log \left (d x + c\right ) + 2 \,{\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \,{\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \,{\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{24 \, b^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")
[Out]
1/24*(6*A*b^4*d^4*g^2*i*x^4 + 2*((4*A - B)*b^4*c*d^3 + (8*A + B)*a*b^3*d^4)*g^2*i*x^3 - (B*b^4*c^2*d^2 - 4*(6*
A - B)*a*b^3*c*d^3 - (12*A + 5*B)*a^2*b^2*d^4)*g^2*i*x^2 + 2*(B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 2*(6*A + B)*a^
2*b^2*c*d^3 + B*a^3*b*d^4)*g^2*i*x + 2*(4*B*a^3*b*c*d^3 - B*a^4*d^4)*g^2*i*log(b*x + a) - 2*(B*b^4*c^4 - 4*B*a
*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*g^2*i*log(d*x + c) + 2*(3*B*b^4*d^4*g^2*i*x^4 + 12*B*a^2*b^2*c*d^3*g^2*i*x +
4*(B*b^4*c*d^3 + 2*B*a*b^3*d^4)*g^2*i*x^3 + 6*(2*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*g^2*i*x^2)*log((b*e*x + a*e)/
(d*x + c)))/(b^2*d^3)
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Sympy [B] time = 6.82122, size = 870, normalized size = 4.83 \begin{align*} \frac{A b^{2} d g^{2} i x^{4}}{4} - \frac{B a^{3} g^{2} i \left (a d - 4 b c\right ) \log{\left (x + \frac{B a^{4} c d^{3} g^{2} i + \frac{B a^{4} d^{3} g^{2} i \left (a d - 4 b c\right )}{b} - 10 B a^{3} b c^{2} d^{2} g^{2} i - B a^{3} c d^{2} g^{2} i \left (a d - 4 b c\right ) + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 b^{2}} - \frac{B c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{B a^{4} c d^{3} g^{2} i - 10 B a^{3} b c^{2} d^{2} g^{2} i + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i + B a b c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) - \frac{B b^{2} c^{3} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 d^{3}} + x^{3} \left (\frac{2 A a b d g^{2} i}{3} + \frac{A b^{2} c g^{2} i}{3} + \frac{B a b d g^{2} i}{12} - \frac{B b^{2} c g^{2} i}{12}\right ) + \left (B a^{2} c g^{2} i x + \frac{B a^{2} d g^{2} i x^{2}}{2} + B a b c g^{2} i x^{2} + \frac{2 B a b d g^{2} i x^{3}}{3} + \frac{B b^{2} c g^{2} i x^{3}}{3} + \frac{B b^{2} d g^{2} i x^{4}}{4}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )} + \frac{x^{2} \left (12 A a^{2} d^{2} g^{2} i + 24 A a b c d g^{2} i + 5 B a^{2} d^{2} g^{2} i - 4 B a b c d g^{2} i - B b^{2} c^{2} g^{2} i\right )}{24 d} + \frac{x \left (12 A a^{2} b c d^{2} g^{2} i + B a^{3} d^{3} g^{2} i + 2 B a^{2} b c d^{2} g^{2} i - 4 B a b^{2} c^{2} d g^{2} i + B b^{3} c^{3} g^{2} i\right )}{12 b d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)**2*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
A*b**2*d*g**2*i*x**4/4 - B*a**3*g**2*i*(a*d - 4*b*c)*log(x + (B*a**4*c*d**3*g**2*i + B*a**4*d**3*g**2*i*(a*d -
4*b*c)/b - 10*B*a**3*b*c**2*d**2*g**2*i - B*a**3*c*d**2*g**2*i*(a*d - 4*b*c) + 4*B*a**2*b**2*c**3*d*g**2*i -
B*a*b**3*c**4*g**2*i)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a**2*b**2*c**2*d**2*g**2*i + 4*B*a*
b**3*c**3*d*g**2*i - B*b**4*c**4*g**2*i))/(12*b**2) - B*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*log(
x + (B*a**4*c*d**3*g**2*i - 10*B*a**3*b*c**2*d**2*g**2*i + 4*B*a**2*b**2*c**3*d*g**2*i - B*a*b**3*c**4*g**2*i
+ B*a*b*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2) - B*b**2*c**3*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2
*c**2)/d)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a**2*b**2*c**2*d**2*g**2*i + 4*B*a*b**3*c**3*d*
g**2*i - B*b**4*c**4*g**2*i))/(12*d**3) + x**3*(2*A*a*b*d*g**2*i/3 + A*b**2*c*g**2*i/3 + B*a*b*d*g**2*i/12 - B
*b**2*c*g**2*i/12) + (B*a**2*c*g**2*i*x + B*a**2*d*g**2*i*x**2/2 + B*a*b*c*g**2*i*x**2 + 2*B*a*b*d*g**2*i*x**3
/3 + B*b**2*c*g**2*i*x**3/3 + B*b**2*d*g**2*i*x**4/4)*log(e*(a + b*x)/(c + d*x)) + x**2*(12*A*a**2*d**2*g**2*i
+ 24*A*a*b*c*d*g**2*i + 5*B*a**2*d**2*g**2*i - 4*B*a*b*c*d*g**2*i - B*b**2*c**2*g**2*i)/(24*d) + x*(12*A*a**2
*b*c*d**2*g**2*i + B*a**3*d**3*g**2*i + 2*B*a**2*b*c*d**2*g**2*i - 4*B*a*b**2*c**2*d*g**2*i + B*b**3*c**3*g**2
*i)/(12*b*d**2)
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Giac [B] time = 14.0868, size = 555, normalized size = 3.08 \begin{align*} \frac{1}{4} \,{\left (A b^{2} d g^{2} i + B b^{2} d g^{2} i\right )} x^{4} + \frac{1}{12} \,{\left (4 \, A b^{2} c g^{2} i + 3 \, B b^{2} c g^{2} i + 8 \, A a b d g^{2} i + 9 \, B a b d g^{2} i\right )} x^{3} - \frac{{\left (B b^{2} c^{2} g^{2} i - 24 \, A a b c d g^{2} i - 20 \, B a b c d g^{2} i - 12 \, A a^{2} d^{2} g^{2} i - 17 \, B a^{2} d^{2} g^{2} i\right )} x^{2}}{24 \, d} + \frac{1}{12} \,{\left (3 \, B b^{2} d g^{2} i x^{4} + 12 \, B a^{2} c g^{2} i x + 4 \,{\left (B b^{2} c g^{2} i + 2 \, B a b d g^{2} i\right )} x^{3} + 6 \,{\left (2 \, B a b c g^{2} i + B a^{2} d g^{2} i\right )} x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right ) + \frac{{\left (4 \, B a^{3} b c g^{2} i - B a^{4} d g^{2} i\right )} \log \left (b x + a\right )}{12 \, b^{2}} + \frac{{\left (B b^{3} c^{3} g^{2} i - 4 \, B a b^{2} c^{2} d g^{2} i + 12 \, A a^{2} b c d^{2} g^{2} i + 14 \, B a^{2} b c d^{2} g^{2} i + B a^{3} d^{3} g^{2} i\right )} x}{12 \, b d^{2}} - \frac{{\left (B b^{2} c^{4} g^{2} i - 4 \, B a b c^{3} d g^{2} i + 6 \, B a^{2} c^{2} d^{2} g^{2} i\right )} \log \left (-d i x - c i\right )}{12 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
[Out]
1/4*(A*b^2*d*g^2*i + B*b^2*d*g^2*i)*x^4 + 1/12*(4*A*b^2*c*g^2*i + 3*B*b^2*c*g^2*i + 8*A*a*b*d*g^2*i + 9*B*a*b*
d*g^2*i)*x^3 - 1/24*(B*b^2*c^2*g^2*i - 24*A*a*b*c*d*g^2*i - 20*B*a*b*c*d*g^2*i - 12*A*a^2*d^2*g^2*i - 17*B*a^2
*d^2*g^2*i)*x^2/d + 1/12*(3*B*b^2*d*g^2*i*x^4 + 12*B*a^2*c*g^2*i*x + 4*(B*b^2*c*g^2*i + 2*B*a*b*d*g^2*i)*x^3 +
6*(2*B*a*b*c*g^2*i + B*a^2*d*g^2*i)*x^2)*log((b*x + a)/(d*x + c)) + 1/12*(4*B*a^3*b*c*g^2*i - B*a^4*d*g^2*i)*
log(b*x + a)/b^2 + 1/12*(B*b^3*c^3*g^2*i - 4*B*a*b^2*c^2*d*g^2*i + 12*A*a^2*b*c*d^2*g^2*i + 14*B*a^2*b*c*d^2*g
^2*i + B*a^3*d^3*g^2*i)*x/(b*d^2) - 1/12*(B*b^2*c^4*g^2*i - 4*B*a*b*c^3*d*g^2*i + 6*B*a^2*c^2*d^2*g^2*i)*log(-
d*i*x - c*i)/d^3