3.1 \(\int (a g+b g x)^3 (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=212 \[ \frac {g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{20 b^2}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b}+\frac {B g^3 i (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}+\frac {B g^3 i (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}-\frac {B g^3 i (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac {B g^3 i x (b c-a d)^4}{20 b d^3} \]

[Out]

-1/20*B*(-a*d+b*c)^4*g^3*i*x/b/d^3+1/40*B*(-a*d+b*c)^3*g^3*i*(b*x+a)^2/b^2/d^2-1/60*B*(-a*d+b*c)^2*g^3*i*(b*x+
a)^3/b^2/d+1/5*g^3*i*(b*x+a)^4*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b+1/20*(-a*d+b*c)*g^3*i*(b*x+a)^4*(A-B+B*ln
(e*(b*x+a)/(d*x+c)))/b^2+1/20*B*(-a*d+b*c)^5*g^3*i*ln(d*x+c)/b^2/d^4

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Rubi [A]  time = 0.35, antiderivative size = 232, normalized size of antiderivative = 1.09, 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} \[ \frac {g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^2}+\frac {d g^3 i (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^2}+\frac {B g^3 i (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g^3 i (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}-\frac {B g^3 i (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac {B g^3 i (a+b x)^4 (b c-a d)}{20 b^2}-\frac {B g^3 i x (b c-a d)^4}{20 b d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

-(B*(b*c - a*d)^4*g^3*i*x)/(20*b*d^3) + (B*(b*c - a*d)^3*g^3*i*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d)^2*g^
3*i*(a + b*x)^3)/(60*b^2*d) - (B*(b*c - a*d)*g^3*i*(a + b*x)^4)/(20*b^2) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2) + (d*g^3*i*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^2)
+ (B*(b*c - a*d)^5*g^3*i*Log[c + d*x])/(20*b^2*d^4)

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 (c+d x) (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac {(b c-a d) (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}+\frac {d (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}\right ) \, dx\\ &=\frac {(b c-a 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}+\frac {d \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}\\ &=\frac {(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac {d g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac {(B d) \int \frac {(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b^2 g^2}-\frac {(B (b c-a d)) \int \frac {(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b^2 g}\\ &=\frac {(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac {d g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac {\left (B d (b c-a d) g^3\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b^2}-\frac {\left (B (b c-a d)^2 g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{4 b^2}\\ &=\frac {(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac {d g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac {\left (B d (b c-a d) g^3\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^2}-\frac {\left (B (b c-a d)^2 g^3\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}{4 b^2}\\ &=-\frac {B (b c-a d)^4 g^3 x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 (a+b x)^3}{60 b^2 d}-\frac {B (b c-a d) g^3 (a+b x)^4}{20 b^2}+\frac {(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac {d g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2}+\frac {B (b c-a d)^5 g^3 \log (c+d x)}{20 b^2 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 261, normalized size = 1.23 \[ \frac {g^3 i \left (24 d (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+30 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {5 B (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{d^4}+\frac {2 B (b c-a d) \left (4 d^3 (a+b x)^3 (b c-a d)-6 d^2 (a+b x)^2 (b c-a d)^2+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )}{d^4}\right )}{120 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(g^3*i*(30*(b*c - a*d)*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*d*(a + b*x)^5*(A + B*Log[(e*(a +
b*x))/(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]))/d^4 + (2*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]))/d^4))/(120*b^
2)

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fricas [B]  time = 2.37, size = 504, normalized size = 2.38 \[ \frac {24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \, {\left ({\left (5 \, A - B\right )} b^{5} c d^{4} + {\left (15 \, A + B\right )} a b^{4} d^{5}\right )} g^{3} i x^{4} - 2 \, {\left (B b^{5} c^{2} d^{3} - 10 \, {\left (6 \, A - B\right )} a b^{4} c d^{4} - {\left (60 \, A + 11 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 3 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 5 \, {\left (12 \, A - B\right )} a^{2} b^{3} c d^{4} + {\left (20 \, A + 9 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{3} i x^{2} - 6 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, {\left (4 \, A + B\right )} a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i x + 6 \, {\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} i \log \left (d x + c\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{120 \, b^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/120*(24*A*b^5*d^5*g^3*i*x^5 + 6*((5*A - B)*b^5*c*d^4 + (15*A + B)*a*b^4*d^5)*g^3*i*x^4 - 2*(B*b^5*c^2*d^3 -
10*(6*A - B)*a*b^4*c*d^4 - (60*A + 11*B)*a^2*b^3*d^5)*g^3*i*x^3 + 3*(B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 + 5*(12
*A - B)*a^2*b^3*c*d^4 + (20*A + 9*B)*a^3*b^2*d^5)*g^3*i*x^2 - 6*(B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2*b^
3*c^2*d^3 - 5*(4*A + B)*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^3*i*x + 6*(5*B*a^4*b*c*d^4 - B*a^5*d^5)*g^3*i*log(b*x +
a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3)*g^3*i*log(d*x + c) + 6*(4*
B*b^5*d^5*g^3*i*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*x + 5*(B*b^5*c*d^4 + 3*B*a*b^4*d^5)*g^3*i*x^4 + 20*(B*a*b^4*c*d
^4 + B*a^2*b^3*d^5)*g^3*i*x^3 + 10*(3*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*x^2)*log((b*e*x + a*e)/(d*x + c))
)/(b^2*d^4)

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giac [B]  time = 1.59, size = 5682, normalized size = 26.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

-1/120*(6*B*b^11*c^6*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 36*B*a*b^10*c^5*d*g^3*i*e^6*log(-b*e +
(b*x*e + a*e)*d/(d*x + c)) + 90*B*a^2*b^9*c^4*d^2*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 120*B*a^3*
b^8*c^3*d^3*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 90*B*a^4*b^7*c^2*d^4*g^3*i*e^6*log(-b*e + (b*x*e
+ a*e)*d/(d*x + c)) - 36*B*a^5*b^6*c*d^5*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 6*B*a^6*b^5*d^6*g^
3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 30*(b*x*e + a*e)*B*b^10*c^6*d*g^3*i*e^5*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c) + 180*(b*x*e + a*e)*B*a*b^9*c^5*d^2*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/
(d*x + c) - 450*(b*x*e + a*e)*B*a^2*b^8*c^4*d^3*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 60
0*(b*x*e + a*e)*B*a^3*b^7*c^3*d^4*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 450*(b*x*e + a*e
)*B*a^4*b^6*c^2*d^5*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 180*(b*x*e + a*e)*B*a^5*b^5*c*
d^6*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 30*(b*x*e + a*e)*B*a^6*b^4*d^7*g^3*i*e^5*log(-
b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 60*(b*x*e + a*e)^2*B*b^9*c^6*d^2*g^3*i*e^4*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c)^2 - 360*(b*x*e + a*e)^2*B*a*b^8*c^5*d^3*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x +
c))/(d*x + c)^2 + 900*(b*x*e + a*e)^2*B*a^2*b^7*c^4*d^4*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^2 - 1200*(b*x*e + a*e)^2*B*a^3*b^6*c^3*d^5*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 9
00*(b*x*e + a*e)^2*B*a^4*b^5*c^2*d^6*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 360*(b*x*e
+ a*e)^2*B*a^5*b^4*c*d^7*g^3*i*e^4*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^
6*b^3*d^8*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 60*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g^3*i
*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 360*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g^3*i*e^3*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 900*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g^3*i*e^3*log(-b*e + (b*x*e
+ a*e)*d/(d*x + c))/(d*x + c)^3 + 1200*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d^6*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^3 - 900*(b*x*e + a*e)^3*B*a^4*b^4*c^2*d^7*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c)^3 + 360*(b*x*e + a*e)^3*B*a^5*b^3*c*d^8*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^
3 - 60*(b*x*e + a*e)^3*B*a^6*b^2*d^9*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 30*(b*x*e +
a*e)^4*B*b^7*c^6*d^4*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 180*(b*x*e + a*e)^4*B*a*b^
6*c^5*d^5*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 450*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*
g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 600*(b*x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g^3*i*e^2*
log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 450*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g^3*i*e^2*log(-b*e +
(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 180*(b*x*e + a*e)^4*B*a^5*b^2*c*d^9*g^3*i*e^2*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B*a^6*b*d^10*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/
(d*x + c)^4 - 6*(b*x*e + a*e)^5*B*b^6*c^6*d^5*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 36*(
b*x*e + a*e)^5*B*a*b^5*c^5*d^6*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 90*(b*x*e + a*e)^5*
B*a^2*b^4*c^4*d^7*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 120*(b*x*e + a*e)^5*B*a^3*b^3*c^
3*d^8*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 90*(b*x*e + a*e)^5*B*a^4*b^2*c^2*d^9*g^3*i*e
*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 36*(b*x*e + a*e)^5*B*a^5*b*c*d^10*g^3*i*e*log(-b*e + (b*x
*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 6*(b*x*e + a*e)^5*B*a^6*d^11*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c)^5 - 30*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 180*(b*x
*e + a*e)^4*B*a*b^6*c^5*d^5*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 450*(b*x*e + a*e)^4*B*a^2*b^5
*c^4*d^6*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 600*(b*x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g^3*i*e^2*
log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 450*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g^3*i*e^2*log((b*x*e + a*e)/(
d*x + c))/(d*x + c)^4 + 180*(b*x*e + a*e)^4*B*a^5*b^2*c*d^9*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4
- 30*(b*x*e + a*e)^4*B*a^6*b*d^10*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*(b*x*e + a*e)^5*B*b^
6*c^6*d^5*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 36*(b*x*e + a*e)^5*B*a*b^5*c^5*d^6*g^3*i*e*log((b
*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 90*(b*x*e + a*e)^5*B*a^2*b^4*c^4*d^7*g^3*i*e*log((b*x*e + a*e)/(d*x + c))
/(d*x + c)^5 - 120*(b*x*e + a*e)^5*B*a^3*b^3*c^3*d^8*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 90*(b*
x*e + a*e)^5*B*a^4*b^2*c^2*d^9*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 36*(b*x*e + a*e)^5*B*a^5*b*c
*d^10*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 6*(b*x*e + a*e)^5*B*a^6*d^11*g^3*i*e*log((b*x*e + a*e
)/(d*x + c))/(d*x + c)^5 + 6*A*b^11*c^6*g^3*i*e^6 + 5*B*b^11*c^6*g^3*i*e^6 - 36*A*a*b^10*c^5*d*g^3*i*e^6 - 30*
B*a*b^10*c^5*d*g^3*i*e^6 + 90*A*a^2*b^9*c^4*d^2*g^3*i*e^6 + 75*B*a^2*b^9*c^4*d^2*g^3*i*e^6 - 120*A*a^3*b^8*c^3
*d^3*g^3*i*e^6 - 100*B*a^3*b^8*c^3*d^3*g^3*i*e^6 + 90*A*a^4*b^7*c^2*d^4*g^3*i*e^6 + 75*B*a^4*b^7*c^2*d^4*g^3*i
*e^6 - 36*A*a^5*b^6*c*d^5*g^3*i*e^6 - 30*B*a^5*b^6*c*d^5*g^3*i*e^6 + 6*A*a^6*b^5*d^6*g^3*i*e^6 + 5*B*a^6*b^5*d
^6*g^3*i*e^6 - 30*(b*x*e + a*e)*A*b^10*c^6*d*g^3*i*e^5/(d*x + c) - 19*(b*x*e + a*e)*B*b^10*c^6*d*g^3*i*e^5/(d*
x + c) + 180*(b*x*e + a*e)*A*a*b^9*c^5*d^2*g^3*i*e^5/(d*x + c) + 114*(b*x*e + a*e)*B*a*b^9*c^5*d^2*g^3*i*e^5/(
d*x + c) - 450*(b*x*e + a*e)*A*a^2*b^8*c^4*d^3*g^3*i*e^5/(d*x + c) - 285*(b*x*e + a*e)*B*a^2*b^8*c^4*d^3*g^3*i
*e^5/(d*x + c) + 600*(b*x*e + a*e)*A*a^3*b^7*c^3*d^4*g^3*i*e^5/(d*x + c) + 380*(b*x*e + a*e)*B*a^3*b^7*c^3*d^4
*g^3*i*e^5/(d*x + c) - 450*(b*x*e + a*e)*A*a^4*b^6*c^2*d^5*g^3*i*e^5/(d*x + c) - 285*(b*x*e + a*e)*B*a^4*b^6*c
^2*d^5*g^3*i*e^5/(d*x + c) + 180*(b*x*e + a*e)*A*a^5*b^5*c*d^6*g^3*i*e^5/(d*x + c) + 114*(b*x*e + a*e)*B*a^5*b
^5*c*d^6*g^3*i*e^5/(d*x + c) - 30*(b*x*e + a*e)*A*a^6*b^4*d^7*g^3*i*e^5/(d*x + c) - 19*(b*x*e + a*e)*B*a^6*b^4
*d^7*g^3*i*e^5/(d*x + c) + 60*(b*x*e + a*e)^2*A*b^9*c^6*d^2*g^3*i*e^4/(d*x + c)^2 + 23*(b*x*e + a*e)^2*B*b^9*c
^6*d^2*g^3*i*e^4/(d*x + c)^2 - 360*(b*x*e + a*e)^2*A*a*b^8*c^5*d^3*g^3*i*e^4/(d*x + c)^2 - 138*(b*x*e + a*e)^2
*B*a*b^8*c^5*d^3*g^3*i*e^4/(d*x + c)^2 + 900*(b*x*e + a*e)^2*A*a^2*b^7*c^4*d^4*g^3*i*e^4/(d*x + c)^2 + 345*(b*
x*e + a*e)^2*B*a^2*b^7*c^4*d^4*g^3*i*e^4/(d*x + c)^2 - 1200*(b*x*e + a*e)^2*A*a^3*b^6*c^3*d^5*g^3*i*e^4/(d*x +
c)^2 - 460*(b*x*e + a*e)^2*B*a^3*b^6*c^3*d^5*g^3*i*e^4/(d*x + c)^2 + 900*(b*x*e + a*e)^2*A*a^4*b^5*c^2*d^6*g^
3*i*e^4/(d*x + c)^2 + 345*(b*x*e + a*e)^2*B*a^4*b^5*c^2*d^6*g^3*i*e^4/(d*x + c)^2 - 360*(b*x*e + a*e)^2*A*a^5*
b^4*c*d^7*g^3*i*e^4/(d*x + c)^2 - 138*(b*x*e + a*e)^2*B*a^5*b^4*c*d^7*g^3*i*e^4/(d*x + c)^2 + 60*(b*x*e + a*e)
^2*A*a^6*b^3*d^8*g^3*i*e^4/(d*x + c)^2 + 23*(b*x*e + a*e)^2*B*a^6*b^3*d^8*g^3*i*e^4/(d*x + c)^2 - 60*(b*x*e +
a*e)^3*A*b^8*c^6*d^3*g^3*i*e^3/(d*x + c)^3 - 3*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g^3*i*e^3/(d*x + c)^3 + 360*(b*x*
e + a*e)^3*A*a*b^7*c^5*d^4*g^3*i*e^3/(d*x + c)^3 + 18*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g^3*i*e^3/(d*x + c)^3 -
900*(b*x*e + a*e)^3*A*a^2*b^6*c^4*d^5*g^3*i*e^3/(d*x + c)^3 - 45*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g^3*i*e^3/(
d*x + c)^3 + 1200*(b*x*e + a*e)^3*A*a^3*b^5*c^3*d^6*g^3*i*e^3/(d*x + c)^3 + 60*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d
^6*g^3*i*e^3/(d*x + c)^3 - 900*(b*x*e + a*e)^3*A*a^4*b^4*c^2*d^7*g^3*i*e^3/(d*x + c)^3 - 45*(b*x*e + a*e)^3*B*
a^4*b^4*c^2*d^7*g^3*i*e^3/(d*x + c)^3 + 360*(b*x*e + a*e)^3*A*a^5*b^3*c*d^8*g^3*i*e^3/(d*x + c)^3 + 18*(b*x*e
+ a*e)^3*B*a^5*b^3*c*d^8*g^3*i*e^3/(d*x + c)^3 - 60*(b*x*e + a*e)^3*A*a^6*b^2*d^9*g^3*i*e^3/(d*x + c)^3 - 3*(b
*x*e + a*e)^3*B*a^6*b^2*d^9*g^3*i*e^3/(d*x + c)^3 - 6*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g^3*i*e^2/(d*x + c)^4 + 36
*(b*x*e + a*e)^4*B*a*b^6*c^5*d^5*g^3*i*e^2/(d*x + c)^4 - 90*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*g^3*i*e^2/(d*x +
c)^4 + 120*(b*x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g^3*i*e^2/(d*x + c)^4 - 90*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g^3
*i*e^2/(d*x + c)^4 + 36*(b*x*e + a*e)^4*B*a^5*b^2*c*d^9*g^3*i*e^2/(d*x + c)^4 - 6*(b*x*e + a*e)^4*B*a^6*b*d^10
*g^3*i*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^7*d^4*e^5 -
5*(b*x*e + a*e)*b^6*d^5*e^4/(d*x + c) + 10*(b*x*e + a*e)^2*b^5*d^6*e^3/(d*x + c)^2 - 10*(b*x*e + a*e)^3*b^4*d
^7*e^2/(d*x + c)^3 + 5*(b*x*e + a*e)^4*b^3*d^8*e/(d*x + c)^4 - (b*x*e + a*e)^5*b^2*d^9/(d*x + c)^5)

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maple [B]  time = 0.18, size = 7284, normalized size = 34.36 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(d*i*x+c*i)*(B*ln((b*x+a)/(d*x+c)*e)+A),x)

[Out]

result too large to display

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maxima [B]  time = 1.39, size = 1022, normalized size = 4.82 \[ \frac {1}{5} \, A b^{3} d g^{3} i x^{5} + \frac {1}{4} \, A b^{3} c g^{3} i x^{4} + \frac {3}{4} \, A a b^{2} d g^{3} i x^{4} + A a b^{2} c g^{3} i x^{3} + A a^{2} b d g^{3} i x^{3} + \frac {3}{2} \, A a^{2} b c g^{3} i x^{2} + \frac {1}{2} \, A a^{3} d g^{3} 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^{3} c g^{3} i + \frac {3}{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} b c g^{3} i + \frac {1}{2} \, {\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^{2} c g^{3} 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^{3} c g^{3} 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^{3} d g^{3} i + \frac {1}{2} \, {\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^{2} b d g^{3} i + \frac {1}{8} \, {\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 a b^{2} d g^{3} i + \frac {1}{60} \, {\left (12 \, x^{5} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{3} d g^{3} i + A a^{3} c g^{3} i x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/5*A*b^3*d*g^3*i*x^5 + 1/4*A*b^3*c*g^3*i*x^4 + 3/4*A*a*b^2*d*g^3*i*x^4 + A*a*b^2*c*g^3*i*x^3 + A*a^2*b*d*g^3*
i*x^3 + 3/2*A*a^2*b*c*g^3*i*x^2 + 1/2*A*a^3*d*g^3*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^3*c*g^3*i + 3/2*(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*b*c*g^3*i + 1/2*(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^2*c*g^3*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*l
og(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^3*c*g^3*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^3*d*g^3*i + 1/2*(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*b*d*g
^3*i + 1/8*(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^2*d*g^
3*i + 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^3*d*g^3*i + A*a^3*c*g^3*i*x

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mupad [B]  time = 5.38, size = 1195, normalized size = 5.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))),x)

[Out]

x*((a*c*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))/5 - (A*b^2*g^3*i*(20*a*d + 20*b
*c))/20))/(20*b*d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d - 2*B*a*b*c
*d))/(4*d) + A*a*b^2*c*g^3*i))/(b*d) - ((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((b^2*g^3*i*
(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^3*i*(24*A*a^2*
d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d - 2*B*a*b*c*d))/(4*d) + A*a*b^2*c*g^3*i))/(20*b*d)
- (a*c*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(b*d) + (a*
g^3*i*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 12*A*a*b*c*d))/d))/(20*b*d) + (a^2*g^3*i*(2*A*a^2*d
^2 + 12*A*b^2*c^2 + B*a^2*d^2 - 3*B*b^2*c^2 + 16*A*a*b*c*d + 2*B*a*b*c*d))/(2*b*d)) + x^4*((b^2*g^3*i*(20*A*a*
d + 10*A*b*c + B*a*d - B*b*c))/20 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/80) - x^3*(((20*a*d + 20*b*c)*((b^2*g^3*i*
(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(60*b*d) - (b*g^3*i*(24*A*a^2*
d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d - 2*B*a*b*c*d))/(12*d) + (A*a*b^2*c*g^3*i)/3) + x^2
*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))/5 - (A*b^2*g^3*i*(
20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*
d - 2*B*a*b*c*d))/(4*d) + A*a*b^2*c*g^3*i))/(40*b*d) - (a*c*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d - B*b*c))
/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(2*b*d) + (a*g^3*i*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^
2 + 12*A*a*b*c*d))/(2*d)) + log((e*(a + b*x))/(c + d*x))*((B*a^2*g^3*i*x^2*(a*d + 3*b*c))/2 + (B*b^2*g^3*i*x^4
*(3*a*d + b*c))/4 + B*a^3*c*g^3*i*x + (B*b^3*d*g^3*i*x^5)/5 + B*a*b*g^3*i*x^3*(a*d + b*c)) - (log(a + b*x)*(B*
a^5*d*g^3*i - 5*B*a^4*b*c*g^3*i))/(20*b^2) + (log(c + d*x)*(B*b^3*c^5*g^3*i - 10*B*a^3*c^2*d^3*g^3*i - 5*B*a*b
^2*c^4*d*g^3*i + 10*B*a^2*b*c^3*d^2*g^3*i))/(20*d^4) + (A*b^3*d*g^3*i*x^5)/5

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sympy [B]  time = 7.95, size = 1158, normalized size = 5.46 \[ \frac {A b^{3} d g^{3} i x^{5}}{5} - \frac {B a^{4} g^{3} i \left (a d - 5 b c\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{3} i + \frac {B a^{5} d^{4} g^{3} i \left (a d - 5 b c\right )}{b} - 15 B a^{4} b c^{2} d^{3} g^{3} i - B a^{4} c d^{3} g^{3} i \left (a d - 5 b c\right ) + 10 B a^{3} b^{2} c^{3} d^{2} g^{3} i - 5 B a^{2} b^{3} c^{4} d g^{3} i + B a b^{4} c^{5} g^{3} i}{B a^{5} d^{5} g^{3} i - 5 B a^{4} b c d^{4} g^{3} i - 10 B a^{3} b^{2} c^{2} d^{3} g^{3} i + 10 B a^{2} b^{3} c^{3} d^{2} g^{3} i - 5 B a b^{4} c^{4} d g^{3} i + B b^{5} c^{5} g^{3} i} \right )}}{20 b^{2}} - \frac {B c^{2} g^{3} i \left (10 a^{3} d^{3} - 10 a^{2} b c d^{2} + 5 a b^{2} c^{2} d - b^{3} c^{3}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{3} i - 15 B a^{4} b c^{2} d^{3} g^{3} i + 10 B a^{3} b^{2} c^{3} d^{2} g^{3} i - 5 B a^{2} b^{3} c^{4} d g^{3} i + B a b^{4} c^{5} g^{3} i + B a b c^{2} g^{3} i \left (10 a^{3} d^{3} - 10 a^{2} b c d^{2} + 5 a b^{2} c^{2} d - b^{3} c^{3}\right ) - \frac {B b^{2} c^{3} g^{3} i \left (10 a^{3} d^{3} - 10 a^{2} b c d^{2} + 5 a b^{2} c^{2} d - b^{3} c^{3}\right )}{d}}{B a^{5} d^{5} g^{3} i - 5 B a^{4} b c d^{4} g^{3} i - 10 B a^{3} b^{2} c^{2} d^{3} g^{3} i + 10 B a^{2} b^{3} c^{3} d^{2} g^{3} i - 5 B a b^{4} c^{4} d g^{3} i + B b^{5} c^{5} g^{3} i} \right )}}{20 d^{4}} + x^{4} \left (\frac {3 A a b^{2} d g^{3} i}{4} + \frac {A b^{3} c g^{3} i}{4} + \frac {B a b^{2} d g^{3} i}{20} - \frac {B b^{3} c g^{3} i}{20}\right ) + x^{3} \left (A a^{2} b d g^{3} i + A a b^{2} c g^{3} i + \frac {11 B a^{2} b d g^{3} i}{60} - \frac {B a b^{2} c g^{3} i}{6} - \frac {B b^{3} c^{2} g^{3} i}{60 d}\right ) + x^{2} \left (\frac {A a^{3} d g^{3} i}{2} + \frac {3 A a^{2} b c g^{3} i}{2} + \frac {9 B a^{3} d g^{3} i}{40} - \frac {B a^{2} b c g^{3} i}{8} - \frac {B a b^{2} c^{2} g^{3} i}{8 d} + \frac {B b^{3} c^{3} g^{3} i}{40 d^{2}}\right ) + x \left (A a^{3} c g^{3} i + \frac {B a^{4} d g^{3} i}{20 b} + \frac {B a^{3} c g^{3} i}{4} - \frac {B a^{2} b c^{2} g^{3} i}{2 d} + \frac {B a b^{2} c^{3} g^{3} i}{4 d^{2}} - \frac {B b^{3} c^{4} g^{3} i}{20 d^{3}}\right ) + \left (B a^{3} c g^{3} i x + \frac {B a^{3} d g^{3} i x^{2}}{2} + \frac {3 B a^{2} b c g^{3} i x^{2}}{2} + B a^{2} b d g^{3} i x^{3} + B a b^{2} c g^{3} i x^{3} + \frac {3 B a b^{2} d g^{3} i x^{4}}{4} + \frac {B b^{3} c g^{3} i x^{4}}{4} + \frac {B b^{3} d g^{3} i 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)**3*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**3*d*g**3*i*x**5/5 - B*a**4*g**3*i*(a*d - 5*b*c)*log(x + (B*a**5*c*d**4*g**3*i + B*a**5*d**4*g**3*i*(a*d -
5*b*c)/b - 15*B*a**4*b*c**2*d**3*g**3*i - B*a**4*c*d**3*g**3*i*(a*d - 5*b*c) + 10*B*a**3*b**2*c**3*d**2*g**3*
i - 5*B*a**2*b**3*c**4*d*g**3*i + B*a*b**4*c**5*g**3*i)/(B*a**5*d**5*g**3*i - 5*B*a**4*b*c*d**4*g**3*i - 10*B*
a**3*b**2*c**2*d**3*g**3*i + 10*B*a**2*b**3*c**3*d**2*g**3*i - 5*B*a*b**4*c**4*d*g**3*i + B*b**5*c**5*g**3*i))
/(20*b**2) - B*c**2*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3)*log(x + (B*a**5*c*d
**4*g**3*i - 15*B*a**4*b*c**2*d**3*g**3*i + 10*B*a**3*b**2*c**3*d**2*g**3*i - 5*B*a**2*b**3*c**4*d*g**3*i + B*
a*b**4*c**5*g**3*i + B*a*b*c**2*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3) - B*b**
2*c**3*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3)/d)/(B*a**5*d**5*g**3*i - 5*B*a**
4*b*c*d**4*g**3*i - 10*B*a**3*b**2*c**2*d**3*g**3*i + 10*B*a**2*b**3*c**3*d**2*g**3*i - 5*B*a*b**4*c**4*d*g**3
*i + B*b**5*c**5*g**3*i))/(20*d**4) + x**4*(3*A*a*b**2*d*g**3*i/4 + A*b**3*c*g**3*i/4 + B*a*b**2*d*g**3*i/20 -
B*b**3*c*g**3*i/20) + x**3*(A*a**2*b*d*g**3*i + A*a*b**2*c*g**3*i + 11*B*a**2*b*d*g**3*i/60 - B*a*b**2*c*g**3
*i/6 - B*b**3*c**2*g**3*i/(60*d)) + x**2*(A*a**3*d*g**3*i/2 + 3*A*a**2*b*c*g**3*i/2 + 9*B*a**3*d*g**3*i/40 - B
*a**2*b*c*g**3*i/8 - B*a*b**2*c**2*g**3*i/(8*d) + B*b**3*c**3*g**3*i/(40*d**2)) + x*(A*a**3*c*g**3*i + B*a**4*
d*g**3*i/(20*b) + B*a**3*c*g**3*i/4 - B*a**2*b*c**2*g**3*i/(2*d) + B*a*b**2*c**3*g**3*i/(4*d**2) - B*b**3*c**4
*g**3*i/(20*d**3)) + (B*a**3*c*g**3*i*x + B*a**3*d*g**3*i*x**2/2 + 3*B*a**2*b*c*g**3*i*x**2/2 + B*a**2*b*d*g**
3*i*x**3 + B*a*b**2*c*g**3*i*x**3 + 3*B*a*b**2*d*g**3*i*x**4/4 + B*b**3*c*g**3*i*x**4/4 + B*b**3*d*g**3*i*x**5
/5)*log(e*(a + b*x)/(c + d*x))

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