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]

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

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Rubi [A]  time = 0.34, 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 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 (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*}

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Mathematica [A]  time = 0.18, size = 216, normalized size = 0.90 \[ \frac {g i^2 \left (6 b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-8 (c+d x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {4 B (b c-a d)^2 \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )}{b^3}-\frac {B (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^3}\right )}{24 d^2} \]

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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fricas [A]  time = 1.00, size = 367, normalized size = 1.54 \[ \frac {6 \, A b^{4} d^{4} g i^{2} x^{4} + 2 \, {\left ({\left (8 \, A - B\right )} b^{4} c d^{3} + {\left (4 \, A + B\right )} a b^{3} d^{4}\right )} g i^{2} x^{3} + {\left ({\left (12 \, A - 5 \, B\right )} b^{4} c^{2} d^{2} + 4 \, {\left (6 \, A + B\right )} a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g i^{2} x^{2} - 2 \, {\left (B b^{4} c^{3} d - 2 \, {\left (6 \, A - B\right )} a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g i^{2} x + 2 \, {\left (6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} g i^{2} \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} g i^{2} \log \left (d x + c\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g i^{2} x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} x + 4 \, {\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b^{3} d^{2}} \]

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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giac [B]  time = 1.06, size = 3856, normalized size = 16.13 \[ \text {result too large to display} \]

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]

1/24*(2*B*b^9*c^5*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 10*B*a*b^8*c^4*d*g*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*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*e^5
*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 10*B*a^4*b^5*c*d^4*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 2*B*
a^5*b^4*d^5*g*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*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*e*log(-b*e + (
b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 12*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d
*x + c)^2 + 60*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 120*(b*x*e + a
*e)^2*B*a^2*b^5*c^3*d^4*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 120*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5
*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 60*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g*e^3*log((b*x*e + a*e)/(
d*x + c))/(d*x + c)^2 + 12*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 8*(b
*x*e + a*e)^3*B*b^6*c^5*d^3*g*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*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*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*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*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*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*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*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*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*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*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*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 2*A*b^9*c^
5*g*e^5 - B*b^9*c^5*g*e^5 - 10*A*a*b^8*c^4*d*g*e^5 + 5*B*a*b^8*c^4*d*g*e^5 + 20*A*a^2*b^7*c^3*d^2*g*e^5 - 10*B
*a^2*b^7*c^3*d^2*g*e^5 - 20*A*a^3*b^6*c^2*d^3*g*e^5 + 10*B*a^3*b^6*c^2*d^3*g*e^5 + 10*A*a^4*b^5*c*d^4*g*e^5 -
5*B*a^4*b^5*c*d^4*g*e^5 - 2*A*a^5*b^4*d^5*g*e^5 + B*a^5*b^4*d^5*g*e^5 - 8*(b*x*e + a*e)*A*b^8*c^5*d*g*e^4/(d*x
+ c) + 6*(b*x*e + a*e)*B*b^8*c^5*d*g*e^4/(d*x + c) + 40*(b*x*e + a*e)*A*a*b^7*c^4*d^2*g*e^4/(d*x + c) - 30*(b
*x*e + a*e)*B*a*b^7*c^4*d^2*g*e^4/(d*x + c) - 80*(b*x*e + a*e)*A*a^2*b^6*c^3*d^3*g*e^4/(d*x + c) + 60*(b*x*e +
a*e)*B*a^2*b^6*c^3*d^3*g*e^4/(d*x + c) + 80*(b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g*e^4/(d*x + c) - 60*(b*x*e + a*e
)*B*a^3*b^5*c^2*d^4*g*e^4/(d*x + c) - 40*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g*e^4/(d*x + c) + 30*(b*x*e + a*e)*B*a^
4*b^4*c*d^5*g*e^4/(d*x + c) + 8*(b*x*e + a*e)*A*a^5*b^3*d^6*g*e^4/(d*x + c) - 6*(b*x*e + a*e)*B*a^5*b^3*d^6*g*
e^4/(d*x + c) - 7*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g*e^3/(d*x + c)^2 + 35*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g*e^3/(
d*x + c)^2 - 70*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g*e^3/(d*x + c)^2 + 70*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g*e
^3/(d*x + c)^2 - 35*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g*e^3/(d*x + c)^2 + 7*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g*e^3/
(d*x + c)^2 + 2*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g*e^2/(d*x + c)^3 - 10*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g*e^2/(d*
x + c)^3 + 20*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g*e^2/(d*x + c)^3 - 20*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*g*e^2
/(d*x + c)^3 + 10*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g*e^2/(d*x + c)^3 - 2*(b*x*e + a*e)^3*B*a^5*b*d^8*g*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^7*d^2*e^4 - 4*(b*x*e + a*e
)*b^6*d^3*e^3/(d*x + c) + 6*(b*x*e + a*e)^2*b^5*d^4*e^2/(d*x + c)^2 - 4*(b*x*e + a*e)^3*b^4*d^5*e/(d*x + c)^3
+ (b*x*e + a*e)^4*b^3*d^6/(d*x + c)^4)

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maple [B]  time = 0.15, size = 3439, normalized size = 14.39 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.25, size = 671, normalized size = 2.81 \[ \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 \]

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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mupad [B]  time = 5.00, size = 636, normalized size = 2.66 \[ x^3\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )-x^2\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{6\,b}+\frac {A\,a\,c\,d\,g\,i^2}{2}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a\,c^2\,g\,i^2\,x+\frac {B\,c\,g\,i^2\,x^2\,\left (2\,a\,d+b\,c\right )}{2}+\frac {B\,d\,g\,i^2\,x^3\,\left (a\,d+2\,b\,c\right )}{3}+\frac {B\,b\,d^2\,g\,i^2\,x^4}{4}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{3\,b}+A\,a\,c\,d\,g\,i^2\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {c\,g\,i^2\,\left (6\,A\,a^2\,d^2+2\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+12\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{2\,b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,g\,a^4\,d^2\,i^2-4\,B\,g\,a^3\,b\,c\,d\,i^2+6\,B\,g\,a^2\,b^2\,c^2\,i^2\right )}{12\,b^3}+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^4\,g\,i^2-4\,B\,a\,c^3\,d\,g\,i^2\right )}{12\,d^2}+\frac {A\,b\,d^2\,g\,i^2\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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sympy [B]  time = 5.07, size = 850, normalized size = 3.56 \[ \frac {A b d^{2} g i^{2} x^{4}}{4} + \frac {B a^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + \frac {B a^{3} d^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{b} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a^{2} c d g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) - B a b^{3} c^{4} g i^{2}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 b^{3}} - \frac {B c^{3} g i^{2} \left (4 a d - b c\right ) \log {\left (x + \frac {B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a b^{3} c^{4} g i^{2} - B a b^{2} c^{3} g i^{2} \left (4 a d - b c\right ) + \frac {B b^{3} c^{4} g i^{2} \left (4 a d - b c\right )}{d}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 d^{2}} + x^{3} \left (\frac {A a d^{2} g i^{2}}{3} + \frac {2 A b c d g i^{2}}{3} + \frac {B a d^{2} g i^{2}}{12} - \frac {B b c d g i^{2}}{12}\right ) + x^{2} \left (A a c d g i^{2} + \frac {A b c^{2} g i^{2}}{2} + \frac {B a^{2} d^{2} g i^{2}}{24 b} + \frac {B a c d g i^{2}}{6} - \frac {5 B b c^{2} g i^{2}}{24}\right ) + x \left (A a c^{2} g i^{2} - \frac {B a^{3} d^{2} g i^{2}}{12 b^{2}} + \frac {B a^{2} c d g i^{2}}{3 b} - \frac {B a c^{2} g i^{2}}{6} - \frac {B b c^{3} g i^{2}}{12 d}\right ) + \left (B a c^{2} g i^{2} x + B a c d g i^{2} x^{2} + \frac {B a d^{2} g i^{2} x^{3}}{3} + \frac {B b c^{2} g i^{2} x^{2}}{2} + \frac {2 B b c d g i^{2} x^{3}}{3} + \frac {B b d^{2} g i^{2} x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

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

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