3.10 \(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\)
Optimal. Leaf size=423 \[ \frac {b^3 g^3 i^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4}-\frac {3 b^2 g^3 i^2 (c+d x)^5 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}-\frac {g^3 i^2 (c+d x)^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^4}+\frac {3 b g^3 i^2 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac {b^2 B g^3 i^2 (c+d x)^5 (b c-a d)}{30 d^4}+\frac {B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}+\frac {B g^3 i^2 (c+d x)^2 (b c-a d)^4}{120 b d^4}-\frac {19 B g^3 i^2 (c+d x)^3 (b c-a d)^3}{180 d^4}+\frac {13 b B g^3 i^2 (c+d x)^4 (b c-a d)^2}{120 d^4} \]
[Out]
1/60*B*(-a*d+b*c)^5*g^3*i^2*x/b^2/d^3+1/120*B*(-a*d+b*c)^4*g^3*i^2*(d*x+c)^2/b/d^4-19/180*B*(-a*d+b*c)^3*g^3*i
^2*(d*x+c)^3/d^4+13/120*b*B*(-a*d+b*c)^2*g^3*i^2*(d*x+c)^4/d^4-1/30*b^2*B*(-a*d+b*c)*g^3*i^2*(d*x+c)^5/d^4+1/6
0*B*(-a*d+b*c)^6*g^3*i^2*ln((b*x+a)/(d*x+c))/b^3/d^4-1/3*(-a*d+b*c)^3*g^3*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x
+c)))/d^4+3/4*b*(-a*d+b*c)^2*g^3*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4-3/5*b^2*(-a*d+b*c)*g^3*i^2*(d*x
+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/6*b^3*g^3*i^2*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/60*B*(-a*d+b
*c)^6*g^3*i^2*ln(d*x+c)/b^3/d^4
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Rubi [A] time = 0.66, antiderivative size = 330, normalized size of antiderivative = 0.78,
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} \[ \frac {d^2 g^3 i^2 (a+b x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^3}+\frac {g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^3}+\frac {2 d g^3 i^2 (a+b x)^5 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^3}-\frac {B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}+\frac {B g^3 i^2 (a+b x)^2 (b c-a d)^4}{120 b^3 d^2}+\frac {B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac {B g^3 i^2 (a+b x)^3 (b c-a d)^3}{180 b^3 d}-\frac {7 B g^3 i^2 (a+b x)^4 (b c-a d)^2}{120 b^3}-\frac {B d g^3 i^2 (a+b x)^5 (b c-a d)}{30 b^3} \]
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
-(B*(b*c - a*d)^5*g^3*i^2*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2)/(120*b^3*d^2) - (B*(b*c - a*
d)^3*g^3*i^2*(a + b*x)^3)/(180*b^3*d) - (7*B*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4)/(120*b^3) - (B*d*(b*c - a*d)*g
^3*i^2*(a + b*x)^5)/(30*b^3) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^3
) + (2*d*(b*c - a*d)*g^3*i^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^3) + (d^2*g^3*i^2*(a + b*x
)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3) + (B*(b*c - a*d)^6*g^3*i^2*Log[c + d*x])/(60*b^3*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 (10 c+10 d x)^2 (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 {100 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {200 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {100 d^2 (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}\right ) \, dx\\ &=\frac {\left (100 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2}+\frac {\left (100 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g^2}+\frac {(200 d (b c-a 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^2 g}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2\right ) \int \frac {(b c-a d) g^6 (a+b x)^5}{c+d x} \, dx}{3 b^3 g^3}-\frac {(40 B d (b c-a d)) \int \frac {(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{b^3 g^2}-\frac {\left (25 B (b c-a d)^2\right ) \int \frac {(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{b^3 g}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2 (b c-a d) g^3\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{3 b^3}-\frac {\left (40 B d (b c-a d)^2 g^3\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{b^3}-\frac {\left (25 B (b c-a d)^3 g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{b^3}\\ &=\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {\left (50 B d^2 (b c-a d) g^3\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{3 b^3}-\frac {\left (40 B d (b c-a d)^2 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}{b^3}-\frac {\left (25 B (b c-a d)^3 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}{b^3}\\ &=-\frac {5 B (b c-a d)^5 g^3 x}{3 b^2 d^3}+\frac {5 B (b c-a d)^4 g^3 (a+b x)^2}{6 b^3 d^2}-\frac {5 B (b c-a d)^3 g^3 (a+b x)^3}{9 b^3 d}-\frac {35 B (b c-a d)^2 g^3 (a+b x)^4}{6 b^3}-\frac {10 B d (b c-a d) g^3 (a+b x)^5}{3 b^3}+\frac {25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}+\frac {5 B (b c-a d)^6 g^3 \log (c+d x)}{3 b^3 d^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 429, normalized size = 1.01 \[ \frac {g^3 i^2 \left (60 d^6 (a+b x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+144 d^5 (a+b x)^5 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+90 d^4 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-15 B (b c-a d)^3 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+12 B (b c-a d)^2 \left (4 d^3 (a+b x)^3 (b c-a d)-6 d^2 (a+b x)^2 (b c-a d)^2+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )-B (b c-a d) \left (15 d^4 (a+b x)^4 (a d-b c)+20 d^3 (a+b x)^3 (b c-a d)^2+30 d^2 (a+b x)^2 (a d-b c)^3+60 b d x (b c-a d)^4-60 (b c-a d)^5 \log (c+d x)+12 d^5 (a+b x)^5\right )\right )}{360 b^3 d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
(g^3*i^2*(90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^5*(b*c - a*d)*(a + b*x
)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 60*d^6*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 15*B*(b*c
- a*d)^3*(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]) + 12*B*(b*c - a*d)^2*(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]) - B*(b*c - a*d)*(60*b*d*(b*c - a*d)^4*x + 30*d
^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*x)^4 + 12*d^
5*(a + b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x])))/(360*b^3*d^4)
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fricas [A] time = 1.38, size = 724, normalized size = 1.71 \[ \frac {60 \, A b^{6} d^{6} g^{3} i^{2} x^{6} + 12 \, {\left ({\left (12 \, A - B\right )} b^{6} c d^{5} + {\left (18 \, A + B\right )} a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 3 \, {\left ({\left (30 \, A - 7 \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (30 \, A - B\right )} a b^{5} c d^{5} + {\left (90 \, A + 13 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} - 2 \, {\left (B b^{6} c^{3} d^{3} - 3 \, {\left (60 \, A - 13 \, B\right )} a b^{5} c^{2} d^{4} - 3 \, {\left (120 \, A + 7 \, B\right )} a^{2} b^{4} c d^{5} - {\left (60 \, A + 19 \, B\right )} a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 3 \, {\left (B b^{6} c^{4} d^{2} - 6 \, B a b^{5} c^{3} d^{3} + 30 \, {\left (6 \, A - B\right )} a^{2} b^{4} c^{2} d^{4} + 2 \, {\left (60 \, A + 17 \, B\right )} a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{3} i^{2} x^{2} - 6 \, {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 15 \, B a^{2} b^{4} c^{3} d^{3} - 5 \, {\left (12 \, A + B\right )} a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{3} i^{2} x + 6 \, {\left (15 \, B a^{4} b^{2} c^{2} d^{4} - 6 \, B a^{5} b c d^{5} + B a^{6} d^{6}\right )} g^{3} i^{2} \log \left (b x + a\right ) + 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2} - 20 \, B a^{3} b^{3} c^{3} d^{3}\right )} g^{3} i^{2} \log \left (d x + c\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{3} i^{2} x^{6} + 60 \, B a^{3} b^{3} c^{2} d^{4} g^{3} i^{2} x + 12 \, {\left (2 \, B b^{6} c d^{5} + 3 \, B a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 15 \, {\left (B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + 3 \, B a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} + 20 \, {\left (3 \, B a b^{5} c^{2} d^{4} + 6 \, B a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 30 \, {\left (3 \, B a^{2} b^{4} c^{2} d^{4} + 2 \, B a^{3} b^{3} c d^{5}\right )} g^{3} i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{360 \, b^{3} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")
[Out]
1/360*(60*A*b^6*d^6*g^3*i^2*x^6 + 12*((12*A - B)*b^6*c*d^5 + (18*A + B)*a*b^5*d^6)*g^3*i^2*x^5 + 3*((30*A - 7*
B)*b^6*c^2*d^4 + 6*(30*A - B)*a*b^5*c*d^5 + (90*A + 13*B)*a^2*b^4*d^6)*g^3*i^2*x^4 - 2*(B*b^6*c^3*d^3 - 3*(60*
A - 13*B)*a*b^5*c^2*d^4 - 3*(120*A + 7*B)*a^2*b^4*c*d^5 - (60*A + 19*B)*a^3*b^3*d^6)*g^3*i^2*x^3 + 3*(B*b^6*c^
4*d^2 - 6*B*a*b^5*c^3*d^3 + 30*(6*A - B)*a^2*b^4*c^2*d^4 + 2*(60*A + 17*B)*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^3*
i^2*x^2 - 6*(B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 15*B*a^2*b^4*c^3*d^3 - 5*(12*A + B)*a^3*b^3*c^2*d^4 - 6*B*a^4*b
^2*c*d^5 + B*a^5*b*d^6)*g^3*i^2*x + 6*(15*B*a^4*b^2*c^2*d^4 - 6*B*a^5*b*c*d^5 + B*a^6*d^6)*g^3*i^2*log(b*x + a
) + 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2 - 20*B*a^3*b^3*c^3*d^3)*g^3*i^2*log(d*x + c) + 6*(10
*B*b^6*d^6*g^3*i^2*x^6 + 60*B*a^3*b^3*c^2*d^4*g^3*i^2*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*x^5 + 15*
(B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + 3*B*a^2*b^4*d^6)*g^3*i^2*x^4 + 20*(3*B*a*b^5*c^2*d^4 + 6*B*a^2*b^4*c*d^5 +
B*a^3*b^3*d^6)*g^3*i^2*x^3 + 30*(3*B*a^2*b^4*c^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*x^2)*log((b*e*x + a*e)/(d*x
+ c)))/(b^3*d^4)
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giac [B] time = 1.82, size = 7651, normalized size = 18.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
[Out]
1/360*(6*B*b^13*c^7*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 42*B*a*b^12*c^6*d*g^3*e^7*log(-b*e + (b*x*
e + a*e)*d/(d*x + c)) + 126*B*a^2*b^11*c^5*d^2*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 210*B*a^3*b^10*
c^4*d^3*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 210*B*a^4*b^9*c^3*d^4*g^3*e^7*log(-b*e + (b*x*e + a*e)
*d/(d*x + c)) - 126*B*a^5*b^8*c^2*d^5*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 42*B*a^6*b^7*c*d^6*g^3*e
^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6*B*a^7*b^6*d^7*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 36*
(b*x*e + a*e)*B*b^12*c^7*d*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 252*(b*x*e + a*e)*B*a*b^1
1*c^6*d^2*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 756*(b*x*e + a*e)*B*a^2*b^10*c^5*d^3*g^3*e
^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 1260*(b*x*e + a*e)*B*a^3*b^9*c^4*d^4*g^3*e^6*log(-b*e + (
b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 1260*(b*x*e + a*e)*B*a^4*b^8*c^3*d^5*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/
(d*x + c))/(d*x + c) + 756*(b*x*e + a*e)*B*a^5*b^7*c^2*d^6*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c) - 252*(b*x*e + a*e)*B*a^6*b^6*c*d^7*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 36*(b*x*e +
a*e)*B*a^7*b^5*d^8*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 90*(b*x*e + a*e)^2*B*b^11*c^7*d^
2*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 630*(b*x*e + a*e)^2*B*a*b^10*c^6*d^3*g^3*e^5*log
(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 1890*(b*x*e + a*e)^2*B*a^2*b^9*c^5*d^4*g^3*e^5*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 3150*(b*x*e + a*e)^2*B*a^3*b^8*c^4*d^5*g^3*e^5*log(-b*e + (b*x*e + a*e)*
d/(d*x + c))/(d*x + c)^2 + 3150*(b*x*e + a*e)^2*B*a^4*b^7*c^3*d^6*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c)^2 - 1890*(b*x*e + a*e)^2*B*a^5*b^6*c^2*d^7*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^2 + 630*(b*x*e + a*e)^2*B*a^6*b^5*c*d^8*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 90*(b*x*e
+ a*e)^2*B*a^7*b^4*d^9*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*(b*x*e + a*e)^3*B*b^10
*c^7*d^3*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 840*(b*x*e + a*e)^3*B*a*b^9*c^6*d^4*g^3*e
^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 2520*(b*x*e + a*e)^3*B*a^2*b^8*c^5*d^5*g^3*e^4*log(-b*e
+ (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 4200*(b*x*e + a*e)^3*B*a^3*b^7*c^4*d^6*g^3*e^4*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c)^3 - 4200*(b*x*e + a*e)^3*B*a^4*b^6*c^3*d^7*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*
x + c))/(d*x + c)^3 + 2520*(b*x*e + a*e)^3*B*a^5*b^5*c^2*d^8*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*
x + c)^3 - 840*(b*x*e + a*e)^3*B*a^6*b^4*c*d^9*g^3*e^4*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^7*b^3*d^10*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 90*(b*x*e + a*e)^4
*B*b^9*c^7*d^4*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 630*(b*x*e + a*e)^4*B*a*b^8*c^6*d^5
*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 1890*(b*x*e + a*e)^4*B*a^2*b^7*c^5*d^6*g^3*e^3*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 3150*(b*x*e + a*e)^4*B*a^3*b^6*c^4*d^7*g^3*e^3*log(-b*e + (b
*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 3150*(b*x*e + a*e)^4*B*a^4*b^5*c^3*d^8*g^3*e^3*log(-b*e + (b*x*e + a*e)
*d/(d*x + c))/(d*x + c)^4 - 1890*(b*x*e + a*e)^4*B*a^5*b^4*c^2*d^9*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c
))/(d*x + c)^4 + 630*(b*x*e + a*e)^4*B*a^6*b^3*c*d^10*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^
4 - 90*(b*x*e + a*e)^4*B*a^7*b^2*d^11*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 36*(b*x*e +
a*e)^5*B*b^8*c^7*d^5*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 252*(b*x*e + a*e)^5*B*a*b^7*c
^6*d^6*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 756*(b*x*e + a*e)^5*B*a^2*b^6*c^5*d^7*g^3*e
^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 1260*(b*x*e + a*e)^5*B*a^3*b^5*c^4*d^8*g^3*e^2*log(-b*e
+ (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 1260*(b*x*e + a*e)^5*B*a^4*b^4*c^3*d^9*g^3*e^2*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c)^5 + 756*(b*x*e + a*e)^5*B*a^5*b^3*c^2*d^10*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*
x + c))/(d*x + c)^5 - 252*(b*x*e + a*e)^5*B*a^6*b^2*c*d^11*g^3*e^2*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^7*b*d^12*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 6*(b*x*e
+ a*e)^6*B*b^7*c^7*d^6*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 42*(b*x*e + a*e)^6*B*a*b^6*c^
6*d^7*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 + 126*(b*x*e + a*e)^6*B*a^2*b^5*c^5*d^8*g^3*e*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 210*(b*x*e + a*e)^6*B*a^3*b^4*c^4*d^9*g^3*e*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c)^6 + 210*(b*x*e + a*e)^6*B*a^4*b^3*c^3*d^10*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d
*x + c))/(d*x + c)^6 - 126*(b*x*e + a*e)^6*B*a^5*b^2*c^2*d^11*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^6 + 42*(b*x*e + a*e)^6*B*a^6*b*c*d^12*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 6*(b*x*e
+ a*e)^6*B*a^7*d^13*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 90*(b*x*e + a*e)^4*B*b^9*c^7*d^
4*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 630*(b*x*e + a*e)^4*B*a*b^8*c^6*d^5*g^3*e^3*log((b*x*e +
a*e)/(d*x + c))/(d*x + c)^4 - 1890*(b*x*e + a*e)^4*B*a^2*b^7*c^5*d^6*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x
+ c)^4 + 3150*(b*x*e + a*e)^4*B*a^3*b^6*c^4*d^7*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 3150*(b*x*
e + a*e)^4*B*a^4*b^5*c^3*d^8*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 1890*(b*x*e + a*e)^4*B*a^5*b^4
*c^2*d^9*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 630*(b*x*e + a*e)^4*B*a^6*b^3*c*d^10*g^3*e^3*log((
b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 90*(b*x*e + a*e)^4*B*a^7*b^2*d^11*g^3*e^3*log((b*x*e + a*e)/(d*x + c))/(
d*x + c)^4 + 36*(b*x*e + a*e)^5*B*b^8*c^7*d^5*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 252*(b*x*e +
a*e)^5*B*a*b^7*c^6*d^6*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 756*(b*x*e + a*e)^5*B*a^2*b^6*c^5*d^
7*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 1260*(b*x*e + a*e)^5*B*a^3*b^5*c^4*d^8*g^3*e^2*log((b*x*e
+ a*e)/(d*x + c))/(d*x + c)^5 + 1260*(b*x*e + a*e)^5*B*a^4*b^4*c^3*d^9*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(
d*x + c)^5 - 756*(b*x*e + a*e)^5*B*a^5*b^3*c^2*d^10*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 252*(b*
x*e + a*e)^5*B*a^6*b^2*c*d^11*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 36*(b*x*e + a*e)^5*B*a^7*b*d^
12*g^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 6*(b*x*e + a*e)^6*B*b^7*c^7*d^6*g^3*e*log((b*x*e + a*e)/
(d*x + c))/(d*x + c)^6 + 42*(b*x*e + a*e)^6*B*a*b^6*c^6*d^7*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 - 1
26*(b*x*e + a*e)^6*B*a^2*b^5*c^5*d^8*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 210*(b*x*e + a*e)^6*B*a^
3*b^4*c^4*d^9*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 - 210*(b*x*e + a*e)^6*B*a^4*b^3*c^3*d^10*g^3*e*lo
g((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 126*(b*x*e + a*e)^6*B*a^5*b^2*c^2*d^11*g^3*e*log((b*x*e + a*e)/(d*x +
c))/(d*x + c)^6 - 42*(b*x*e + a*e)^6*B*a^6*b*c*d^12*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 6*(b*x*e
+ a*e)^6*B*a^7*d^13*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 6*A*b^13*c^7*g^3*e^7 + 2*B*b^13*c^7*g^3*
e^7 - 42*A*a*b^12*c^6*d*g^3*e^7 - 14*B*a*b^12*c^6*d*g^3*e^7 + 126*A*a^2*b^11*c^5*d^2*g^3*e^7 + 42*B*a^2*b^11*c
^5*d^2*g^3*e^7 - 210*A*a^3*b^10*c^4*d^3*g^3*e^7 - 70*B*a^3*b^10*c^4*d^3*g^3*e^7 + 210*A*a^4*b^9*c^3*d^4*g^3*e^
7 + 70*B*a^4*b^9*c^3*d^4*g^3*e^7 - 126*A*a^5*b^8*c^2*d^5*g^3*e^7 - 42*B*a^5*b^8*c^2*d^5*g^3*e^7 + 42*A*a^6*b^7
*c*d^6*g^3*e^7 + 14*B*a^6*b^7*c*d^6*g^3*e^7 - 6*A*a^7*b^6*d^7*g^3*e^7 - 2*B*a^7*b^6*d^7*g^3*e^7 - 36*(b*x*e +
a*e)*A*b^12*c^7*d*g^3*e^6/(d*x + c) - 6*(b*x*e + a*e)*B*b^12*c^7*d*g^3*e^6/(d*x + c) + 252*(b*x*e + a*e)*A*a*b
^11*c^6*d^2*g^3*e^6/(d*x + c) + 42*(b*x*e + a*e)*B*a*b^11*c^6*d^2*g^3*e^6/(d*x + c) - 756*(b*x*e + a*e)*A*a^2*
b^10*c^5*d^3*g^3*e^6/(d*x + c) - 126*(b*x*e + a*e)*B*a^2*b^10*c^5*d^3*g^3*e^6/(d*x + c) + 1260*(b*x*e + a*e)*A
*a^3*b^9*c^4*d^4*g^3*e^6/(d*x + c) + 210*(b*x*e + a*e)*B*a^3*b^9*c^4*d^4*g^3*e^6/(d*x + c) - 1260*(b*x*e + a*e
)*A*a^4*b^8*c^3*d^5*g^3*e^6/(d*x + c) - 210*(b*x*e + a*e)*B*a^4*b^8*c^3*d^5*g^3*e^6/(d*x + c) + 756*(b*x*e + a
*e)*A*a^5*b^7*c^2*d^6*g^3*e^6/(d*x + c) + 126*(b*x*e + a*e)*B*a^5*b^7*c^2*d^6*g^3*e^6/(d*x + c) - 252*(b*x*e +
a*e)*A*a^6*b^6*c*d^7*g^3*e^6/(d*x + c) - 42*(b*x*e + a*e)*B*a^6*b^6*c*d^7*g^3*e^6/(d*x + c) + 36*(b*x*e + a*e
)*A*a^7*b^5*d^8*g^3*e^6/(d*x + c) + 6*(b*x*e + a*e)*B*a^7*b^5*d^8*g^3*e^6/(d*x + c) + 90*(b*x*e + a*e)^2*A*b^1
1*c^7*d^2*g^3*e^5/(d*x + c)^2 - 3*(b*x*e + a*e)^2*B*b^11*c^7*d^2*g^3*e^5/(d*x + c)^2 - 630*(b*x*e + a*e)^2*A*a
*b^10*c^6*d^3*g^3*e^5/(d*x + c)^2 + 21*(b*x*e + a*e)^2*B*a*b^10*c^6*d^3*g^3*e^5/(d*x + c)^2 + 1890*(b*x*e + a*
e)^2*A*a^2*b^9*c^5*d^4*g^3*e^5/(d*x + c)^2 - 63*(b*x*e + a*e)^2*B*a^2*b^9*c^5*d^4*g^3*e^5/(d*x + c)^2 - 3150*(
b*x*e + a*e)^2*A*a^3*b^8*c^4*d^5*g^3*e^5/(d*x + c)^2 + 105*(b*x*e + a*e)^2*B*a^3*b^8*c^4*d^5*g^3*e^5/(d*x + c)
^2 + 3150*(b*x*e + a*e)^2*A*a^4*b^7*c^3*d^6*g^3*e^5/(d*x + c)^2 - 105*(b*x*e + a*e)^2*B*a^4*b^7*c^3*d^6*g^3*e^
5/(d*x + c)^2 - 1890*(b*x*e + a*e)^2*A*a^5*b^6*c^2*d^7*g^3*e^5/(d*x + c)^2 + 63*(b*x*e + a*e)^2*B*a^5*b^6*c^2*
d^7*g^3*e^5/(d*x + c)^2 + 630*(b*x*e + a*e)^2*A*a^6*b^5*c*d^8*g^3*e^5/(d*x + c)^2 - 21*(b*x*e + a*e)^2*B*a^6*b
^5*c*d^8*g^3*e^5/(d*x + c)^2 - 90*(b*x*e + a*e)^2*A*a^7*b^4*d^9*g^3*e^5/(d*x + c)^2 + 3*(b*x*e + a*e)^2*B*a^7*
b^4*d^9*g^3*e^5/(d*x + c)^2 - 120*(b*x*e + a*e)^3*A*b^10*c^7*d^3*g^3*e^4/(d*x + c)^3 + 34*(b*x*e + a*e)^3*B*b^
10*c^7*d^3*g^3*e^4/(d*x + c)^3 + 840*(b*x*e + a*e)^3*A*a*b^9*c^6*d^4*g^3*e^4/(d*x + c)^3 - 238*(b*x*e + a*e)^3
*B*a*b^9*c^6*d^4*g^3*e^4/(d*x + c)^3 - 2520*(b*x*e + a*e)^3*A*a^2*b^8*c^5*d^5*g^3*e^4/(d*x + c)^3 + 714*(b*x*e
+ a*e)^3*B*a^2*b^8*c^5*d^5*g^3*e^4/(d*x + c)^3 + 4200*(b*x*e + a*e)^3*A*a^3*b^7*c^4*d^6*g^3*e^4/(d*x + c)^3 -
1190*(b*x*e + a*e)^3*B*a^3*b^7*c^4*d^6*g^3*e^4/(d*x + c)^3 - 4200*(b*x*e + a*e)^3*A*a^4*b^6*c^3*d^7*g^3*e^4/(
d*x + c)^3 + 1190*(b*x*e + a*e)^3*B*a^4*b^6*c^3*d^7*g^3*e^4/(d*x + c)^3 + 2520*(b*x*e + a*e)^3*A*a^5*b^5*c^2*d
^8*g^3*e^4/(d*x + c)^3 - 714*(b*x*e + a*e)^3*B*a^5*b^5*c^2*d^8*g^3*e^4/(d*x + c)^3 - 840*(b*x*e + a*e)^3*A*a^6
*b^4*c*d^9*g^3*e^4/(d*x + c)^3 + 238*(b*x*e + a*e)^3*B*a^6*b^4*c*d^9*g^3*e^4/(d*x + c)^3 + 120*(b*x*e + a*e)^3
*A*a^7*b^3*d^10*g^3*e^4/(d*x + c)^3 - 34*(b*x*e + a*e)^3*B*a^7*b^3*d^10*g^3*e^4/(d*x + c)^3 - 33*(b*x*e + a*e)
^4*B*b^9*c^7*d^4*g^3*e^3/(d*x + c)^4 + 231*(b*x*e + a*e)^4*B*a*b^8*c^6*d^5*g^3*e^3/(d*x + c)^4 - 693*(b*x*e +
a*e)^4*B*a^2*b^7*c^5*d^6*g^3*e^3/(d*x + c)^4 + 1155*(b*x*e + a*e)^4*B*a^3*b^6*c^4*d^7*g^3*e^3/(d*x + c)^4 - 11
55*(b*x*e + a*e)^4*B*a^4*b^5*c^3*d^8*g^3*e^3/(d*x + c)^4 + 693*(b*x*e + a*e)^4*B*a^5*b^4*c^2*d^9*g^3*e^3/(d*x
+ c)^4 - 231*(b*x*e + a*e)^4*B*a^6*b^3*c*d^10*g^3*e^3/(d*x + c)^4 + 33*(b*x*e + a*e)^4*B*a^7*b^2*d^11*g^3*e^3/
(d*x + c)^4 + 6*(b*x*e + a*e)^5*B*b^8*c^7*d^5*g^3*e^2/(d*x + c)^5 - 42*(b*x*e + a*e)^5*B*a*b^7*c^6*d^6*g^3*e^2
/(d*x + c)^5 + 126*(b*x*e + a*e)^5*B*a^2*b^6*c^5*d^7*g^3*e^2/(d*x + c)^5 - 210*(b*x*e + a*e)^5*B*a^3*b^5*c^4*d
^8*g^3*e^2/(d*x + c)^5 + 210*(b*x*e + a*e)^5*B*a^4*b^4*c^3*d^9*g^3*e^2/(d*x + c)^5 - 126*(b*x*e + a*e)^5*B*a^5
*b^3*c^2*d^10*g^3*e^2/(d*x + c)^5 + 42*(b*x*e + a*e)^5*B*a^6*b^2*c*d^11*g^3*e^2/(d*x + c)^5 - 6*(b*x*e + a*e)^
5*B*a^7*b*d^12*g^3*e^2/(d*x + c)^5)*(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^4*e^6 - 6*(b*x*e + a*e)*b^8*d^5*e^5/(d*x + c) + 15*(b*x*e + a*e)^2*b^7*d^6*e^4/(d*x + c)^2 - 20*(b*x*e +
a*e)^3*b^6*d^7*e^3/(d*x + c)^3 + 15*(b*x*e + a*e)^4*b^5*d^8*e^2/(d*x + c)^4 - 6*(b*x*e + a*e)^5*b^4*d^9*e/(d*x
+ c)^5 + (b*x*e + a*e)^6*b^3*d^10/(d*x + c)^6)
________________________________________________________________________________________
maple [B] time = 0.19, size = 9298, normalized size = 21.98 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(B*ln((b*x+a)/(d*x+c)*e)+A),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [B] time = 1.69, size = 1789, normalized size = 4.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")
[Out]
1/6*A*b^3*d^2*g^3*i^2*x^6 + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^3*i^2*x^5 + 1/4*A*b^3*c^2*g^3*i^2*x^
4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4 + 3/4*A*a^2*b*d^2*g^3*i^2*x^4 + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2*b*c*d*g^3*i^
2*x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d*g^3*i^2*x^2 + (x*log(b*e*x/(d*x +
c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^3*c^2*g^3*i^2 + 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^2*g^3*i^2 + 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^2*g^3*i^2 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(
d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d -
a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^3*c^2*g^3*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^3*c*d*g^3*i^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*c*d*g^3*i^2 + 1/4*(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*c*d*g^3*i^2 + 1/30*(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*c*d*g^3*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^3*d^2*g^3*i^2 + 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^2*b*d^2*g^3*i^2 + 1/20*(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*a*b^2*d^2*g^3*i^2 +
1/360*(60*x^6*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 60*a^6*log(b*x + a)/b^6 + 60*c^6*log(d*x + c)/d^6 - (12*(
b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5
*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5))*B*b^3*d^2*g^3*i^2 + A*a^3*c^2*g^3*i^2*x
________________________________________________________________________________________
mupad [B] time = 5.89, size = 2473, normalized size = 5.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))),x)
[Out]
x^3*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*
B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(12*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d -
B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 +
15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(180*b*d) - (a
*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(3*b*d)
) - x^4*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(
60*a*d + 60*b*c))/(240*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c
*d - B*a*b*c*d))/20 + (A*a*b^2*c*d*g^3*i^2)/4) + x^2*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*
b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*
A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(2*b*d) - ((60*a*
d + 60*b*c)*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*
d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c +
B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2
*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d
) - (a*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(
b*d)))/(120*b*d) + (a*g^3*i^2*(3*A*a^3*d^3 + 12*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 54*A*a*b^2*c^2*d + 36*A*
a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 5*B*a^2*b*c*d^2))/(6*b*d)) + log((e*(a + b*x))/(c + d*x))*(B*a^3*c^2*g^3*i^2*x
+ (B*a*g^3*i^2*x^3*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/3 + (B*b*g^3*i^2*x^4*(3*a^2*d^2 + b^2*c^2 + 6*a*b*c*d))
/4 + (B*b^3*d^2*g^3*i^2*x^6)/6 + (B*a^2*c*g^3*i^2*x^2*(2*a*d + 3*b*c))/2 + (B*b^2*d*g^3*i^2*x^5*(3*a*d + 2*b*c
))/5) + x^5*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/30 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/30
0) - x*(((60*a*d + 60*b*c)*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2
*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 -
2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(b*d) - ((60*a*d + 60*b*c)*((g^3*i^2*(16*A*
a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*
a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d
*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^
2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d) - (a*c*((b^2*d*g^3*i^2*(24
*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(b*d)))/(60*b*d) + (a*g^3*i^2
*(3*A*a^3*d^3 + 12*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 54*A*a*b^2*c^2*d + 36*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d
+ 5*B*a^2*b*c*d^2))/(3*b*d)))/(60*b*d) + (a*c*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3
+ 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^
2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*
c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d)
)/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d) - (a*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*
d*g^3*i^2*(60*a*d + 60*b*c))/60))/(b*d)))/(b*d) - (a^2*c*g^3*i^2*(6*A*a^2*d^2 + 12*A*b^2*c^2 + 2*B*a^2*d^2 - 3
*B*b^2*c^2 + 24*A*a*b*c*d + B*a*b*c*d))/(2*b*d)) + (log(a + b*x)*(B*a^6*d^2*g^3*i^2 + 15*B*a^4*b^2*c^2*g^3*i^2
- 6*B*a^5*b*c*d*g^3*i^2))/(60*b^3) + (log(c + d*x)*(B*b^3*c^6*g^3*i^2 - 20*B*a^3*c^3*d^3*g^3*i^2 - 6*B*a*b^2*
c^5*d*g^3*i^2 + 15*B*a^2*b*c^4*d^2*g^3*i^2))/(60*d^4) + (A*b^3*d^2*g^3*i^2*x^6)/6
________________________________________________________________________________________
sympy [B] time = 13.45, size = 1727, normalized size = 4.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)**3*(d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
A*b**3*d**2*g**3*i**2*x**6/6 + B*a**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)*log(x + (B*a**6*c*d**5*
g**3*i**2 - 6*B*a**5*b*c**2*d**4*g**3*i**2 + B*a**5*d**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)/b +
35*B*a**4*b**2*c**3*d**3*g**3*i**2 - B*a**4*c*d**3*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2) - 15*B*a**
3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2)/(B*a**6*d**6*g**3*i**2
- 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**3*b**3*c**3*d**3*g**3*i**2 - 15*B
*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g**3*i**2))/(60*b**3) - B*c**3*g**3
*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)*log(x + (B*a**6*c*d**5*g**3*i**2 - 6*B*a
**5*b*c**2*d**4*g**3*i**2 + 35*B*a**4*b**2*c**3*d**3*g**3*i**2 - 15*B*a**3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2
*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2 - B*a*b**2*c**3*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 +
6*a*b**2*c**2*d - b**3*c**3) + B*b**3*c**4*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3
*c**3)/d)/(B*a**6*d**6*g**3*i**2 - 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**
3*b**3*c**3*d**3*g**3*i**2 - 15*B*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g*
*3*i**2))/(60*d**4) + x**5*(3*A*a*b**2*d**2*g**3*i**2/5 + 2*A*b**3*c*d*g**3*i**2/5 + B*a*b**2*d**2*g**3*i**2/3
0 - B*b**3*c*d*g**3*i**2/30) + x**4*(3*A*a**2*b*d**2*g**3*i**2/4 + 3*A*a*b**2*c*d*g**3*i**2/2 + A*b**3*c**2*g*
*3*i**2/4 + 13*B*a**2*b*d**2*g**3*i**2/120 - B*a*b**2*c*d*g**3*i**2/20 - 7*B*b**3*c**2*g**3*i**2/120) + x**3*(
A*a**3*d**2*g**3*i**2/3 + 2*A*a**2*b*c*d*g**3*i**2 + A*a*b**2*c**2*g**3*i**2 + 19*B*a**3*d**2*g**3*i**2/180 +
7*B*a**2*b*c*d*g**3*i**2/60 - 13*B*a*b**2*c**2*g**3*i**2/60 - B*b**3*c**3*g**3*i**2/(180*d)) + x**2*(A*a**3*c*
d*g**3*i**2 + 3*A*a**2*b*c**2*g**3*i**2/2 + B*a**4*d**2*g**3*i**2/(120*b) + 17*B*a**3*c*d*g**3*i**2/60 - B*a**
2*b*c**2*g**3*i**2/4 - B*a*b**2*c**3*g**3*i**2/(20*d) + B*b**3*c**4*g**3*i**2/(120*d**2)) + x*(A*a**3*c**2*g**
3*i**2 - B*a**5*d**2*g**3*i**2/(60*b**2) + B*a**4*c*d*g**3*i**2/(10*b) + B*a**3*c**2*g**3*i**2/12 - B*a**2*b*c
**3*g**3*i**2/(4*d) + B*a*b**2*c**4*g**3*i**2/(10*d**2) - B*b**3*c**5*g**3*i**2/(60*d**3)) + (B*a**3*c**2*g**3
*i**2*x + B*a**3*c*d*g**3*i**2*x**2 + B*a**3*d**2*g**3*i**2*x**3/3 + 3*B*a**2*b*c**2*g**3*i**2*x**2/2 + 2*B*a*
*2*b*c*d*g**3*i**2*x**3 + 3*B*a**2*b*d**2*g**3*i**2*x**4/4 + B*a*b**2*c**2*g**3*i**2*x**3 + 3*B*a*b**2*c*d*g**
3*i**2*x**4/2 + 3*B*a*b**2*d**2*g**3*i**2*x**5/5 + B*b**3*c**2*g**3*i**2*x**4/4 + 2*B*b**3*c*d*g**3*i**2*x**5/
5 + B*b**3*d**2*g**3*i**2*x**6/6)*log(e*(a + b*x)/(c + d*x))
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