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

Optimal. Leaf size=457 \[ -\frac{b^2 g^3 i^3 (c+d x)^6 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^4}+\frac{b^3 g^3 i^3 (c+d x)^7 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{7 d^4}-\frac{g^3 i^3 (c+d x)^4 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac{3 b g^3 i^3 (c+d x)^5 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}+\frac{B g^3 i^3 x (b c-a d)^6}{140 b^3 d^3}+\frac{B g^3 i^3 (c+d x)^2 (b c-a d)^5}{280 b^2 d^4}-\frac{b^2 B g^3 i^3 (c+d x)^6 (b c-a d)}{42 d^4}+\frac{B g^3 i^3 (b c-a d)^7 \log \left (\frac{a+b x}{c+d x}\right )}{140 b^4 d^4}+\frac{B g^3 i^3 (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}+\frac{B g^3 i^3 (c+d x)^3 (b c-a d)^4}{420 b d^4}-\frac{17 B g^3 i^3 (c+d x)^4 (b c-a d)^3}{280 d^4}+\frac{b B g^3 i^3 (c+d x)^5 (b c-a d)^2}{14 d^4} \]

[Out]

(B*(b*c - a*d)^6*g^3*i^3*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2)/(280*b^2*d^4) + (B*(b*c - a*
d)^4*g^3*i^3*(c + d*x)^3)/(420*b*d^4) - (17*B*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4)/(280*d^4) + (b*B*(b*c - a*d)^
2*g^3*i^3*(c + d*x)^5)/(14*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*(c + d*x)^6)/(42*d^4) + (B*(b*c - a*d)^7*g^3*i^3*
Log[(a + b*x)/(c + d*x)])/(140*b^4*d^4) - ((b*c - a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x
)]))/(4*d^4) + (3*b*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^4) - (b^2*(b*
c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4) + (b^3*g^3*i^3*(c + d*x)^7*(A + B*L
og[(e*(a + b*x))/(c + d*x)]))/(7*d^4) + (B*(b*c - a*d)^7*g^3*i^3*Log[c + d*x])/(140*b^4*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.944784, antiderivative size = 416, normalized size of antiderivative = 0.91, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 43} \[ \frac{d^2 g^3 i^3 (a+b x)^6 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4}+\frac{d^3 g^3 i^3 (a+b x)^7 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{7 b^4}+\frac{g^3 i^3 (a+b x)^4 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^4}+\frac{3 d g^3 i^3 (a+b x)^5 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^4}-\frac{B g^3 i^3 x (b c-a d)^6}{140 b^3 d^3}+\frac{B g^3 i^3 (a+b x)^2 (b c-a d)^5}{280 b^4 d^2}-\frac{B d^2 g^3 i^3 (a+b x)^6 (b c-a d)}{42 b^4}+\frac{B g^3 i^3 (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}-\frac{B g^3 i^3 (a+b x)^3 (b c-a d)^4}{420 b^4 d}-\frac{17 B g^3 i^3 (a+b x)^4 (b c-a d)^3}{280 b^4}-\frac{B d g^3 i^3 (a+b x)^5 (b c-a d)^2}{14 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)^6*g^3*i^3*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2)/(280*b^4*d^2) - (B*(b*c - a
*d)^4*g^3*i^3*(a + b*x)^3)/(420*b^4*d) - (17*B*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4)/(280*b^4) - (B*d*(b*c - a*d)
^2*g^3*i^3*(a + b*x)^5)/(14*b^4) - (B*d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6)/(42*b^4) + ((b*c - a*d)^3*g^3*i^3*(
a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^4) + (3*d*(b*c - a*d)^2*g^3*i^3*(a + b*x)^5*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(5*b^4) + (d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(2*b^4) + (d^3*g^3*i^3*(a + b*x)^7*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(7*b^4) + (B*(b*c - a*d)^7*g^3*i^3*L
og[c + d*x])/(140*b^4*d^4)

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]

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 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])

Rubi steps

\begin{align*} \int (20 c+20 d x)^3 (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)^3 g^3 (20 c+20 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3}+\frac{3 b (b c-a d)^2 g^3 (20 c+20 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{20 d^3}-\frac{3 b^2 (b c-a d) g^3 (20 c+20 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{400 d^3}+\frac{b^3 g^3 (20 c+20 d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8000 d^3}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int (20 c+20 d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{8000 d^3}-\frac{\left (3 b^2 (b c-a d) g^3\right ) \int (20 c+20 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{400 d^3}+\frac{\left (3 b (b c-a d)^2 g^3\right ) \int (20 c+20 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{20 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int (20 c+20 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d^3}\\ &=-\frac{2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac{4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{\left (b^3 B g^3\right ) \int \frac{1280000000 (b c-a d) (c+d x)^6}{a+b x} \, dx}{1120000 d^4}+\frac{\left (b^2 B (b c-a d) g^3\right ) \int \frac{64000000 (b c-a d) (c+d x)^5}{a+b x} \, dx}{16000 d^4}-\frac{\left (3 b B (b c-a d)^2 g^3\right ) \int \frac{3200000 (b c-a d) (c+d x)^4}{a+b x} \, dx}{2000 d^4}+\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{160000 (b c-a d) (c+d x)^3}{a+b x} \, dx}{80 d^4}\\ &=-\frac{2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac{4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{\left (8000 b^3 B (b c-a d) g^3\right ) \int \frac{(c+d x)^6}{a+b x} \, dx}{7 d^4}+\frac{\left (4000 b^2 B (b c-a d)^2 g^3\right ) \int \frac{(c+d x)^5}{a+b x} \, dx}{d^4}-\frac{\left (4800 b B (b c-a d)^3 g^3\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{d^4}+\frac{\left (2000 B (b c-a d)^4 g^3\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^4}\\ &=-\frac{2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac{4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{\left (8000 b^3 B (b c-a d) g^3\right ) \int \left (\frac{d (b c-a d)^5}{b^6}+\frac{(b c-a d)^6}{b^6 (a+b x)}+\frac{d (b c-a d)^4 (c+d x)}{b^5}+\frac{d (b c-a d)^3 (c+d x)^2}{b^4}+\frac{d (b c-a d)^2 (c+d x)^3}{b^3}+\frac{d (b c-a d) (c+d x)^4}{b^2}+\frac{d (c+d x)^5}{b}\right ) \, dx}{7 d^4}+\frac{\left (4000 b^2 B (b c-a d)^2 g^3\right ) \int \left (\frac{d (b c-a d)^4}{b^5}+\frac{(b c-a d)^5}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x)}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2}{b^3}+\frac{d (b c-a d) (c+d x)^3}{b^2}+\frac{d (c+d x)^4}{b}\right ) \, dx}{d^4}-\frac{\left (4800 b B (b c-a d)^3 g^3\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{d^4}+\frac{\left (2000 B (b c-a d)^4 g^3\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^4}\\ &=\frac{400 B (b c-a d)^6 g^3 x}{7 b^3 d^3}+\frac{200 B (b c-a d)^5 g^3 (c+d x)^2}{7 b^2 d^4}+\frac{400 B (b c-a d)^4 g^3 (c+d x)^3}{21 b d^4}-\frac{3400 B (b c-a d)^3 g^3 (c+d x)^4}{7 d^4}+\frac{4000 b B (b c-a d)^2 g^3 (c+d x)^5}{7 d^4}-\frac{4000 b^2 B (b c-a d) g^3 (c+d x)^6}{21 d^4}+\frac{400 B (b c-a d)^7 g^3 \log (a+b x)}{7 b^4 d^4}-\frac{2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac{4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac{8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}\\ \end{align*}

Mathematica [A]  time = 0.590608, size = 586, normalized size = 1.28 \[ \frac{g^3 i^3 \left (420 d^2 (a+b x)^6 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+120 d^3 (a+b x)^7 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+210 (a+b x)^4 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+504 d (a+b x)^5 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{120 b^2 B c x (b c-a d)^5}{d^3}-\frac{126 b B x (b c-a d)^6}{d^3}+\frac{63 B (a+b x)^2 (b c-a d)^5}{d^2}-\frac{60 b B c (a+b x)^2 (b c-a d)^4}{d^2}-20 b B c d^2 (a+b x)^6+24 a B d^2 (a+b x)^5 (a d-b c)+\frac{120 a b B x (a d-b c)^5}{d^2}+\frac{126 B (b c-a d)^7 \log (c+d x)}{d^4}+\frac{120 a B (b c-a d)^6 \log (c+d x)}{d^3}-\frac{120 b B c (b c-a d)^6 \log (c+d x)}{d^4}-\frac{42 B (a+b x)^3 (b c-a d)^4}{d}+\frac{60 a B (a+b x)^2 (b c-a d)^4}{d}+\frac{40 b B c (a+b x)^3 (b c-a d)^3}{d}-84 B d (a+b x)^5 (b c-a d)^2-30 b B c (a+b x)^4 (b c-a d)^2+30 a B d (a+b x)^4 (b c-a d)^2+24 b B c d (a+b x)^5 (b c-a d)+21 B (a+b x)^4 (a d-b c)^3+40 a B (a+b x)^3 (a d-b c)^3+20 a B d^3 (a+b x)^6\right )}{840 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i^3*((120*b^2*B*c*(b*c - a*d)^5*x)/d^3 - (126*b*B*(b*c - a*d)^6*x)/d^3 + (120*a*b*B*(-(b*c) + a*d)^5*x)/d
^2 - (60*b*B*c*(b*c - a*d)^4*(a + b*x)^2)/d^2 + (60*a*B*(b*c - a*d)^4*(a + b*x)^2)/d + (63*B*(b*c - a*d)^5*(a
+ b*x)^2)/d^2 + (40*b*B*c*(b*c - a*d)^3*(a + b*x)^3)/d - (42*B*(b*c - a*d)^4*(a + b*x)^3)/d + 40*a*B*(-(b*c) +
a*d)^3*(a + b*x)^3 - 30*b*B*c*(b*c - a*d)^2*(a + b*x)^4 + 30*a*B*d*(b*c - a*d)^2*(a + b*x)^4 + 21*B*(-(b*c) +
a*d)^3*(a + b*x)^4 + 24*b*B*c*d*(b*c - a*d)*(a + b*x)^5 - 84*B*d*(b*c - a*d)^2*(a + b*x)^5 + 24*a*B*d^2*(-(b*
c) + a*d)*(a + b*x)^5 - 20*b*B*c*d^2*(a + b*x)^6 + 20*a*B*d^3*(a + b*x)^6 + 210*(b*c - a*d)^3*(a + b*x)^4*(A +
B*Log[(e*(a + b*x))/(c + d*x)]) + 504*d*(b*c - a*d)^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 420*
d^2*(b*c - a*d)*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 120*d^3*(a + b*x)^7*(A + B*Log[(e*(a + b*x)
)/(c + d*x)]) - (120*b*B*c*(b*c - a*d)^6*Log[c + d*x])/d^4 + (120*a*B*(b*c - a*d)^6*Log[c + d*x])/d^3 + (126*B
*(b*c - a*d)^7*Log[c + d*x])/d^4))/(840*b^4)

________________________________________________________________________________________

Maple [B]  time = 0.222, size = 11172, normalized size = 24.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 1.52151, size = 3560, normalized size = 7.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*A*b^3*d^3*g^3*i^3*x^7 + 1/2*A*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A*a*b^2*d^3*g^3*i^3*x^6 + 3/5*A*b^3*c^2*d*g^3*i^
3*x^5 + 9/5*A*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A*a^2*b*d^3*g^3*i^3*x^5 + 1/4*A*b^3*c^3*g^3*i^3*x^4 + 9/4*A*a*b^2*
c^2*d*g^3*i^3*x^4 + 9/4*A*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A*a^3*d^3*g^3*i^3*x^4 + A*a*b^2*c^3*g^3*i^3*x^3 + 3*A*
a^2*b*c^2*d*g^3*i^3*x^3 + A*a^3*c*d^2*g^3*i^3*x^3 + 3/2*A*a^2*b*c^3*g^3*i^3*x^2 + 3/2*A*a^3*c^2*d*g^3*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^3*c^3*g^3*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^
2*b*c^3*g^3*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*b^2*c^3*g^3*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*b^3*c^3*g^3*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^3*c^2*
d*g^3*i^3 + 3/2*(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^2*d*g^3*i^3 + 3/8*(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*c^2*d*g^3*i^3 + 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*b^3*c^2*d*g^3*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*l
og(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*c*d^2*g^3*i^3 + 3/8*(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*c*d^2*g^3*i^3 + 3/
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*c*d^2*g^3*i^3 + 1/120*(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*c*d^2*g^3*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^3*d^3*g^3*i^3 + 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^2*b*d^3*g^3*i^3 + 1/120*(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*a*b^2*d^3*g^3*i^3 + 1/420*(60*x^7*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 60*a^7*log(b*x +
a)/b^7 - 60*c^7*log(d*x + c)/d^7 - (10*(b^6*c*d^5 - a*b^5*d^6)*x^6 - 12*(b^6*c^2*d^4 - a^2*b^4*d^6)*x^5 + 15*(
b^6*c^3*d^3 - a^3*b^3*d^6)*x^4 - 20*(b^6*c^4*d^2 - a^4*b^2*d^6)*x^3 + 30*(b^6*c^5*d - a^5*b*d^6)*x^2 - 60*(b^6
*c^6 - a^6*d^6)*x)/(b^6*d^6))*B*b^3*d^3*g^3*i^3 + A*a^3*c^3*g^3*i^3*x

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Fricas [B]  time = 1.79378, size = 1894, normalized size = 4.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(120*A*b^7*d^7*g^3*i^3*x^7 + 20*((21*A - B)*b^7*c*d^6 + (21*A + B)*a*b^6*d^7)*g^3*i^3*x^6 + 12*((42*A -
5*B)*b^7*c^2*d^5 + 126*A*a*b^6*c*d^6 + (42*A + 5*B)*a^2*b^5*d^7)*g^3*i^3*x^5 + 3*((70*A - 17*B)*b^7*c^3*d^4 +
7*(90*A - 7*B)*a*b^6*c^2*d^5 + 7*(90*A + 7*B)*a^2*b^5*c*d^6 + (70*A + 17*B)*a^3*b^4*d^7)*g^3*i^3*x^4 - 2*(B*b^
7*c^4*d^3 - 14*(30*A - 7*B)*a*b^6*c^3*d^4 - 1260*A*a^2*b^5*c^2*d^5 - 14*(30*A + 7*B)*a^3*b^4*c*d^6 - B*a^4*b^3
*d^7)*g^3*i^3*x^3 + 3*(B*b^7*c^5*d^2 - 7*B*a*b^6*c^4*d^3 + 84*(5*A - B)*a^2*b^5*c^3*d^4 + 84*(5*A + B)*a^3*b^4
*c^2*d^5 + 7*B*a^4*b^3*c*d^6 - B*a^5*b^2*d^7)*g^3*i^3*x^2 - 6*(B*b^7*c^6*d - 7*B*a*b^6*c^5*d^2 + 21*B*a^2*b^5*
c^4*d^3 - 140*A*a^3*b^4*c^3*d^4 - 21*B*a^4*b^3*c^2*d^5 + 7*B*a^5*b^2*c*d^6 - B*a^6*b*d^7)*g^3*i^3*x + 6*(35*B*
a^4*b^3*c^3*d^4 - 21*B*a^5*b^2*c^2*d^5 + 7*B*a^6*b*c*d^6 - B*a^7*d^7)*g^3*i^3*log(b*x + a) + 6*(B*b^7*c^7 - 7*
B*a*b^6*c^6*d + 21*B*a^2*b^5*c^5*d^2 - 35*B*a^3*b^4*c^4*d^3)*g^3*i^3*log(d*x + c) + 6*(20*B*b^7*d^7*g^3*i^3*x^
7 + 140*B*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(B*b^7*c*d^6 + B*a*b^6*d^7)*g^3*i^3*x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b
^6*c*d^6 + B*a^2*b^5*d^7)*g^3*i^3*x^5 + 35*(B*b^7*c^3*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B*a^2*b^5*c*d^6 + B*a^3*b^4*
d^7)*g^3*i^3*x^4 + 140*(B*a*b^6*c^3*d^4 + 3*B*a^2*b^5*c^2*d^5 + B*a^3*b^4*c*d^6)*g^3*i^3*x^3 + 210*(B*a^2*b^5*
c^3*d^4 + B*a^3*b^4*c^2*d^5)*g^3*i^3*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^4*d^4)

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Sympy [B]  time = 20.5444, size = 2188, normalized size = 4.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*d**3*g**3*i**3*x**7/7 - B*a**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3
)*log(x + (B*a**7*c*d**6*g**3*i**3 - 7*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2*c**3*d**4*g**3*i**3 + B*a
**5*d**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3)/b - 70*B*a**4*b**3*c**4*d**
3*g**3*i**3 - B*a**4*c*d**3*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3) + 21*B*a
**3*b**4*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6*c**7*g**3*i**3)/(B*a**7*d**7*g**3*i**
3 - 7*B*a**6*b*c*d**6*g**3*i**3 + 21*B*a**5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c**3*d**4*g**3*i**3 - 35
*B*a**3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5*c**5*d**2*g**3*i**3 - 7*B*a*b**6*c**6*d*g**3*i**3 + B*b**7*c
**7*g**3*i**3))/(140*b**4) - B*c**4*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3)*
log(x + (B*a**7*c*d**6*g**3*i**3 - 7*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2*c**3*d**4*g**3*i**3 - 70*B*
a**4*b**3*c**4*d**3*g**3*i**3 + 21*B*a**3*b**4*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6
*c**7*g**3*i**3 + B*a*b**3*c**4*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3) - B*
b**4*c**5*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3)/d)/(B*a**7*d**7*g**3*i**3
- 7*B*a**6*b*c*d**6*g**3*i**3 + 21*B*a**5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c**3*d**4*g**3*i**3 - 35*B
*a**3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5*c**5*d**2*g**3*i**3 - 7*B*a*b**6*c**6*d*g**3*i**3 + B*b**7*c**
7*g**3*i**3))/(140*d**4) + x**6*(A*a*b**2*d**3*g**3*i**3/2 + A*b**3*c*d**2*g**3*i**3/2 + B*a*b**2*d**3*g**3*i*
*3/42 - B*b**3*c*d**2*g**3*i**3/42) + x**5*(3*A*a**2*b*d**3*g**3*i**3/5 + 9*A*a*b**2*c*d**2*g**3*i**3/5 + 3*A*
b**3*c**2*d*g**3*i**3/5 + B*a**2*b*d**3*g**3*i**3/14 - B*b**3*c**2*d*g**3*i**3/14) + x**4*(A*a**3*d**3*g**3*i*
*3/4 + 9*A*a**2*b*c*d**2*g**3*i**3/4 + 9*A*a*b**2*c**2*d*g**3*i**3/4 + A*b**3*c**3*g**3*i**3/4 + 17*B*a**3*d**
3*g**3*i**3/280 + 7*B*a**2*b*c*d**2*g**3*i**3/40 - 7*B*a*b**2*c**2*d*g**3*i**3/40 - 17*B*b**3*c**3*g**3*i**3/2
80) + (B*a**3*c**3*g**3*i**3*x + 3*B*a**3*c**2*d*g**3*i**3*x**2/2 + B*a**3*c*d**2*g**3*i**3*x**3 + B*a**3*d**3
*g**3*i**3*x**4/4 + 3*B*a**2*b*c**3*g**3*i**3*x**2/2 + 3*B*a**2*b*c**2*d*g**3*i**3*x**3 + 9*B*a**2*b*c*d**2*g*
*3*i**3*x**4/4 + 3*B*a**2*b*d**3*g**3*i**3*x**5/5 + B*a*b**2*c**3*g**3*i**3*x**3 + 9*B*a*b**2*c**2*d*g**3*i**3
*x**4/4 + 9*B*a*b**2*c*d**2*g**3*i**3*x**5/5 + B*a*b**2*d**3*g**3*i**3*x**6/2 + B*b**3*c**3*g**3*i**3*x**4/4 +
3*B*b**3*c**2*d*g**3*i**3*x**5/5 + B*b**3*c*d**2*g**3*i**3*x**6/2 + B*b**3*d**3*g**3*i**3*x**7/7)*log(e*(a +
b*x)/(c + d*x)) + x**3*(420*A*a**3*b*c*d**3*g**3*i**3 + 1260*A*a**2*b**2*c**2*d**2*g**3*i**3 + 420*A*a*b**3*c*
*3*d*g**3*i**3 + B*a**4*d**4*g**3*i**3 + 98*B*a**3*b*c*d**3*g**3*i**3 - 98*B*a*b**3*c**3*d*g**3*i**3 - B*b**4*
c**4*g**3*i**3)/(420*b*d) - x**2*(-420*A*a**3*b**2*c**2*d**3*g**3*i**3 - 420*A*a**2*b**3*c**3*d**2*g**3*i**3 +
B*a**5*d**5*g**3*i**3 - 7*B*a**4*b*c*d**4*g**3*i**3 - 84*B*a**3*b**2*c**2*d**3*g**3*i**3 + 84*B*a**2*b**3*c**
3*d**2*g**3*i**3 + 7*B*a*b**4*c**4*d*g**3*i**3 - B*b**5*c**5*g**3*i**3)/(280*b**2*d**2) + x*(140*A*a**3*b**3*c
**3*d**3*g**3*i**3 + B*a**6*d**6*g**3*i**3 - 7*B*a**5*b*c*d**5*g**3*i**3 + 21*B*a**4*b**2*c**2*d**4*g**3*i**3
- 21*B*a**2*b**4*c**4*d**2*g**3*i**3 + 7*B*a*b**5*c**5*d*g**3*i**3 - B*b**6*c**6*g**3*i**3)/(140*b**3*d**3)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out