3.1.29 \(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^6} \, dx\) [29]

Optimal. Leaf size=181 \[ \frac {B d i^3 (c+d x)^4}{16 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b B i^3 (c+d x)^5}{25 (b c-a d)^2 g^6 (a+b x)^5}+\frac {d i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^2 g^6 (a+b x)^5} \]

[Out]

1/16*B*d*i^3*(d*x+c)^4/(-a*d+b*c)^2/g^6/(b*x+a)^4-1/25*b*B*i^3*(d*x+c)^5/(-a*d+b*c)^2/g^6/(b*x+a)^5+1/4*d*i^3*
(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^2/g^6/(b*x+a)^4-1/5*b*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)
))/(-a*d+b*c)^2/g^6/(b*x+a)^5

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Rubi [A]
time = 0.11, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 45, 2372, 12} \begin {gather*} -\frac {b i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^2}+\frac {d i^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^2}-\frac {b B i^3 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^2}+\frac {B d i^3 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

(B*d*i^3*(c + d*x)^4)/(16*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*B*i^3*(c + d*x)^5)/(25*(b*c - a*d)^2*g^6*(a + b*
x)^5) + (d*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*i^3*(c
+ d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c - a*d)^2*g^6*(a + b*x)^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {(29 c+29 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx &=\int \left (\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^6}+\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^5}+\frac {73167 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^4}+\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (24389 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^6}+\frac {\left (73167 d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^6}+\frac {\left (73167 d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^6}+\frac {\left (24389 (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{b^3 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^3\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^4\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac {\left (24389 B d^3 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^4 g^6}+\frac {\left (24389 B d^2 (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^4 g^6}+\frac {\left (73167 B d (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b^4 g^6}+\frac {\left (24389 B (b c-a d)^4\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^4 g^6}\\ &=-\frac {24389 B (b c-a d)^3}{25 b^4 g^6 (a+b x)^5}-\frac {268279 B d (b c-a d)^2}{80 b^4 g^6 (a+b x)^4}-\frac {73167 B d^2 (b c-a d)}{20 b^4 g^6 (a+b x)^3}-\frac {24389 B d^3}{40 b^4 g^6 (a+b x)^2}+\frac {24389 B d^4}{20 b^4 (b c-a d) g^6 (a+b x)}+\frac {24389 B d^5 \log (a+b x)}{20 b^4 (b c-a d)^2 g^6}-\frac {24389 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac {73167 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac {24389 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac {24389 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}-\frac {24389 B d^5 \log (c+d x)}{20 b^4 (b c-a d)^2 g^6}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(181)=362\).
time = 0.39, size = 608, normalized size = 3.36 \begin {gather*} -\frac {i^3 \left (80 A b^5 c^5+16 b^5 B c^5-100 a A b^4 c^4 d-25 a b^4 B c^4 d+20 a^5 A d^5+9 a^5 B d^5+300 A b^5 c^4 d x+55 b^5 B c^4 d x-400 a A b^4 c^3 d^2 x-100 a b^4 B c^3 d^2 x+100 a^4 A b d^5 x+45 a^4 b B d^5 x+400 A b^5 c^3 d^2 x^2+60 b^5 B c^3 d^2 x^2-600 a A b^4 c^2 d^3 x^2-150 a b^4 B c^2 d^3 x^2+200 a^3 A b^2 d^5 x^2+90 a^3 b^2 B d^5 x^2+200 A b^5 c^2 d^3 x^3+10 b^5 B c^2 d^3 x^3-400 a A b^4 c d^4 x^3-100 a b^4 B c d^4 x^3+200 a^2 A b^3 d^5 x^3+90 a^2 b^3 B d^5 x^3-20 b^5 B c d^4 x^4+20 a b^4 B d^5 x^4-20 B d^5 (a+b x)^5 \log (a+b x)+20 B (b c-a d)^2 \left (a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+20 a^5 B d^5 \log (c+d x)+100 a^4 b B d^5 x \log (c+d x)+200 a^3 b^2 B d^5 x^2 \log (c+d x)+200 a^2 b^3 B d^5 x^3 \log (c+d x)+100 a b^4 B d^5 x^4 \log (c+d x)+20 b^5 B d^5 x^5 \log (c+d x)\right )}{400 b^4 (b c-a d)^2 g^6 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

-1/400*(i^3*(80*A*b^5*c^5 + 16*b^5*B*c^5 - 100*a*A*b^4*c^4*d - 25*a*b^4*B*c^4*d + 20*a^5*A*d^5 + 9*a^5*B*d^5 +
300*A*b^5*c^4*d*x + 55*b^5*B*c^4*d*x - 400*a*A*b^4*c^3*d^2*x - 100*a*b^4*B*c^3*d^2*x + 100*a^4*A*b*d^5*x + 45
*a^4*b*B*d^5*x + 400*A*b^5*c^3*d^2*x^2 + 60*b^5*B*c^3*d^2*x^2 - 600*a*A*b^4*c^2*d^3*x^2 - 150*a*b^4*B*c^2*d^3*
x^2 + 200*a^3*A*b^2*d^5*x^2 + 90*a^3*b^2*B*d^5*x^2 + 200*A*b^5*c^2*d^3*x^3 + 10*b^5*B*c^2*d^3*x^3 - 400*a*A*b^
4*c*d^4*x^3 - 100*a*b^4*B*c*d^4*x^3 + 200*a^2*A*b^3*d^5*x^3 + 90*a^2*b^3*B*d^5*x^3 - 20*b^5*B*c*d^4*x^4 + 20*a
*b^4*B*d^5*x^4 - 20*B*d^5*(a + b*x)^5*Log[a + b*x] + 20*B*(b*c - a*d)^2*(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a
*b^2*d*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3))*Log[(e*(a + b*x
))/(c + d*x)] + 20*a^5*B*d^5*Log[c + d*x] + 100*a^4*b*B*d^5*x*Log[c + d*x] + 200*a^3*b^2*B*d^5*x^2*Log[c + d*x
] + 200*a^2*b^3*B*d^5*x^3*Log[c + d*x] + 100*a*b^4*B*d^5*x^4*Log[c + d*x] + 20*b^5*B*d^5*x^5*Log[c + d*x]))/(b
^4*(b*c - a*d)^2*g^6*(a + b*x)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(173)=346\).
time = 0.75, size = 357, normalized size = 1.97 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(1/5*i^3*d^2*e^4/(a*d-b*c)^3/g^6*A*b/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5-1/4*i^3*d^3*e^3/(a*d-b
*c)^3/g^6*A/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4-i^3*d^2*e^4/(a*d-b*c)^3/g^6*B*b*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5)+i^3*d^3*e^3/(a*d-b*c)^3/g^6*B*(-1/4/
(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4202 vs. \(2 (163) = 326\).
time = 0.63, size = 4202, normalized size = 23.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

1/1200*I*B*d^3*(60*(10*b^3*x^3 + 10*a*b^2*x^2 + 5*a^2*b*x + a^3)*log(b*x*e/(d*x + c) + a*e/(d*x + c))/(b^9*g^6
*x^5 + 5*a*b^8*g^6*x^4 + 10*a^2*b^7*g^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) + (77*a^3*b^
4*c^4 - 548*a^4*b^3*c^3*d + 352*a^5*b^2*c^2*d^2 - 148*a^6*b*c*d^3 + 27*a^7*d^4 - 60*(10*b^7*c^3*d - 10*a*b^6*c
^2*d^2 + 5*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^4 + 30*(10*b^7*c^4 - 100*a*b^6*c^3*d + 95*a^2*b^5*c^2*d^2 - 46*a^3*b
^4*c*d^3 + 9*a^4*b^3*d^4)*x^3 + 10*(50*a*b^6*c^4 - 410*a^2*b^5*c^3*d + 337*a^3*b^4*c^2*d^2 - 148*a^4*b^3*c*d^3
+ 27*a^5*b^2*d^4)*x^2 + 5*(65*a^2*b^5*c^4 - 488*a^3*b^4*c^3*d + 352*a^4*b^3*c^2*d^2 - 148*a^5*b^2*c*d^3 + 27*
a^6*b*d^4)*x)/((b^13*c^4 - 4*a*b^12*c^3*d + 6*a^2*b^11*c^2*d^2 - 4*a^3*b^10*c*d^3 + a^4*b^9*d^4)*g^6*x^5 + 5*(
a*b^12*c^4 - 4*a^2*b^11*c^3*d + 6*a^3*b^10*c^2*d^2 - 4*a^4*b^9*c*d^3 + a^5*b^8*d^4)*g^6*x^4 + 10*(a^2*b^11*c^4
- 4*a^3*b^10*c^3*d + 6*a^4*b^9*c^2*d^2 - 4*a^5*b^8*c*d^3 + a^6*b^7*d^4)*g^6*x^3 + 10*(a^3*b^10*c^4 - 4*a^4*b^
9*c^3*d + 6*a^5*b^8*c^2*d^2 - 4*a^6*b^7*c*d^3 + a^7*b^6*d^4)*g^6*x^2 + 5*(a^4*b^9*c^4 - 4*a^5*b^8*c^3*d + 6*a^
6*b^7*c^2*d^2 - 4*a^7*b^6*c*d^3 + a^8*b^5*d^4)*g^6*x + (a^5*b^8*c^4 - 4*a^6*b^7*c^3*d + 6*a^7*b^6*c^2*d^2 - 4*
a^8*b^5*c*d^3 + a^9*b^4*d^4)*g^6) - 60*(10*b^3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 - a^3*d^5)*log(b*x +
a)/((b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*g^6)
+ 60*(10*b^3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 - a^3*d^5)*log(d*x + c)/((b^9*c^5 - 5*a*b^8*c^4*d + 10
*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*g^6)) + 1/600*I*B*c*d^2*(60*(10*b^2*x^2
+ 5*a*b*x + a^2)*log(b*x*e/(d*x + c) + a*e/(d*x + c))/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 1
0*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) + (47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2
- 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 9
5*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d
^2 - 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278
*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8
*d^4)*g^6*x^5 + 5*(a*b^11*c^4 - 4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4
+ 10*(a^2*b^10*c^4 - 4*a^3*b^9*c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^
9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*
b^7*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*
b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 + a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/(
(b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*
(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*
b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6)) + 1/400*I*B*c^2*d*(60*(5*b*x + a)*log(b*x*e/(d*x + c) + a*e
/(d*x + c))/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b
^2*g^6) + (27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c*d^3 + 77*a^5*d^4 - 60*(5*b^5*c
*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3 - 10*(10*b^5*c^3*d - 67*a*b^4*
c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^3*d + 232*a^2*b^3*c^2*d^2 - 428
*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*
d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*g^6*x^4 +
10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c
^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6
*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4
*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x + a)/((b^7*c^5 - 5*a*b^6*c^4*d
+ 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6) + 60*(5*b*c*d^4 - a*d^5)*log(
d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*
g^6)) + 1/300*I*B*c^3*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 +
137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5
*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*
c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*
b^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (163) = 326\).
time = 0.41, size = 620, normalized size = 3.43 \begin {gather*} -\frac {16 \, {\left (-5 i \, A - i \, B\right )} b^{5} c^{5} + 25 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{4} d - {\left (20 i \, A + 9 i \, B\right )} a^{5} d^{5} + 20 \, {\left (i \, B b^{5} c d^{4} - i \, B a b^{4} d^{5}\right )} x^{4} + 10 \, {\left ({\left (-20 i \, A - i \, B\right )} b^{5} c^{2} d^{3} + 10 \, {\left (4 i \, A + i \, B\right )} a b^{4} c d^{4} + {\left (-20 i \, A - 9 i \, B\right )} a^{2} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (2 \, {\left (-20 i \, A - 3 i \, B\right )} b^{5} c^{3} d^{2} + 15 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{2} d^{3} + {\left (-20 i \, A - 9 i \, B\right )} a^{3} b^{2} d^{5}\right )} x^{2} + 5 \, {\left ({\left (-60 i \, A - 11 i \, B\right )} b^{5} c^{4} d + 20 \, {\left (4 i \, A + i \, B\right )} a b^{4} c^{3} d^{2} + {\left (-20 i \, A - 9 i \, B\right )} a^{4} b d^{5}\right )} x + 20 \, {\left (i \, B b^{5} d^{5} x^{5} + 5 i \, B a b^{4} d^{5} x^{4} - 4 i \, B b^{5} c^{5} + 5 i \, B a b^{4} c^{4} d + 10 \, {\left (-i \, B b^{5} c^{2} d^{3} + 2 i \, B a b^{4} c d^{4}\right )} x^{3} + 10 \, {\left (-2 i \, B b^{5} c^{3} d^{2} + 3 i \, B a b^{4} c^{2} d^{3}\right )} x^{2} + 5 \, {\left (-3 i \, B b^{5} c^{4} d + 4 i \, B a b^{4} c^{3} d^{2}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{400 \, {\left ({\left (b^{11} c^{2} - 2 \, a b^{10} c d + a^{2} b^{9} d^{2}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{2} - 2 \, a^{2} b^{9} c d + a^{3} b^{8} d^{2}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{2} - 2 \, a^{3} b^{8} c d + a^{4} b^{7} d^{2}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{2} - 2 \, a^{4} b^{7} c d + a^{5} b^{6} d^{2}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{2} - 2 \, a^{5} b^{6} c d + a^{6} b^{5} d^{2}\right )} g^{6} x + {\left (a^{5} b^{6} c^{2} - 2 \, a^{6} b^{5} c d + a^{7} b^{4} d^{2}\right )} g^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

-1/400*(16*(-5*I*A - I*B)*b^5*c^5 + 25*(4*I*A + I*B)*a*b^4*c^4*d - (20*I*A + 9*I*B)*a^5*d^5 + 20*(I*B*b^5*c*d^
4 - I*B*a*b^4*d^5)*x^4 + 10*((-20*I*A - I*B)*b^5*c^2*d^3 + 10*(4*I*A + I*B)*a*b^4*c*d^4 + (-20*I*A - 9*I*B)*a^
2*b^3*d^5)*x^3 + 10*(2*(-20*I*A - 3*I*B)*b^5*c^3*d^2 + 15*(4*I*A + I*B)*a*b^4*c^2*d^3 + (-20*I*A - 9*I*B)*a^3*
b^2*d^5)*x^2 + 5*((-60*I*A - 11*I*B)*b^5*c^4*d + 20*(4*I*A + I*B)*a*b^4*c^3*d^2 + (-20*I*A - 9*I*B)*a^4*b*d^5)
*x + 20*(I*B*b^5*d^5*x^5 + 5*I*B*a*b^4*d^5*x^4 - 4*I*B*b^5*c^5 + 5*I*B*a*b^4*c^4*d + 10*(-I*B*b^5*c^2*d^3 + 2*
I*B*a*b^4*c*d^4)*x^3 + 10*(-2*I*B*b^5*c^3*d^2 + 3*I*B*a*b^4*c^2*d^3)*x^2 + 5*(-3*I*B*b^5*c^4*d + 4*I*B*a*b^4*c
^3*d^2)*x)*log((b*x + a)*e/(d*x + c)))/((b^11*c^2 - 2*a*b^10*c*d + a^2*b^9*d^2)*g^6*x^5 + 5*(a*b^10*c^2 - 2*a^
2*b^9*c*d + a^3*b^8*d^2)*g^6*x^4 + 10*(a^2*b^9*c^2 - 2*a^3*b^8*c*d + a^4*b^7*d^2)*g^6*x^3 + 10*(a^3*b^8*c^2 -
2*a^4*b^7*c*d + a^5*b^6*d^2)*g^6*x^2 + 5*(a^4*b^7*c^2 - 2*a^5*b^6*c*d + a^6*b^5*d^2)*g^6*x + (a^5*b^6*c^2 - 2*
a^6*b^5*c*d + a^7*b^4*d^2)*g^6)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**6,x)

[Out]

Timed out

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Giac [A]
time = 4.00, size = 238, normalized size = 1.31 \begin {gather*} -\frac {{\left (-80 i \, B b e^{6} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {100 i \, {\left (b x e + a e\right )} B d e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - 80 i \, A b e^{6} - 16 i \, B b e^{6} + \frac {100 i \, {\left (b x e + a e\right )} A d e^{5}}{d x + c} + \frac {25 i \, {\left (b x e + a e\right )} B d e^{5}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{400 \, {\left (\frac {{\left (b x e + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b x e + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

-1/400*(-80*I*B*b*e^6*log((b*x*e + a*e)/(d*x + c)) + 100*I*(b*x*e + a*e)*B*d*e^5*log((b*x*e + a*e)/(d*x + c))/
(d*x + c) - 80*I*A*b*e^6 - 16*I*B*b*e^6 + 100*I*(b*x*e + a*e)*A*d*e^5/(d*x + c) + 25*I*(b*x*e + a*e)*B*d*e^5/(
d*x + c))*(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*x*e + a*e)^5*b*c*g^6/(d*
x + c)^5 - (b*x*e + a*e)^5*a*d*g^6/(d*x + c)^5)

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Mupad [B]
time = 8.31, size = 1053, normalized size = 5.82 \begin {gather*} -\frac {\frac {20\,A\,a^4\,d^4\,i^3-80\,A\,b^4\,c^4\,i^3+9\,B\,a^4\,d^4\,i^3-16\,B\,b^4\,c^4\,i^3+20\,A\,a^2\,b^2\,c^2\,d^2\,i^3+9\,B\,a^2\,b^2\,c^2\,d^2\,i^3+20\,A\,a\,b^3\,c^3\,d\,i^3+20\,A\,a^3\,b\,c\,d^3\,i^3+9\,B\,a\,b^3\,c^3\,d\,i^3+9\,B\,a^3\,b\,c\,d^3\,i^3}{20\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (20\,A\,a^2\,b^2\,d^4\,i^3+9\,B\,a^2\,b^2\,d^4\,i^3-40\,A\,b^4\,c^2\,d^2\,i^3-6\,B\,b^4\,c^2\,d^2\,i^3+20\,A\,a\,b^3\,c\,d^3\,i^3+9\,B\,a\,b^3\,c\,d^3\,i^3\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (20\,A\,a^3\,b\,d^4\,i^3+9\,B\,a^3\,b\,d^4\,i^3-60\,A\,b^4\,c^3\,d\,i^3-11\,B\,b^4\,c^3\,d\,i^3+20\,A\,a\,b^3\,c^2\,d^2\,i^3+20\,A\,a^2\,b^2\,c\,d^3\,i^3+9\,B\,a\,b^3\,c^2\,d^2\,i^3+9\,B\,a^2\,b^2\,c\,d^3\,i^3\right )}{4\,\left (a\,d-b\,c\right )}+\frac {x^3\,\left (20\,A\,a\,b^3\,d^4\,i^3+9\,B\,a\,b^3\,d^4\,i^3-20\,A\,b^4\,c\,d^3\,i^3-B\,b^4\,c\,d^3\,i^3\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^4\,d^4\,i^3\,x^4}{a\,d-b\,c}}{20\,a^5\,b^4\,g^6+100\,a^4\,b^5\,g^6\,x+200\,a^3\,b^6\,g^6\,x^2+200\,a^2\,b^7\,g^6\,x^3+100\,a\,b^8\,g^6\,x^4+20\,b^9\,g^6\,x^5}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x^2\,\left (b\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{20\,b^4\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{10\,b^3\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{10\,b^3\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{5\,b^2\,g^6}\right )+x\,\left (b\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{20\,b^3\,g^6}\right )+a\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,a\,d^3\,i^3}{20\,b^4\,g^6}+\frac {3\,B\,c\,d^2\,i^3}{10\,b^3\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{5\,b^2\,g^6}\right )+a\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{20\,b^5\,g^6}+\frac {B\,c\,d^2\,i^3}{10\,b^4\,g^6}\right )+\frac {3\,B\,c^2\,d\,i^3}{20\,b^3\,g^6}\right )+\frac {B\,c^3\,i^3}{5\,b^2\,g^6}+\frac {B\,d^3\,i^3\,x^3}{2\,b^2\,g^6}\right )}{5\,a^4\,x+\frac {a^5}{b}+b^4\,x^5+10\,a^3\,b\,x^2+5\,a\,b^3\,x^4+10\,a^2\,b^2\,x^3}-\frac {B\,d^5\,i^3\,\mathrm {atanh}\left (\frac {20\,b^6\,c^2\,g^6-20\,a^2\,b^4\,d^2\,g^6}{20\,b^4\,g^6\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{10\,b^4\,g^6\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^6,x)

[Out]

- ((20*A*a^4*d^4*i^3 - 80*A*b^4*c^4*i^3 + 9*B*a^4*d^4*i^3 - 16*B*b^4*c^4*i^3 + 20*A*a^2*b^2*c^2*d^2*i^3 + 9*B*
a^2*b^2*c^2*d^2*i^3 + 20*A*a*b^3*c^3*d*i^3 + 20*A*a^3*b*c*d^3*i^3 + 9*B*a*b^3*c^3*d*i^3 + 9*B*a^3*b*c*d^3*i^3)
/(20*(a*d - b*c)) + (x^2*(20*A*a^2*b^2*d^4*i^3 + 9*B*a^2*b^2*d^4*i^3 - 40*A*b^4*c^2*d^2*i^3 - 6*B*b^4*c^2*d^2*
i^3 + 20*A*a*b^3*c*d^3*i^3 + 9*B*a*b^3*c*d^3*i^3))/(2*(a*d - b*c)) + (x*(20*A*a^3*b*d^4*i^3 + 9*B*a^3*b*d^4*i^
3 - 60*A*b^4*c^3*d*i^3 - 11*B*b^4*c^3*d*i^3 + 20*A*a*b^3*c^2*d^2*i^3 + 20*A*a^2*b^2*c*d^3*i^3 + 9*B*a*b^3*c^2*
d^2*i^3 + 9*B*a^2*b^2*c*d^3*i^3))/(4*(a*d - b*c)) + (x^3*(20*A*a*b^3*d^4*i^3 + 9*B*a*b^3*d^4*i^3 - 20*A*b^4*c*
d^3*i^3 - B*b^4*c*d^3*i^3))/(2*(a*d - b*c)) + (B*b^4*d^4*i^3*x^4)/(a*d - b*c))/(20*a^5*b^4*g^6 + 20*b^9*g^6*x^
5 + 100*a^4*b^5*g^6*x + 100*a*b^8*g^6*x^4 + 200*a^3*b^6*g^6*x^2 + 200*a^2*b^7*g^6*x^3) - (log((e*(a + b*x))/(c
+ d*x))*(x^2*(b*(b*((B*a*d^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*a*d^3*i^3)/(20*b^4*g^6) +
(3*B*c*d^2*i^3)/(10*b^3*g^6)) + (3*B*a*d^3*i^3)/(10*b^3*g^6) + (3*B*c*d^2*i^3)/(5*b^2*g^6)) + x*(b*(a*((B*a*d
^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*c^2*d*i^3)/(20*b^3*g^6)) + a*(b*((B*a*d^3*i^3)/(20*b
^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*a*d^3*i^3)/(20*b^4*g^6) + (3*B*c*d^2*i^3)/(10*b^3*g^6)) + (3*B*c^
2*d*i^3)/(5*b^2*g^6)) + a*(a*((B*a*d^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*c^2*d*i^3)/(20*b
^3*g^6)) + (B*c^3*i^3)/(5*b^2*g^6) + (B*d^3*i^3*x^3)/(2*b^2*g^6)))/(5*a^4*x + a^5/b + b^4*x^5 + 10*a^3*b*x^2 +
5*a*b^3*x^4 + 10*a^2*b^2*x^3) - (B*d^5*i^3*atanh((20*b^6*c^2*g^6 - 20*a^2*b^4*d^2*g^6)/(20*b^4*g^6*(a*d - b*c
)^2) - (2*b*d*x)/(a*d - b*c)))/(10*b^4*g^6*(a*d - b*c)^2)

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