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

Optimal. Leaf size=271 \[ -\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2}+\frac {B g i^3 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B g i^3 (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac {B g i^3 x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d)}{20 d^2} \]

[Out]

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

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Rubi [A]  time = 0.34, antiderivative size = 232, normalized size of antiderivative = 0.86, 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} \[ -\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2}+\frac {B g i^3 (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 (b c-a d)^5 \log (a+b x)}{20 b^4 d^2}+\frac {B g i^3 x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d)}{20 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^4*g*i^3*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2*g*i
^3*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4)/(20*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[a + b*x])
/(20*b^4*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2) + (b*g*i^3*(c + d
*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*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 (22 c+22 d x)^3 (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 (22 c+22 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {b g (22 c+22 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{22 d}\right ) \, dx\\ &=\frac {(b g) \int (22 c+22 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{22 d}+\frac {((-b c+a d) g) \int (22 c+22 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d}\\ &=-\frac {2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {10648 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac {(b B g) \int \frac {5153632 (b c-a d) (c+d x)^4}{a+b x} \, dx}{2420 d^2}+\frac {(B (b c-a d) g) \int \frac {234256 (b c-a d) (c+d x)^3}{a+b x} \, dx}{88 d^2}\\ &=-\frac {2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {10648 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac {(10648 b B (b c-a d) g) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^2}+\frac {\left (2662 B (b c-a d)^2 g\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^2}\\ &=-\frac {2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {10648 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}-\frac {(10648 b B (b c-a d) g) \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^2}+\frac {\left (2662 B (b c-a d)^2 g\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}{d^2}\\ &=\frac {2662 B (b c-a d)^4 g x}{5 b^3 d}+\frac {1331 B (b c-a d)^3 g (c+d x)^2}{5 b^2 d^2}+\frac {2662 B (b c-a d)^2 g (c+d x)^3}{15 b d^2}-\frac {2662 B (b c-a d) g (c+d x)^4}{5 d^2}+\frac {2662 B (b c-a d)^5 g \log (a+b x)}{5 b^4 d^2}-\frac {2662 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {10648 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 261, normalized size = 0.96 \[ \frac {g i^3 \left (24 b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-30 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {5 B (b c-a d)^2 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^4}-\frac {2 B (b c-a d) \left (4 b^3 (c+d x)^3 (b c-a d)+6 b^2 (c+d x)^2 (b c-a d)^2+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{b^4}\right )}{120 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*((5*B*(b*c - a*d)^2*(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^4 - (2*B*(b*c - a*d)*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 + 4*
b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]))/b^4 - 30*(b*c - a*d)*(c + d*
x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*b*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(120*d^2)

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fricas [A]  time = 0.96, size = 502, normalized size = 1.85 \[ \frac {24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \, {\left ({\left (15 \, A - B\right )} b^{5} c d^{4} + {\left (5 \, A + B\right )} a b^{4} d^{5}\right )} g i^{3} x^{4} + 2 \, {\left ({\left (60 \, A - 11 \, B\right )} b^{5} c^{2} d^{3} + 10 \, {\left (6 \, A + B\right )} a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g i^{3} x^{3} + 3 \, {\left ({\left (20 \, A - 9 \, B\right )} b^{5} c^{3} d^{2} + 5 \, {\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g i^{3} x^{2} - 6 \, {\left (B b^{5} c^{4} d - 5 \, {\left (4 \, A - B\right )} a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} x + 6 \, {\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} \log \left (d x + c\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{120 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.44, size = 5710, normalized size = 21.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(6*B*b^11*c^6*g*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 36*B*a*b^10*c^5*d*g*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*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*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*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*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 6*B*a^6*b^5*d^6*g*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*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*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*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*B*a^3
*b^7*c^3*d^4*g*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*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*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*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*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 3
60*(b*x*e + a*e)^2*B*a*b^8*c^5*d^3*g*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*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*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^4*b^5*c^2*d^6*
g*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*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*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*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*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 90
0*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g*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*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*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
*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*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*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*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*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 6
00*(b*x*e + a*e)^4*B*a^3*b^4*c^3*d^7*g*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*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*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*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*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*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*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*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*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*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*i*e*log(-b*e +
(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 60*(b*x*e + a*e)^2*B*b^9*c^6*d^2*g*i*e^4*log((b*x*e + a*e)/(d*x + c))
/(d*x + c)^2 + 360*(b*x*e + a*e)^2*B*a*b^8*c^5*d^3*g*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 900*(b*x
*e + a*e)^2*B*a^2*b^7*c^4*d^4*g*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 1200*(b*x*e + a*e)^2*B*a^3*b^
6*c^3*d^5*g*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 900*(b*x*e + a*e)^2*B*a^4*b^5*c^2*d^6*g*i*e^4*log
((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 360*(b*x*e + a*e)^2*B*a^5*b^4*c*d^7*g*i*e^4*log((b*x*e + a*e)/(d*x + c
))/(d*x + c)^2 - 60*(b*x*e + a*e)^2*B*a^6*b^3*d^8*g*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 60*(b*x*e
+ a*e)^3*B*b^8*c^6*d^3*g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 360*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4
*g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 900*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g*i*e^3*log((b*x*e +
a*e)/(d*x + c))/(d*x + c)^3 - 1200*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d^6*g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*
x + c)^3 + 900*(b*x*e + a*e)^3*B*a^4*b^4*c^2*d^7*g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 360*(b*x*e
+ a*e)^3*B*a^5*b^3*c*d^8*g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 60*(b*x*e + a*e)^3*B*a^6*b^2*d^9*
g*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 30*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g*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*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*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*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*
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*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*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*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*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*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*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*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*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*i*e*log((b*x*
e + a*e)/(d*x + c))/(d*x + c)^5 + 6*A*b^11*c^6*g*i*e^6 - 5*B*b^11*c^6*g*i*e^6 - 36*A*a*b^10*c^5*d*g*i*e^6 + 30
*B*a*b^10*c^5*d*g*i*e^6 + 90*A*a^2*b^9*c^4*d^2*g*i*e^6 - 75*B*a^2*b^9*c^4*d^2*g*i*e^6 - 120*A*a^3*b^8*c^3*d^3*
g*i*e^6 + 100*B*a^3*b^8*c^3*d^3*g*i*e^6 + 90*A*a^4*b^7*c^2*d^4*g*i*e^6 - 75*B*a^4*b^7*c^2*d^4*g*i*e^6 - 36*A*a
^5*b^6*c*d^5*g*i*e^6 + 30*B*a^5*b^6*c*d^5*g*i*e^6 + 6*A*a^6*b^5*d^6*g*i*e^6 - 5*B*a^6*b^5*d^6*g*i*e^6 - 30*(b*
x*e + a*e)*A*b^10*c^6*d*g*i*e^5/(d*x + c) + 31*(b*x*e + a*e)*B*b^10*c^6*d*g*i*e^5/(d*x + c) + 180*(b*x*e + a*e
)*A*a*b^9*c^5*d^2*g*i*e^5/(d*x + c) - 186*(b*x*e + a*e)*B*a*b^9*c^5*d^2*g*i*e^5/(d*x + c) - 450*(b*x*e + a*e)*
A*a^2*b^8*c^4*d^3*g*i*e^5/(d*x + c) + 465*(b*x*e + a*e)*B*a^2*b^8*c^4*d^3*g*i*e^5/(d*x + c) + 600*(b*x*e + a*e
)*A*a^3*b^7*c^3*d^4*g*i*e^5/(d*x + c) - 620*(b*x*e + a*e)*B*a^3*b^7*c^3*d^4*g*i*e^5/(d*x + c) - 450*(b*x*e + a
*e)*A*a^4*b^6*c^2*d^5*g*i*e^5/(d*x + c) + 465*(b*x*e + a*e)*B*a^4*b^6*c^2*d^5*g*i*e^5/(d*x + c) + 180*(b*x*e +
a*e)*A*a^5*b^5*c*d^6*g*i*e^5/(d*x + c) - 186*(b*x*e + a*e)*B*a^5*b^5*c*d^6*g*i*e^5/(d*x + c) - 30*(b*x*e + a*
e)*A*a^6*b^4*d^7*g*i*e^5/(d*x + c) + 31*(b*x*e + a*e)*B*a^6*b^4*d^7*g*i*e^5/(d*x + c) - 47*(b*x*e + a*e)^2*B*b
^9*c^6*d^2*g*i*e^4/(d*x + c)^2 + 282*(b*x*e + a*e)^2*B*a*b^8*c^5*d^3*g*i*e^4/(d*x + c)^2 - 705*(b*x*e + a*e)^2
*B*a^2*b^7*c^4*d^4*g*i*e^4/(d*x + c)^2 + 940*(b*x*e + a*e)^2*B*a^3*b^6*c^3*d^5*g*i*e^4/(d*x + c)^2 - 705*(b*x*
e + a*e)^2*B*a^4*b^5*c^2*d^6*g*i*e^4/(d*x + c)^2 + 282*(b*x*e + a*e)^2*B*a^5*b^4*c*d^7*g*i*e^4/(d*x + c)^2 - 4
7*(b*x*e + a*e)^2*B*a^6*b^3*d^8*g*i*e^4/(d*x + c)^2 + 27*(b*x*e + a*e)^3*B*b^8*c^6*d^3*g*i*e^3/(d*x + c)^3 - 1
62*(b*x*e + a*e)^3*B*a*b^7*c^5*d^4*g*i*e^3/(d*x + c)^3 + 405*(b*x*e + a*e)^3*B*a^2*b^6*c^4*d^5*g*i*e^3/(d*x +
c)^3 - 540*(b*x*e + a*e)^3*B*a^3*b^5*c^3*d^6*g*i*e^3/(d*x + c)^3 + 405*(b*x*e + a*e)^3*B*a^4*b^4*c^2*d^7*g*i*e
^3/(d*x + c)^3 - 162*(b*x*e + a*e)^3*B*a^5*b^3*c*d^8*g*i*e^3/(d*x + c)^3 + 27*(b*x*e + a*e)^3*B*a^6*b^2*d^9*g*
i*e^3/(d*x + c)^3 - 6*(b*x*e + a*e)^4*B*b^7*c^6*d^4*g*i*e^2/(d*x + c)^4 + 36*(b*x*e + a*e)^4*B*a*b^6*c^5*d^5*g
*i*e^2/(d*x + c)^4 - 90*(b*x*e + a*e)^4*B*a^2*b^5*c^4*d^6*g*i*e^2/(d*x + c)^4 + 120*(b*x*e + a*e)^4*B*a^3*b^4*
c^3*d^7*g*i*e^2/(d*x + c)^4 - 90*(b*x*e + a*e)^4*B*a^4*b^3*c^2*d^8*g*i*e^2/(d*x + c)^4 + 36*(b*x*e + a*e)^4*B*
a^5*b^2*c*d^9*g*i*e^2/(d*x + c)^4 - 6*(b*x*e + a*e)^4*B*a^6*b*d^10*g*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^9*d^2*e^5 - 5*(b*x*e + a*e)*b^8*d^3*e^4/(d*x + c) + 10*(b
*x*e + a*e)^2*b^7*d^4*e^3/(d*x + c)^2 - 10*(b*x*e + a*e)^3*b^6*d^5*e^2/(d*x + c)^3 + 5*(b*x*e + a*e)^4*b^5*d^6
*e/(d*x + c)^4 - (b*x*e + a*e)^5*b^4*d^7/(d*x + c)^5)

________________________________________________________________________________________

maple [B]  time = 0.16, size = 4481, normalized size = 16.54 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*e^5*B*g*i^3*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)^5*a^2*c^8/(d*x+c)^5+9*e^4*
B*g*i^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*c^7/(d*x+c)^4*a^2*b^3+1/d*e^5*B*g*
i^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)^5*a*c^4*b^5-1/2*e*B*g*i^3/(1/(d*x+c)*a*d
*e-1/(d*x+c)*b*c*e)*a^2*c^3-d^2*e^5*B*g*i^3*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)^5*a^4*c-5/4*d^2*e^4*B*g*i^3*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)^4*a^4*c+5/
2*d*e^4*B*g*i^3*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)^4*a^3*c^2+42*d^4*e^5*B*g
*i^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)^5*a^6*c^4/(d*x+c)^5+1/4/d*B*g*i^3*ln(-b
*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*c^4*a-1/20/d^2*B*g*i^3*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*c^5*b-1/20
/d^2*e*B*g*i^3*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^5-1/12*d^2*e^3*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e
)^3*a^4*c+1/2*d*B*g*i^3/b^2*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^3*c^2-1/4*d^2*B*g*i^3/b^3*ln(-b*e+(b/d*
e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^4*c-1/2*B*g*i^3/b*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^2*c^3+1/20*d^3*B*g*
i^3/b^4*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^5+1/4*d^3*e^4*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a
^5+1/20*d^3*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^5+2*d*e^5*A*g*i^3*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)
*b*c*e)^5*a^3*c^2+5/2*d*e^4*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^3*b^2*c^2+1/12/d*e^3*B*g*i^3*b^3/(1/
(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^4*a-5/4*d^2*e^4*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^4*b*c+1/8*d^2
*e^2*B*g*i^3/b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^4*c-1/8/d*e^2*B*g*i^3*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*
e)^2*a*c^4+1/6*d*e^3*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^3*c^2*b-d^2*e^5*A*g*i^3*b^2/(1/(d*x+c)*a*d*
e-1/(d*x+c)*b*c*e)^5*a^4*c-1/4*d^2*e*B*g*i^3/b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^4*c+1/2*d*e*B*g*i^3/b/(1/
(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^3*c^2+1/d*e^5*A*g*i^3*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^5*c^4*a+5/4/d*e^4
*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*b^4*c^4*a-1/4*d^2*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4
*a^4*b*c+1/2*d*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^3*b^2*c^2+1/4/d*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-
1/(d*x+c)*b*c*e)^4*b^4*c^4*a+1/4/d*e*B*g*i^3*b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^4*a-1/5/d^2*e^5*B*g*i^3*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)^5*c^5-1/4/d^2*e^4*B*g*i^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*c^5*b^5+1/5*d^3*e^5*B*g*i^3*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)^5*a^5-5/2*e^4*B*g*i^3*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)^4*a^2*c^3-2*e^5*B*g*i^3*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)^5*a^2*c^3-9*d^5*e^4*B*g*i^3*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)^4*a^7*c^2
/(d*x+c)^4-24*d*e^5*B*g*i^3*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)^5*a^3*c^7/(d
*x+c)^5-2/d*e^5*B*g*i^3*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)^5*a*c^9/(d*x+c)^
5-2*d^7*e^5*B*g*i^3/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)^5*a^9*c/(d*x+c)^5+9*
d^6*e^5*B*g*i^3/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)^5*a^8*c^2/(d*x+c)^5+42*d
^2*e^5*B*g*i^3*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)^5*a^4*c^6/(d*x+c)^5-24*d^
5*e^5*B*g*i^3/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)^5*a^7*c^3/(d*x+c)^5-252/5*d^
3*e^5*B*g*i^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)^5*a^5*c^5/(d*x+c)^5*b+63/2*d^2
*e^4*B*g*i^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*c^5/(d*x+c)^4*a^4*b-21*d*e^4*
B*g*i^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*c^6/(d*x+c)^4*a^3*b^2-9/4/d*e^4*B*
g*i^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*c^8/(d*x+c)^4*a*b^4+21*d^4*e^4*B*g*i
^3*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)^4*a^6*c^3/(d*x+c)^4+9/4*d^6*e^4*B*g*i^3
*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)^4*a^8/(d*x+c)^4*c-1/4*d^7*e^4*B*g*i^3*l
n(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)^4*a^9/(d*x+c)^4+5/4/d*e^4*B*g*i^3*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)^4*a*c^4+2*d*e^5*B*g*i^3*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)^5*a^3*c^2+1/5/d^2*e^5*B*g*i^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)^5*c^10/(d*x+c)^5*b^6+1/4/d^2*e^4*B*g*i^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*c^9/(d*x+c)^4*b^5-63/2*d^3*e^4*B*g*i^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^5*c^4/(d*x+c)^4+1/5*d^8*e^5*B*g*i^3/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)^5*a^10/(d*x+c)^5-1/4/d^2*e^4*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*b^5*c^5-1/
5/d^2*e^5*A*g*i^3*b^6/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^5*c^5-2*e^5*A*g*i^3*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c
*e)^5*a^2*c^3-5/2*e^4*A*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^2*b^3*c^3+1/4*e^2*B*g*i^3*b/(1/(d*x+c)*a*d
*e-1/(d*x+c)*b*c*e)^2*a^2*c^3-1/2*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*a^2*b^3*c^3-1/6*e^3*B*g*i^3*
b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c^3+1/20*d^3*e*B*g*i^3/b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^5+1
/40/d^2*e^2*B*g*i^3*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^5-1/4*d*e^2*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b
*c*e)^2*a^3*c^2-1/60/d^2*e^3*B*g*i^3*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^5-1/40*d^3*e^2*B*g*i^3/b^2/(1/(
d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^5+1/4*d^3*e^4*B*g*i^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^5+1/60*d^3*e^3*B*g*i^3/b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^5+1/5*d^3*e^5*A*g*i^3*b/(1/(d*x
+c)*a*d*e-1/(d*x+c)*b*c*e)^5*a^5-1/20/d^2*e^4*B*g*i^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^4*b^5*c^5

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maxima [B]  time = 1.34, size = 1022, normalized size = 3.77 \[ \frac {1}{5} \, A b d^{3} g i^{3} x^{5} + \frac {3}{4} \, A b c d^{2} g i^{3} x^{4} + \frac {1}{4} \, A a d^{3} g i^{3} x^{4} + A b c^{2} d g i^{3} x^{3} + A a c d^{2} g i^{3} x^{3} + \frac {1}{2} \, A b c^{3} g i^{3} x^{2} + \frac {3}{2} \, A a c^{2} d g i^{3} 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^{3} g i^{3} + \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^{3} g i^{3} + \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 c^{2} d g i^{3} + \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 b c^{2} d g i^{3} + \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 c d^{2} g i^{3} + \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 b c d^{2} g i^{3} + \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 a d^{3} g i^{3} + \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 d^{3} g i^{3} + A a c^{3} g i^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.44, size = 1192, normalized size = 4.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 7.31, size = 1158, normalized size = 4.27 \[ \frac {A b d^{3} g i^{3} x^{5}}{5} - \frac {B a^{2} g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g i^{3} - 5 B a^{4} b c^{2} d^{3} g i^{3} + 10 B a^{3} b^{2} c^{3} d^{2} g i^{3} + \frac {B a^{3} d^{2} g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right )}{b} - 15 B a^{2} b^{3} c^{4} d g i^{3} - B a^{2} c d g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right ) + B a b^{4} c^{5} g i^{3}}{B a^{5} d^{5} g i^{3} - 5 B a^{4} b c d^{4} g i^{3} + 10 B a^{3} b^{2} c^{2} d^{3} g i^{3} - 10 B a^{2} b^{3} c^{3} d^{2} g i^{3} - 5 B a b^{4} c^{4} d g i^{3} + B b^{5} c^{5} g i^{3}} \right )}}{20 b^{4}} - \frac {B c^{4} g i^{3} \left (5 a d - b c\right ) \log {\left (x + \frac {B a^{5} c d^{4} g i^{3} - 5 B a^{4} b c^{2} d^{3} g i^{3} + 10 B a^{3} b^{2} c^{3} d^{2} g i^{3} - 15 B a^{2} b^{3} c^{4} d g i^{3} + B a b^{4} c^{5} g i^{3} + B a b^{3} c^{4} g i^{3} \left (5 a d - b c\right ) - \frac {B b^{4} c^{5} g i^{3} \left (5 a d - b c\right )}{d}}{B a^{5} d^{5} g i^{3} - 5 B a^{4} b c d^{4} g i^{3} + 10 B a^{3} b^{2} c^{2} d^{3} g i^{3} - 10 B a^{2} b^{3} c^{3} d^{2} g i^{3} - 5 B a b^{4} c^{4} d g i^{3} + B b^{5} c^{5} g i^{3}} \right )}}{20 d^{2}} + x^{4} \left (\frac {A a d^{3} g i^{3}}{4} + \frac {3 A b c d^{2} g i^{3}}{4} + \frac {B a d^{3} g i^{3}}{20} - \frac {B b c d^{2} g i^{3}}{20}\right ) + x^{3} \left (A a c d^{2} g i^{3} + A b c^{2} d g i^{3} + \frac {B a^{2} d^{3} g i^{3}}{60 b} + \frac {B a c d^{2} g i^{3}}{6} - \frac {11 B b c^{2} d g i^{3}}{60}\right ) + x^{2} \left (\frac {3 A a c^{2} d g i^{3}}{2} + \frac {A b c^{3} g i^{3}}{2} - \frac {B a^{3} d^{3} g i^{3}}{40 b^{2}} + \frac {B a^{2} c d^{2} g i^{3}}{8 b} + \frac {B a c^{2} d g i^{3}}{8} - \frac {9 B b c^{3} g i^{3}}{40}\right ) + x \left (A a c^{3} g i^{3} + \frac {B a^{4} d^{3} g i^{3}}{20 b^{3}} - \frac {B a^{3} c d^{2} g i^{3}}{4 b^{2}} + \frac {B a^{2} c^{2} d g i^{3}}{2 b} - \frac {B a c^{3} g i^{3}}{4} - \frac {B b c^{4} g i^{3}}{20 d}\right ) + \left (B a c^{3} g i^{3} x + \frac {3 B a c^{2} d g i^{3} x^{2}}{2} + B a c d^{2} g i^{3} x^{3} + \frac {B a d^{3} g i^{3} x^{4}}{4} + \frac {B b c^{3} g i^{3} x^{2}}{2} + B b c^{2} d g i^{3} x^{3} + \frac {3 B b c d^{2} g i^{3} x^{4}}{4} + \frac {B b d^{3} g i^{3} 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)*(d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

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

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