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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.29, 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 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])

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 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]

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 = 0.16, 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)

________________________________________________________________________________________

fricas [B]  time = 1.05, size = 370, normalized size = 2.06 \[ \frac {6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \, {\left ({\left (4 \, A - B\right )} b^{4} c d^{3} + {\left (8 \, A + B\right )} a b^{3} d^{4}\right )} g^{2} i x^{3} - {\left (B b^{4} c^{2} d^{2} - 4 \, {\left (6 \, A - B\right )} a b^{3} c d^{3} - {\left (12 \, A + 5 \, B\right )} a^{2} b^{2} d^{4}\right )} g^{2} i x^{2} + 2 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \, {\left (6 \, A + B\right )} a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i x + 2 \, {\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i \log \left (b x + a\right ) - 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i \log \left (d x + c\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b^{2} d^{3}} \]

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)

________________________________________________________________________________________

giac [B]  time = 1.19, size = 3993, normalized size = 22.18 \[ \text {result too large to display} \]

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]

1/24*(2*B*b^9*c^5*g^2*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 10*B*a*b^8*c^4*d*g^2*i*e^5*log(-b*e + (b*x
*e + a*e)*d/(d*x + c)) + 20*B*a^2*b^7*c^3*d^2*g^2*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 20*B*a^3*b^6*c
^2*d^3*g^2*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 10*B*a^4*b^5*c*d^4*g^2*i*e^5*log(-b*e + (b*x*e + a*e)
*d/(d*x + c)) - 2*B*a^5*b^4*d^5*g^2*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*(b*x*e + a*e)*B*b^8*c^5*d*
g^2*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 40*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*i*e^4*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 80*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*i*e^4*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c) + 80*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))
/(d*x + c) - 40*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 8*(b
*x*e + a*e)*B*a^5*b^3*d^6*g^2*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 12*(b*x*e + a*e)^2*B*b^7
*c^5*d^2*g^2*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 60*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^2*
i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 120*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^2*i*e^3*log(
-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^2*i*e^3*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 60*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^2*i*e^3*log(-b*e + (b*x*e + a*e)*d/
(d*x + c))/(d*x + c)^2 - 12*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g^2*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^2 - 8*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 40*(b*
x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 80*(b*x*e + a*e)^3*
B*a^2*b^4*c^3*d^5*g^2*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 80*(b*x*e + a*e)^3*B*a^3*b^3*c
^2*d^6*g^2*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 40*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^2*i*
e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 8*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*i*e^2*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 2*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^4 - 10*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^4 + 20*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^2*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 20*(b*x*
e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 10*(b*x*e + a*e)^4*B*
a^4*b*c*d^8*g^2*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 2*(b*x*e + a*e)^4*B*a^5*d^9*g^2*i*e*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 8*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*i*e^2*log((b*x*e + a*e)/
(d*x + c))/(d*x + c)^3 - 40*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3
+ 80*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^2*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 80*(b*x*e + a*e)^
3*B*a^3*b^3*c^2*d^6*g^2*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 40*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^
2*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 8*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*i*e^2*log((b*x*e + a*e)/(
d*x + c))/(d*x + c)^3 - 2*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 10*
(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 20*(b*x*e + a*e)^4*B*a^2*b^
3*c^3*d^6*g^2*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 20*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*i*e*log(
(b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 10*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^2*i*e*log((b*x*e + a*e)/(d*x + c))/(
d*x + c)^4 + 2*(b*x*e + a*e)^4*B*a^5*d^9*g^2*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 2*A*b^9*c^5*g^2*i*
e^5 + B*b^9*c^5*g^2*i*e^5 - 10*A*a*b^8*c^4*d*g^2*i*e^5 - 5*B*a*b^8*c^4*d*g^2*i*e^5 + 20*A*a^2*b^7*c^3*d^2*g^2*
i*e^5 + 10*B*a^2*b^7*c^3*d^2*g^2*i*e^5 - 20*A*a^3*b^6*c^2*d^3*g^2*i*e^5 - 10*B*a^3*b^6*c^2*d^3*g^2*i*e^5 + 10*
A*a^4*b^5*c*d^4*g^2*i*e^5 + 5*B*a^4*b^5*c*d^4*g^2*i*e^5 - 2*A*a^5*b^4*d^5*g^2*i*e^5 - B*a^5*b^4*d^5*g^2*i*e^5
- 8*(b*x*e + a*e)*A*b^8*c^5*d*g^2*i*e^4/(d*x + c) - 2*(b*x*e + a*e)*B*b^8*c^5*d*g^2*i*e^4/(d*x + c) + 40*(b*x*
e + a*e)*A*a*b^7*c^4*d^2*g^2*i*e^4/(d*x + c) + 10*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*i*e^4/(d*x + c) - 80*(b*x*
e + a*e)*A*a^2*b^6*c^3*d^3*g^2*i*e^4/(d*x + c) - 20*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*i*e^4/(d*x + c) + 80*(
b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g^2*i*e^4/(d*x + c) + 20*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*i*e^4/(d*x + c) -
40*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g^2*i*e^4/(d*x + c) - 10*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*i*e^4/(d*x + c) +
8*(b*x*e + a*e)*A*a^5*b^3*d^6*g^2*i*e^4/(d*x + c) + 2*(b*x*e + a*e)*B*a^5*b^3*d^6*g^2*i*e^4/(d*x + c) + 12*(b*
x*e + a*e)^2*A*b^7*c^5*d^2*g^2*i*e^3/(d*x + c)^2 - (b*x*e + a*e)^2*B*b^7*c^5*d^2*g^2*i*e^3/(d*x + c)^2 - 60*(b
*x*e + a*e)^2*A*a*b^6*c^4*d^3*g^2*i*e^3/(d*x + c)^2 + 5*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^2*i*e^3/(d*x + c)^2
+ 120*(b*x*e + a*e)^2*A*a^2*b^5*c^3*d^4*g^2*i*e^3/(d*x + c)^2 - 10*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^2*i*e^3
/(d*x + c)^2 - 120*(b*x*e + a*e)^2*A*a^3*b^4*c^2*d^5*g^2*i*e^3/(d*x + c)^2 + 10*(b*x*e + a*e)^2*B*a^3*b^4*c^2*
d^5*g^2*i*e^3/(d*x + c)^2 + 60*(b*x*e + a*e)^2*A*a^4*b^3*c*d^6*g^2*i*e^3/(d*x + c)^2 - 5*(b*x*e + a*e)^2*B*a^4
*b^3*c*d^6*g^2*i*e^3/(d*x + c)^2 - 12*(b*x*e + a*e)^2*A*a^5*b^2*d^7*g^2*i*e^3/(d*x + c)^2 + (b*x*e + a*e)^2*B*
a^5*b^2*d^7*g^2*i*e^3/(d*x + c)^2 + 2*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*i*e^2/(d*x + c)^3 - 10*(b*x*e + a*e)^3
*B*a*b^5*c^4*d^4*g^2*i*e^2/(d*x + c)^3 + 20*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^2*i*e^2/(d*x + c)^3 - 20*(b*x*
e + a*e)^3*B*a^3*b^3*c^2*d^6*g^2*i*e^2/(d*x + c)^3 + 10*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^2*i*e^2/(d*x + c)^3
- 2*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*i*e^2/(d*x + c)^3)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*
e)*(b*c - a*d)))/(b^6*d^3*e^4 - 4*(b*x*e + a*e)*b^5*d^4*e^3/(d*x + c) + 6*(b*x*e + a*e)^2*b^4*d^5*e^2/(d*x + c
)^2 - 4*(b*x*e + a*e)^3*b^3*d^6*e/(d*x + c)^3 + (b*x*e + a*e)^4*b^2*d^7/(d*x + c)^4)

________________________________________________________________________________________

maple [B]  time = 0.16, size = 4593, normalized size = 25.52 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/d^3*B*g^2*i*b^2*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*c^4+1/12*d*B*g^2*i/b^2*ln(-b*e+(b/d*e+(a*d-b*c)/
(d*x+c)/d*e)*d)*a^4+1/2/d*B*g^2*i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^2*c^2-1/3*B*g^2*i/b*ln(-b*e+(b/d*
e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^3*c-1/3*e*B*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^3*c+5/24*d*e^2*B*g^2*i/(1/
(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^4+1/2*d*e^2*A*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^4+1/12*d*e*B*g^2*
i/b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^4+1/12*d*e^3*B*g^2*i*b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^4+1/12/d^
3*e*B*g^2*i*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^4+2/3*d*e^3*A*g^2*i*b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*
a^4+2/3/d^3*e^3*A*g^2*i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^4+1/4/d^3*e^4*A*g^2*i*b^6/(1/(d*x+c)*a*d*e-1
/(d*x+c)*b*c*e)^4*c^4+5/24/d^3*e^2*B*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*b^4*c^4+1/2/d^3*e^2*A*g^2*i/(1/
(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*b^4*c^4+1/12/d^3*e^3*B*g^2*i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^4+1/4*
d*e^4*A*g^2*i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^4-1/3*e^3*B*g^2*i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e
)^3*a^3*c-5/6*e^2*B*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^3*b*c-2*e^2*A*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)
*b*c*e)^2*a^3*b*c-8/3*e^3*A*g^2*i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^3*c-e^4*A*g^2*i*b^3/(1/(d*x+c)*a*d
*e-1/(d*x+c)*b*c*e)^4*a^3*c+1/2*d*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e
)^2*a^4-1/3/d^2*B*g^2*i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a*c^3*b-1/2/d^3*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c
)/(d*x+c)/d*e)*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^6/(d*x+c)^2+14*e^4*B*g^2*i*b^3*ln(b/d*e+(a*d-b*c)/(d*
x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^3*c^5/(d*x+c)^4+10*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^3*c^3/(d*x+c)^2*b+70/3*e^3*B*g^2*i*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/
(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^3*c^4/(d*x+c)^3-2/3/d^3*e^3*B*g^2*i*b^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/
(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^7/(d*x+c)^3+4/d*e^3*B*g^2*i*b^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+
c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c^2-8/3/d^2*e^3*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*
x+c)*b*c*e)^3*a*b^4*c^3-1/d^2*e^4*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*
a*b^5*c^3+3/d*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*c^2+2/3*
d^4*e^3*B*g^2*i/b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^7/(d*x+c)^3-1/4*d^5*
e^4*B*g^2*i/b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^8/(d*x+c)^4+3/2/d*e^4*B*
g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^2*c^2*b^4-14/d*e^3*B*g^2*i*b^3*ln(
b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c^5/(d*x+c)^3-7/d*e^4*B*g^2*i*b^4*ln(b/d*
e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*c^6/(d*x+c)^4*a^2-15/2/d*e^2*B*g^2*i*ln(b/d*e+(a*
d-b*c)/(d*x+c)/d*e)*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^4/(d*x+c)^2*a^2+14*d^2*e^4*B*g^2*i*ln(b/d*e+(a*d
-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^5*c^3/(d*x+c)^4*b-35/2*d*e^4*B*g^2*i*b^2*ln(b/d*e+(a*
d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^4*c^4/(d*x+c)^4+3*d^2*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)
/(d*x+c)/d*e)/b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^5*c/(d*x+c)^2+2/d^2*e^4*B*g^2*i*b^5*ln(b/d*e+(a*d-b*c)/(
d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*c^7/(d*x+c)^4*a+3/d^2*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d
*e)*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^5/(d*x+c)^2*a+14/3/d^2*e^3*B*g^2*i*b^4*ln(b/d*e+(a*d-b*c)/(d*x+c
)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^6/(d*x+c)^3*a-70/3*d*e^3*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/
(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^4*c^3/(d*x+c)^3*b-14/3*d^3*e^3*B*g^2*i/b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^6/(d*x+c)^3*c+2*d^4*e^4*B*g^2*i/b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d
*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^7/(d*x+c)^4*c+14*d^2*e^3*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a
*d*e-1/(d*x+c)*b*c*e)^3*a^5*c^2/(d*x+c)^3-7*d^3*e^4*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1
/(d*x+c)*b*c*e)^4*a^6*c^2/(d*x+c)^4-2/d^2*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^3/(1/(d*x+c)*a*d*e-1/(
d*x+c)*b*c*e)^2*c^3*a-1/2*d^3*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e
)^2*a^6/(d*x+c)^2-15/2*d*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^4*c
^2/(d*x+c)^2-1/4/d^3*e^4*B*g^2*i*b^6*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*c^8/(
d*x+c)^4+5/4/d*e^2*B*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*b^2*c^2+1/2/d*e^3*B*g^2*i*b^3/(1/(d*x+c)*a*
d*e-1/(d*x+c)*b*c*e)^3*a^2*c^2-8/3*e^3*B*g^2*i*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*
b*c*e)^3*a^3*c+2/3/d^3*e^3*B*g^2*i*b^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^4
+3/2/d*e^4*A*g^2*i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^2*c^2+3/d*e^2*A*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*
b*c*e)^2*a^2*c^2*b^2-5/6/d^2*e^2*B*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*b^3*c^3*a-2*e^2*B*g^2*i*ln(b/d*e+
(a*d-b*c)/(d*x+c)/d*e)*b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^3*c-1/3/d^2*e*B*g^2*i*b^2/(1/(d*x+c)*a*d*e-1/(d
*x+c)*b*c*e)*a*c^3-8/3/d^2*e^3*A*g^2*i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a*c^3+1/2/d*e*B*g^2*i*b/(1/(d*x
+c)*a*d*e-1/(d*x+c)*b*c*e)*a^2*c^2-1/3/d^2*e^3*B*g^2*i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a*c^3-1/d^2*e^4
*A*g^2*i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a*c^3-2/d^2*e^2*A*g^2*i/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a
*b^3*c^3+4/d*e^3*A*g^2*i*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c^2-e^4*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c
)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^3*c*b^3+2/3*d*e^3*B*g^2*i*b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d
*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^4+1/4*d*e^4*B*g^2*i*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d
*x+c)*b*c*e)^4*a^4+1/4/d^3*e^4*B*g^2*i*b^6*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4
*c^4+1/2/d^3*e^2*B*g^2*i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^4

________________________________________________________________________________________

maxima [B]  time = 1.23, size = 671, normalized size = 3.73 \[ \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 \]

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

________________________________________________________________________________________

mupad [B]  time = 5.11, size = 638, normalized size = 3.54 \[ x^3\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )-x^2\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{6\,d}+\frac {A\,a\,b\,c\,g^2\,i}{2}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,c\,g^2\,i\,x+\frac {B\,a\,g^2\,i\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {B\,b\,g^2\,i\,x^3\,\left (2\,a\,d+b\,c\right )}{3}+\frac {B\,b^2\,d\,g^2\,i\,x^4}{4}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{3\,d}+A\,a\,b\,c\,g^2\,i\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {a\,g^2\,i\,\left (2\,A\,a^2\,d^2+6\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+12\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{2\,b\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,i\,a^2\,c^2\,d^2\,g^2-4\,B\,i\,a\,b\,c^3\,d\,g^2+B\,i\,b^2\,c^4\,g^2\right )}{12\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d\,g^2\,i-4\,B\,a^3\,b\,c\,g^2\,i\right )}{12\,b^2}+\frac {A\,b^2\,d\,g^2\,i\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

x^3*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/12 - (A*b*g^2*i*(12*a*d + 12*b*c))/36) - x^2*((((b*g^2*i*(
12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(24*b*d) - (g^2*
i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d - B*a*b*c*d))/(6*d) + (A*a*b*c*g^2*i)/2)
+ log((e*(a + b*x))/(c + d*x))*(B*a^2*c*g^2*i*x + (B*a*g^2*i*x^2*(a*d + 2*b*c))/2 + (B*b*g^2*i*x^3*(2*a*d + b
*c))/3 + (B*b^2*d*g^2*i*x^4)/4) + x*(((12*a*d + 12*b*c)*((((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 -
(A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(12*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^
2 - B*b^2*c^2 + 18*A*a*b*c*d - B*a*b*c*d))/(3*d) + A*a*b*c*g^2*i))/(12*b*d) - (a*c*((b*g^2*i*(12*A*a*d + 8*A*b
*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(b*d) + (a*g^2*i*(2*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^
2*d^2 - 2*B*b^2*c^2 + 12*A*a*b*c*d + B*a*b*c*d))/(2*b*d)) - (log(c + d*x)*(B*b^2*c^4*g^2*i + 6*B*a^2*c^2*d^2*g
^2*i - 4*B*a*b*c^3*d*g^2*i))/(12*d^3) - (log(a + b*x)*(B*a^4*d*g^2*i - 4*B*a^3*b*c*g^2*i))/(12*b^2) + (A*b^2*d
*g^2*i*x^4)/4

________________________________________________________________________________________

sympy [B]  time = 5.11, size = 850, normalized size = 4.72 \[ \frac {A b^{2} d g^{2} i x^{4}}{4} - \frac {B a^{3} g^{2} i \left (a d - 4 b c\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i + \frac {B a^{4} d^{3} g^{2} i \left (a d - 4 b c\right )}{b} - 10 B a^{3} b c^{2} d^{2} g^{2} i - B a^{3} c d^{2} g^{2} i \left (a d - 4 b c\right ) + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 b^{2}} - \frac {B c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i - 10 B a^{3} b c^{2} d^{2} g^{2} i + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i + B a b c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) - \frac {B b^{2} c^{3} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 d^{3}} + x^{3} \left (\frac {2 A a b d g^{2} i}{3} + \frac {A b^{2} c g^{2} i}{3} + \frac {B a b d g^{2} i}{12} - \frac {B b^{2} c g^{2} i}{12}\right ) + x^{2} \left (\frac {A a^{2} d g^{2} i}{2} + A a b c g^{2} i + \frac {5 B a^{2} d g^{2} i}{24} - \frac {B a b c g^{2} i}{6} - \frac {B b^{2} c^{2} g^{2} i}{24 d}\right ) + x \left (A a^{2} c g^{2} i + \frac {B a^{3} d g^{2} i}{12 b} + \frac {B a^{2} c g^{2} i}{6} - \frac {B a b c^{2} g^{2} i}{3 d} + \frac {B b^{2} c^{3} g^{2} i}{12 d^{2}}\right ) + \left (B a^{2} c g^{2} i x + \frac {B a^{2} d g^{2} i x^{2}}{2} + B a b c g^{2} i x^{2} + \frac {2 B a b d g^{2} i x^{3}}{3} + \frac {B b^{2} c g^{2} i x^{3}}{3} + \frac {B b^{2} d g^{2} i x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

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

________________________________________________________________________________________