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

Optimal. Leaf size=337 \[ \frac {b^2 g^2 i^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3}+\frac {g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^3}-\frac {b g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3}-\frac {B g^2 i^2 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}-\frac {B g^2 i^2 (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac {B g^2 i^2 x (b c-a d)^4}{30 b^2 d^2}-\frac {B g^2 i^2 (c+d x)^2 (b c-a d)^3}{60 b d^3}+\frac {B g^2 i^2 (c+d x)^3 (b c-a d)^2}{10 d^3}-\frac {b B g^2 i^2 (c+d x)^4 (b c-a d)}{20 d^3} \]

[Out]

-1/30*B*(-a*d+b*c)^4*g^2*i^2*x/b^2/d^2-1/60*B*(-a*d+b*c)^3*g^2*i^2*(d*x+c)^2/b/d^3+1/10*B*(-a*d+b*c)^2*g^2*i^2
*(d*x+c)^3/d^3-1/20*b*B*(-a*d+b*c)*g^2*i^2*(d*x+c)^4/d^3-1/30*B*(-a*d+b*c)^5*g^2*i^2*ln((b*x+a)/(d*x+c))/b^3/d
^3+1/3*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/2*b*(-a*d+b*c)*g^2*i^2*(d*x+c)^4*(A+B*
ln(e*(b*x+a)/(d*x+c)))/d^3+1/5*b^2*g^2*i^2*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/30*B*(-a*d+b*c)^5*g^2*i
^2*ln(d*x+c)/b^3/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.51, antiderivative size = 296, normalized size of antiderivative = 0.88, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 43} \[ \frac {d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^3}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^3}+\frac {d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^3}+\frac {B g^2 i^2 x (b c-a d)^4}{30 b^2 d^2}-\frac {B g^2 i^2 (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac {B g^2 i^2 (a+b x)^2 (b c-a d)^3}{60 b^3 d}-\frac {B g^2 i^2 (a+b x)^3 (b c-a d)^2}{10 b^3}-\frac {B d g^2 i^2 (a+b x)^4 (b c-a d)}{20 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^4*g^2*i^2*x)/(30*b^2*d^2) - (B*(b*c - a*d)^3*g^2*i^2*(a + b*x)^2)/(60*b^3*d) - (B*(b*c - a*d)^2
*g^2*i^2*(a + b*x)^3)/(10*b^3) - (B*d*(b*c - a*d)*g^2*i^2*(a + b*x)^4)/(20*b^3) + ((b*c - a*d)^2*g^2*i^2*(a +
b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3) + (d*(b*c - a*d)*g^2*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b
*x))/(c + d*x)]))/(2*b^3) + (d^2*g^2*i^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^3) - (B*(b*c -
a*d)^5*g^2*i^2*Log[c + d*x])/(30*b^3*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 (11 c+11 d x)^2 (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 {(-b c+a d)^2 g^2 (11 c+11 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac {2 b (b c-a d) g^2 (11 c+11 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{11 d^2}+\frac {b^2 g^2 (11 c+11 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{121 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (11 c+11 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{121 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int (11 c+11 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{11 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (11 c+11 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}\\ &=\frac {121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac {121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac {121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac {\left (b^2 B g^2\right ) \int \frac {161051 (b c-a d) (c+d x)^4}{a+b x} \, dx}{6655 d^3}+\frac {\left (b B (b c-a d) g^2\right ) \int \frac {14641 (b c-a d) (c+d x)^3}{a+b x} \, dx}{242 d^3}-\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {1331 (b c-a d) (c+d x)^2}{a+b x} \, dx}{33 d^3}\\ &=\frac {121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac {121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac {121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac {\left (121 b^2 B (b c-a d) g^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^3}+\frac {\left (121 b B (b c-a d)^2 g^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{2 d^3}-\frac {\left (121 B (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 d^3}\\ &=\frac {121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac {121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac {121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac {\left (121 b^2 B (b c-a d) g^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{5 d^3}+\frac {\left (121 b B (b c-a d)^2 g^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{2 d^3}-\frac {\left (121 B (b c-a d)^3 g^2\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}{3 d^3}\\ &=-\frac {121 B (b c-a d)^4 g^2 x}{30 b^2 d^2}-\frac {121 B (b c-a d)^3 g^2 (c+d x)^2}{60 b d^3}+\frac {121 B (b c-a d)^2 g^2 (c+d x)^3}{10 d^3}-\frac {121 b B (b c-a d) g^2 (c+d x)^4}{20 d^3}-\frac {121 B (b c-a d)^5 g^2 \log (a+b x)}{30 b^3 d^3}+\frac {121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac {121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac {121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 362, normalized size = 1.07 \[ \frac {g^2 i^2 \left (12 d^5 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+30 d^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+20 d^3 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+10 B (b c-a d)^3 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-5 B (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+B (b c-a d) \left (4 d^3 (a+b x)^3 (b c-a d)-6 d^2 (a+b x)^2 (b c-a d)^2+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )\right )}{60 b^3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 30*d^4*(b*c - a*d)*(a + b*x)
^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 12*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 10*B*(b*c
- a*d)^3*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) - 5*B*(b*c - a*d)^2*(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]) + B*(b*c -
a*d)*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b
*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x])))/(60*b^3*d^3)

________________________________________________________________________________________

fricas [A]  time = 1.07, size = 534, normalized size = 1.58 \[ \frac {12 \, A b^{5} d^{5} g^{2} i^{2} x^{5} + 3 \, {\left ({\left (10 \, A - B\right )} b^{5} c d^{4} + {\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 2 \, {\left ({\left (10 \, A - 3 \, B\right )} b^{5} c^{2} d^{3} + 40 \, A a b^{4} c d^{4} + {\left (10 \, A + 3 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} - {\left (B b^{5} c^{3} d^{2} - 15 \, {\left (4 \, A - B\right )} a b^{4} c^{2} d^{3} - 15 \, {\left (4 \, A + B\right )} a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 30 \, A a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{2} i^{2} x + 2 \, {\left (10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{2} i^{2} \log \left (b x + a\right ) - 2 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} g^{2} i^{2} \log \left (d x + c\right ) + 2 \, {\left (6 \, B b^{5} d^{5} g^{2} i^{2} x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} x + 15 \, {\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 10 \, {\left (B b^{5} c^{2} d^{3} + 4 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} + 30 \, {\left (B a b^{4} c^{2} d^{3} + B a^{2} b^{3} c d^{4}\right )} g^{2} i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{60 \, b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*A*b^5*d^5*g^2*i^2*x^5 + 3*((10*A - B)*b^5*c*d^4 + (10*A + B)*a*b^4*d^5)*g^2*i^2*x^4 + 2*((10*A - 3*B)
*b^5*c^2*d^3 + 40*A*a*b^4*c*d^4 + (10*A + 3*B)*a^2*b^3*d^5)*g^2*i^2*x^3 - (B*b^5*c^3*d^2 - 15*(4*A - B)*a*b^4*
c^2*d^3 - 15*(4*A + B)*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g^2*i^2*x^2 + 2*(B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 30*A*
a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^2*i^2*x + 2*(10*B*a^3*b^2*c^2*d^3 - 5*B*a^4*b*c*d^4 + B*a
^5*d^5)*g^2*i^2*log(b*x + a) - 2*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*g^2*i^2*log(d*x + c) + 2
*(6*B*b^5*d^5*g^2*i^2*x^5 + 30*B*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(B*b^5*c*d^4 + B*a*b^4*d^5)*g^2*i^2*x^4 + 10*(
B*b^5*c^2*d^3 + 4*B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^2*i^2*x^3 + 30*(B*a*b^4*c^2*d^3 + B*a^2*b^3*c*d^4)*g^2*i^2*
x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^3*d^3)

________________________________________________________________________________________

giac [B]  time = 1.36, size = 5571, normalized size = 16.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(2*B*b^11*c^6*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 12*B*a*b^10*c^5*d*g^2*e^6*log(-b*e + (b*x*
e + a*e)*d/(d*x + c)) + 30*B*a^2*b^9*c^4*d^2*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 40*B*a^3*b^8*c^3*
d^3*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 30*B*a^4*b^7*c^2*d^4*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d
*x + c)) - 12*B*a^5*b^6*c*d^5*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 2*B*a^6*b^5*d^6*g^2*e^6*log(-b*e
+ (b*x*e + a*e)*d/(d*x + c)) - 10*(b*x*e + a*e)*B*b^10*c^6*d*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d
*x + c) + 60*(b*x*e + a*e)*B*a*b^9*c^5*d^2*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 150*(b*x*
e + a*e)*B*a^2*b^8*c^4*d^3*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 200*(b*x*e + a*e)*B*a^3*b
^7*c^3*d^4*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 150*(b*x*e + a*e)*B*a^4*b^6*c^2*d^5*g^2*e
^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 60*(b*x*e + a*e)*B*a^5*b^5*c*d^6*g^2*e^5*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c) - 10*(b*x*e + a*e)*B*a^6*b^4*d^7*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))
/(d*x + c) + 20*(b*x*e + a*e)^2*B*b^9*c^6*d^2*g^2*e^4*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*b^8*c^5*d^3*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 300*(b*x*e + a*e)^
2*B*a^2*b^7*c^4*d^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 400*(b*x*e + a*e)^2*B*a^3*b^6*
c^3*d^5*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 300*(b*x*e + a*e)^2*B*a^4*b^5*c^2*d^6*g^2*
e^4*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^5*b^4*c*d^7*g^2*e^4*log(-b*e +
(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 20*(b*x*e + a*e)^2*B*a^6*b^3*d^8*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/
(d*x + c))/(d*x + c)^2 - 20*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^3 + 120*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 300*(b
*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 400*(b*x*e + a*e)^
3*B*a^3*b^5*c^3*d^6*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 300*(b*x*e + a*e)^3*B*a^4*b^4*
c^2*d^7*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120*(b*x*e + a*e)^3*B*a^5*b^3*c*d^8*g^2*e^
3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 20*(b*x*e + a*e)^3*B*a^6*b^2*d^9*g^2*e^3*log(-b*e + (b*x
*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 10*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a*b^6*c^5*d^5*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^4 + 150*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 200*(b*
x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 150*(b*x*e + a*e)^4
*B*a^4*b^3*c^2*d^8*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^5*b^2*c*
d^9*g^2*e^2*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^6*b*d^10*g^2*e^2*log(-b
*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 2*(b*x*e + a*e)^5*B*b^6*c^6*d^5*g^2*e*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^5 + 12*(b*x*e + a*e)^5*B*a*b^5*c^5*d^6*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^5 - 30*(b*x*e + a*e)^5*B*a^2*b^4*c^4*d^7*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 40*(b*
x*e + a*e)^5*B*a^3*b^3*c^3*d^8*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 30*(b*x*e + a*e)^5*B*
a^4*b^2*c^2*d^9*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 12*(b*x*e + a*e)^5*B*a^5*b*c*d^10*g^
2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 2*(b*x*e + a*e)^5*B*a^6*d^11*g^2*e*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c)^5 + 20*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x +
c)^3 - 120*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 300*(b*x*e + a*e
)^3*B*a^2*b^6*c^4*d^5*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 400*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d^6
*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 300*(b*x*e + a*e)^3*B*a^4*b^4*c^2*d^7*g^2*e^3*log((b*x*e +
a*e)/(d*x + c))/(d*x + c)^3 - 120*(b*x*e + a*e)^3*B*a^5*b^3*c*d^8*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x +
c)^3 + 20*(b*x*e + a*e)^3*B*a^6*b^2*d^9*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 10*(b*x*e + a*e)^4
*B*b^7*c^6*d^4*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a*b^6*c^5*d^5*g^2*e^2*l
og((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 150*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*g^2*e^2*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^4 + 200*(b*x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 -
150*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*
a^5*b^2*c*d^9*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 10*(b*x*e + a*e)^4*B*a^6*b*d^10*g^2*e^2*log((
b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 2*(b*x*e + a*e)^5*B*b^6*c^6*d^5*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x
+ c)^5 - 12*(b*x*e + a*e)^5*B*a*b^5*c^5*d^6*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 30*(b*x*e + a*e)^
5*B*a^2*b^4*c^4*d^7*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 40*(b*x*e + a*e)^5*B*a^3*b^3*c^3*d^8*g^2*
e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 30*(b*x*e + a*e)^5*B*a^4*b^2*c^2*d^9*g^2*e*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^5 - 12*(b*x*e + a*e)^5*B*a^5*b*c*d^10*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 2*(b*x
*e + a*e)^5*B*a^6*d^11*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 2*A*b^11*c^6*g^2*e^6 - 12*A*a*b^10*c^5
*d*g^2*e^6 + 30*A*a^2*b^9*c^4*d^2*g^2*e^6 - 40*A*a^3*b^8*c^3*d^3*g^2*e^6 + 30*A*a^4*b^7*c^2*d^4*g^2*e^6 - 12*A
*a^5*b^6*c*d^5*g^2*e^6 + 2*A*a^6*b^5*d^6*g^2*e^6 - 10*(b*x*e + a*e)*A*b^10*c^6*d*g^2*e^5/(d*x + c) + 2*(b*x*e
+ a*e)*B*b^10*c^6*d*g^2*e^5/(d*x + c) + 60*(b*x*e + a*e)*A*a*b^9*c^5*d^2*g^2*e^5/(d*x + c) - 12*(b*x*e + a*e)*
B*a*b^9*c^5*d^2*g^2*e^5/(d*x + c) - 150*(b*x*e + a*e)*A*a^2*b^8*c^4*d^3*g^2*e^5/(d*x + c) + 30*(b*x*e + a*e)*B
*a^2*b^8*c^4*d^3*g^2*e^5/(d*x + c) + 200*(b*x*e + a*e)*A*a^3*b^7*c^3*d^4*g^2*e^5/(d*x + c) - 40*(b*x*e + a*e)*
B*a^3*b^7*c^3*d^4*g^2*e^5/(d*x + c) - 150*(b*x*e + a*e)*A*a^4*b^6*c^2*d^5*g^2*e^5/(d*x + c) + 30*(b*x*e + a*e)
*B*a^4*b^6*c^2*d^5*g^2*e^5/(d*x + c) + 60*(b*x*e + a*e)*A*a^5*b^5*c*d^6*g^2*e^5/(d*x + c) - 12*(b*x*e + a*e)*B
*a^5*b^5*c*d^6*g^2*e^5/(d*x + c) - 10*(b*x*e + a*e)*A*a^6*b^4*d^7*g^2*e^5/(d*x + c) + 2*(b*x*e + a*e)*B*a^6*b^
4*d^7*g^2*e^5/(d*x + c) + 20*(b*x*e + a*e)^2*A*b^9*c^6*d^2*g^2*e^4/(d*x + c)^2 - 9*(b*x*e + a*e)^2*B*b^9*c^6*d
^2*g^2*e^4/(d*x + c)^2 - 120*(b*x*e + a*e)^2*A*a*b^8*c^5*d^3*g^2*e^4/(d*x + c)^2 + 54*(b*x*e + a*e)^2*B*a*b^8*
c^5*d^3*g^2*e^4/(d*x + c)^2 + 300*(b*x*e + a*e)^2*A*a^2*b^7*c^4*d^4*g^2*e^4/(d*x + c)^2 - 135*(b*x*e + a*e)^2*
B*a^2*b^7*c^4*d^4*g^2*e^4/(d*x + c)^2 - 400*(b*x*e + a*e)^2*A*a^3*b^6*c^3*d^5*g^2*e^4/(d*x + c)^2 + 180*(b*x*e
+ a*e)^2*B*a^3*b^6*c^3*d^5*g^2*e^4/(d*x + c)^2 + 300*(b*x*e + a*e)^2*A*a^4*b^5*c^2*d^6*g^2*e^4/(d*x + c)^2 -
135*(b*x*e + a*e)^2*B*a^4*b^5*c^2*d^6*g^2*e^4/(d*x + c)^2 - 120*(b*x*e + a*e)^2*A*a^5*b^4*c*d^7*g^2*e^4/(d*x +
c)^2 + 54*(b*x*e + a*e)^2*B*a^5*b^4*c*d^7*g^2*e^4/(d*x + c)^2 + 20*(b*x*e + a*e)^2*A*a^6*b^3*d^8*g^2*e^4/(d*x
+ c)^2 - 9*(b*x*e + a*e)^2*B*a^6*b^3*d^8*g^2*e^4/(d*x + c)^2 + 9*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g^2*e^3/(d*x +
c)^3 - 54*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g^2*e^3/(d*x + c)^3 + 135*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g^2*e^3
/(d*x + c)^3 - 180*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d^6*g^2*e^3/(d*x + c)^3 + 135*(b*x*e + a*e)^3*B*a^4*b^4*c^2*d
^7*g^2*e^3/(d*x + c)^3 - 54*(b*x*e + a*e)^3*B*a^5*b^3*c*d^8*g^2*e^3/(d*x + c)^3 + 9*(b*x*e + a*e)^3*B*a^6*b^2*
d^9*g^2*e^3/(d*x + c)^3 - 2*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g^2*e^2/(d*x + c)^4 + 12*(b*x*e + a*e)^4*B*a*b^6*c^5
*d^5*g^2*e^2/(d*x + c)^4 - 30*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*g^2*e^2/(d*x + c)^4 + 40*(b*x*e + a*e)^4*B*a^3
*b^4*c^3*d^7*g^2*e^2/(d*x + c)^4 - 30*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g^2*e^2/(d*x + c)^4 + 12*(b*x*e + a*e)
^4*B*a^5*b^2*c*d^9*g^2*e^2/(d*x + c)^4 - 2*(b*x*e + a*e)^4*B*a^6*b*d^10*g^2*e^2/(d*x + c)^4)*(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^8*d^3*e^5 - 5*(b*x*e + a*e)*b^7*d^4*e^4/(d*x + c) +
10*(b*x*e + a*e)^2*b^6*d^5*e^3/(d*x + c)^2 - 10*(b*x*e + a*e)^3*b^5*d^6*e^2/(d*x + c)^3 + 5*(b*x*e + a*e)^4*b^
4*d^7*e/(d*x + c)^4 - (b*x*e + a*e)^5*b^3*d^8/(d*x + c)^5)

________________________________________________________________________________________

maple [B]  time = 0.17, size = 6116, normalized size = 18.15 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [B]  time = 1.47, size = 1200, normalized size = 3.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.34, size = 1287, normalized size = 3.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((e*(a + b*x))/(c + d*x))*((B*g^2*i^2*x^3*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/3 + B*a^2*c^2*g^2*i^2*x + (B*b^2
*d^2*g^2*i^2*x^5)/5 + B*a*c*g^2*i^2*x^2*(a*d + b*c) + (B*b*d*g^2*i^2*x^4*(a*d + b*c))/2) - x^3*(((30*a*d + 30*
b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(90*b*d)
- (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/6 + (A*a*b*c*d*g^2*i^2)/3) + x*
((a*c*(((30*a*d + 30*b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30
*b*c))/30))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/2 + A*a*b*
c*d*g^2*i^2))/(b*d) - ((30*a*d + 30*b*c)*(((30*a*d + 30*b*c)*(((30*a*d + 30*b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*
A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*
c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/2 + A*a*b*c*d*g^2*i^2))/(30*b*d) + (g^2*i^2*(3*A*a^3*d^3 + 3*A*b^
3*c^3 + B*a^3*d^3 - B*b^3*c^3 + 27*A*a*b^2*c^2*d + 27*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(3*b
*d) - (a*c*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(b*
d)))/(30*b*d) + (a*c*g^2*i^2*(3*A*a^2*d^2 + 3*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 9*A*a*b*c*d))/(b*d)) + x^2*(
((30*a*d + 30*b*c)*(((30*a*d + 30*b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2
*(30*a*d + 30*b*c))/30))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d
))/2 + A*a*b*c*d*g^2*i^2))/(60*b*d) + (g^2*i^2*(3*A*a^3*d^3 + 3*A*b^3*c^3 + B*a^3*d^3 - B*b^3*c^3 + 27*A*a*b^2
*c^2*d + 27*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(6*b*d) - (a*c*((b*d*g^2*i^2*(15*A*a*d + 15*A*
b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(2*b*d)) + x^4*((b*d*g^2*i^2*(15*A*a*d + 15*A
*b*c + B*a*d - B*b*c))/20 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/120) + (log(a + b*x)*(B*a^5*d^2*g^2*i^2 + 10*B*a
^3*b^2*c^2*g^2*i^2 - 5*B*a^4*b*c*d*g^2*i^2))/(30*b^3) - (log(c + d*x)*(B*b^2*c^5*g^2*i^2 + 10*B*a^2*c^3*d^2*g^
2*i^2 - 5*B*a*b*c^4*d*g^2*i^2))/(30*d^3) + (A*b^2*d^2*g^2*i^2*x^5)/5

________________________________________________________________________________________

sympy [B]  time = 7.81, size = 1266, normalized size = 3.76 \[ \frac {A b^{2} d^{2} g^{2} i^{2} x^{5}}{5} + \frac {B a^{3} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{2} i^{2} - 5 B a^{4} b c^{2} d^{3} g^{2} i^{2} + \frac {B a^{4} d^{3} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right )}{b} + 20 B a^{3} b^{2} c^{3} d^{2} g^{2} i^{2} - B a^{3} c d^{2} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right ) - 5 B a^{2} b^{3} c^{4} d g^{2} i^{2} + B a b^{4} c^{5} g^{2} i^{2}}{B a^{5} d^{5} g^{2} i^{2} - 5 B a^{4} b c d^{4} g^{2} i^{2} + 10 B a^{3} b^{2} c^{2} d^{3} g^{2} i^{2} + 10 B a^{2} b^{3} c^{3} d^{2} g^{2} i^{2} - 5 B a b^{4} c^{4} d g^{2} i^{2} + B b^{5} c^{5} g^{2} i^{2}} \right )}}{30 b^{3}} - \frac {B c^{3} g^{2} i^{2} \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{2} i^{2} - 5 B a^{4} b c^{2} d^{3} g^{2} i^{2} + 20 B a^{3} b^{2} c^{3} d^{2} g^{2} i^{2} - 5 B a^{2} b^{3} c^{4} d g^{2} i^{2} + B a b^{4} c^{5} g^{2} i^{2} - B a b^{2} c^{3} g^{2} i^{2} \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) + \frac {B b^{3} c^{4} g^{2} i^{2} \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right )}{d}}{B a^{5} d^{5} g^{2} i^{2} - 5 B a^{4} b c d^{4} g^{2} i^{2} + 10 B a^{3} b^{2} c^{2} d^{3} g^{2} i^{2} + 10 B a^{2} b^{3} c^{3} d^{2} g^{2} i^{2} - 5 B a b^{4} c^{4} d g^{2} i^{2} + B b^{5} c^{5} g^{2} i^{2}} \right )}}{30 d^{3}} + x^{4} \left (\frac {A a b d^{2} g^{2} i^{2}}{2} + \frac {A b^{2} c d g^{2} i^{2}}{2} + \frac {B a b d^{2} g^{2} i^{2}}{20} - \frac {B b^{2} c d g^{2} i^{2}}{20}\right ) + x^{3} \left (\frac {A a^{2} d^{2} g^{2} i^{2}}{3} + \frac {4 A a b c d g^{2} i^{2}}{3} + \frac {A b^{2} c^{2} g^{2} i^{2}}{3} + \frac {B a^{2} d^{2} g^{2} i^{2}}{10} - \frac {B b^{2} c^{2} g^{2} i^{2}}{10}\right ) + x^{2} \left (A a^{2} c d g^{2} i^{2} + A a b c^{2} g^{2} i^{2} + \frac {B a^{3} d^{2} g^{2} i^{2}}{60 b} + \frac {B a^{2} c d g^{2} i^{2}}{4} - \frac {B a b c^{2} g^{2} i^{2}}{4} - \frac {B b^{2} c^{3} g^{2} i^{2}}{60 d}\right ) + x \left (A a^{2} c^{2} g^{2} i^{2} - \frac {B a^{4} d^{2} g^{2} i^{2}}{30 b^{2}} + \frac {B a^{3} c d g^{2} i^{2}}{6 b} - \frac {B a b c^{3} g^{2} i^{2}}{6 d} + \frac {B b^{2} c^{4} g^{2} i^{2}}{30 d^{2}}\right ) + \left (B a^{2} c^{2} g^{2} i^{2} x + B a^{2} c d g^{2} i^{2} x^{2} + \frac {B a^{2} d^{2} g^{2} i^{2} x^{3}}{3} + B a b c^{2} g^{2} i^{2} x^{2} + \frac {4 B a b c d g^{2} i^{2} x^{3}}{3} + \frac {B a b d^{2} g^{2} i^{2} x^{4}}{2} + \frac {B b^{2} c^{2} g^{2} i^{2} x^{3}}{3} + \frac {B b^{2} c d g^{2} i^{2} x^{4}}{2} + \frac {B b^{2} d^{2} g^{2} i^{2} x^{5}}{5}\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)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**2*d**2*g**2*i**2*x**5/5 + B*a**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)*log(x + (B*a**5*c*d**4*
g**2*i**2 - 5*B*a**4*b*c**2*d**3*g**2*i**2 + B*a**4*d**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)/b +
20*B*a**3*b**2*c**3*d**2*g**2*i**2 - B*a**3*c*d**2*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2) - 5*B*a**2
*b**3*c**4*d*g**2*i**2 + B*a*b**4*c**5*g**2*i**2)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c*d**4*g**2*i**2 + 10*B*
a**3*b**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3*d**2*g**2*i**2 - 5*B*a*b**4*c**4*d*g**2*i**2 + B*b**5*c**5
*g**2*i**2))/(30*b**3) - B*c**3*g**2*i**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)*log(x + (B*a**5*c*d**4*g**2*i
**2 - 5*B*a**4*b*c**2*d**3*g**2*i**2 + 20*B*a**3*b**2*c**3*d**2*g**2*i**2 - 5*B*a**2*b**3*c**4*d*g**2*i**2 + B
*a*b**4*c**5*g**2*i**2 - B*a*b**2*c**3*g**2*i**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2) + B*b**3*c**4*g**2*i**
2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/d)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c*d**4*g**2*i**2 + 10*B*a**3*b
**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3*d**2*g**2*i**2 - 5*B*a*b**4*c**4*d*g**2*i**2 + B*b**5*c**5*g**2*
i**2))/(30*d**3) + x**4*(A*a*b*d**2*g**2*i**2/2 + A*b**2*c*d*g**2*i**2/2 + B*a*b*d**2*g**2*i**2/20 - B*b**2*c*
d*g**2*i**2/20) + x**3*(A*a**2*d**2*g**2*i**2/3 + 4*A*a*b*c*d*g**2*i**2/3 + A*b**2*c**2*g**2*i**2/3 + B*a**2*d
**2*g**2*i**2/10 - B*b**2*c**2*g**2*i**2/10) + x**2*(A*a**2*c*d*g**2*i**2 + A*a*b*c**2*g**2*i**2 + B*a**3*d**2
*g**2*i**2/(60*b) + B*a**2*c*d*g**2*i**2/4 - B*a*b*c**2*g**2*i**2/4 - B*b**2*c**3*g**2*i**2/(60*d)) + x*(A*a**
2*c**2*g**2*i**2 - B*a**4*d**2*g**2*i**2/(30*b**2) + B*a**3*c*d*g**2*i**2/(6*b) - B*a*b*c**3*g**2*i**2/(6*d) +
B*b**2*c**4*g**2*i**2/(30*d**2)) + (B*a**2*c**2*g**2*i**2*x + B*a**2*c*d*g**2*i**2*x**2 + B*a**2*d**2*g**2*i*
*2*x**3/3 + B*a*b*c**2*g**2*i**2*x**2 + 4*B*a*b*c*d*g**2*i**2*x**3/3 + B*a*b*d**2*g**2*i**2*x**4/2 + B*b**2*c*
*2*g**2*i**2*x**3/3 + B*b**2*c*d*g**2*i**2*x**4/2 + B*b**2*d**2*g**2*i**2*x**5/5)*log(e*(a + b*x)/(c + d*x))

________________________________________________________________________________________