3.1.9 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^5} \, dx\) [9]

Optimal. Leaf size=269 \[ -\frac {B d^2 i (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b B d i (c+d x)^3}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i (c+d x)^4}{16 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4} \]

[Out]

-1/4*B*d^2*i*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+2/9*b*B*d*i*(d*x+c)^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/16*b^2*B*
i*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b*x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a
)^2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/4*b^2*i*(d*x+c)^4*(A+B*ln(e*(
b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a)^4

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Rubi [A]
time = 0.12, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2562, 45, 2372, 12, 14} \begin {gather*} -\frac {b^2 i (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B i (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B d^2 i (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d i (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5,x]

[Out]

-1/4*(B*d^2*i*(c + d*x)^2)/((b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*B*d*i*(c + d*x)^3)/(9*(b*c - a*d)^3*g^5*(a +
b*x)^3) - (b^2*B*i*(c + d*x)^4)/(16*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*
x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])
)/(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^
3*g^5*(a + b*x)^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 {(9 c+9 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^5}+\frac {9 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac {(9 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^5}+\frac {(9 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac {(9 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d (b c-a d)) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac {\left (9 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {(3 B d (b c-a d)) \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^2 g^5}+\frac {\left (9 B (b c-a d)^2\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^2 g^5}\\ &=-\frac {9 B (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac {B d}{4 b^2 g^5 (a+b x)^3}+\frac {3 B d^2}{8 b^2 (b c-a d) g^5 (a+b x)^2}-\frac {3 B d^3}{4 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac {3 B d^4 \log (a+b x)}{4 b^2 (b c-a d)^3 g^5}-\frac {9 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac {3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac {3 B d^4 \log (c+d x)}{4 b^2 (b c-a d)^3 g^5}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 210, normalized size = 0.78 \begin {gather*} -\frac {i \left (\frac {36 A b c}{(a+b x)^4}+\frac {9 b B c}{(a+b x)^4}-\frac {36 a A d}{(a+b x)^4}-\frac {9 a B d}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}+\frac {4 B d}{(a+b x)^3}-\frac {6 B d^2}{(b c-a d) (a+b x)^2}+\frac {12 B d^3}{(b c-a d)^2 (a+b x)}+\frac {12 B d^4 \log (a+b x)}{(b c-a d)^3}+\frac {12 B (3 b c+a d+4 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4}-\frac {12 B d^4 \log (c+d x)}{(b c-a d)^3}\right )}{144 b^2 g^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5,x]

[Out]

-1/144*(i*((36*A*b*c)/(a + b*x)^4 + (9*b*B*c)/(a + b*x)^4 - (36*a*A*d)/(a + b*x)^4 - (9*a*B*d)/(a + b*x)^4 + (
48*A*d)/(a + b*x)^3 + (4*B*d)/(a + b*x)^3 - (6*B*d^2)/((b*c - a*d)*(a + b*x)^2) + (12*B*d^3)/((b*c - a*d)^2*(a
+ b*x)) + (12*B*d^4*Log[a + b*x])/(b*c - a*d)^3 + (12*B*(3*b*c + a*d + 4*b*d*x)*Log[(e*(a + b*x))/(c + d*x)])
/(a + b*x)^4 - (12*B*d^4*Log[c + d*x])/(b*c - a*d)^3))/(b^2*g^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(257)=514\).
time = 0.58, size = 516, normalized size = 1.92 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(-1/4*i*d^2*e^3/(a*d-b*c)^4/g^5*A*b^2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4+2/3*i*d^3*e^2/(a*d-b*
c)^4/g^5*A*b/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3-1/2*i*d^4*e/(a*d-b*c)^4/g^5*A/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+i*d
^2*e^3/(a*d-b*c)^4/g^5*B*b^2*(-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)-2*i*d^3*e^2/(a*d-b*c)^4/g^5*B*b*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d
-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)+i*d^4*e/(a*d-b*c)^4/g^5*B*(-1/2/(b*e/d+(a*d-b*c)*e/d/(
d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2))

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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. 1386 vs. \(2 (254) = 508\).
time = 0.37, size = 1386, normalized size = 5.15 \begin {gather*} -\frac {1}{144} i \, B d {\left (\frac {12 \, {\left (4 \, b x + a\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{b^{6} g^{5} x^{4} + 4 \, a b^{5} g^{5} x^{3} + 6 \, a^{2} b^{4} g^{5} x^{2} + 4 \, a^{3} b^{3} g^{5} x + a^{4} b^{2} g^{5}} + \frac {7 \, a b^{3} c^{3} - 33 \, a^{2} b^{2} c^{2} d + 75 \, a^{3} b c d^{2} - 13 \, a^{4} d^{3} + 12 \, {\left (4 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} - 6 \, {\left (4 \, b^{4} c^{2} d - 29 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )} x^{2} + 4 \, {\left (4 \, b^{4} c^{3} - 21 \, a b^{3} c^{2} d + 57 \, a^{2} b^{2} c d^{2} - 13 \, a^{3} b d^{3}\right )} x}{{\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}} + \frac {12 \, {\left (4 \, b c d^{3} - a d^{4}\right )} \log \left (b x + a\right )}{{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} g^{5}} - \frac {12 \, {\left (4 \, b c d^{3} - a d^{4}\right )} \log \left (d x + c\right )}{{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} g^{5}}\right )} + \frac {1}{48} i \, B c {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac {12 \, \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {i \, {\left (4 \, b x + a\right )} A d}{12 \, {\left (b^{6} g^{5} x^{4} + 4 \, a b^{5} g^{5} x^{3} + 6 \, a^{2} b^{4} g^{5} x^{2} + 4 \, a^{3} b^{3} g^{5} x + a^{4} b^{2} g^{5}\right )}} - \frac {i \, A c}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/144*I*B*d*(12*(4*b*x + a)*log(b*x*e/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g
^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(
4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*
c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4
+ 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d
+ 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g
^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x +
a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^
4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/48*I*
B*c*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*
x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^
3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*
b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*
b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*x*e/(d*x + c) +
a*e/(d*x + c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*log
(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x +
c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/12*I*(4*b*x + a)*A*
d/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*I*A*c/(b^5*g^5*x^4
+ 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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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. 603 vs. \(2 (254) = 508\).
time = 0.38, size = 603, normalized size = 2.24 \begin {gather*} \frac {9 \, {\left (-4 i \, A - i \, B\right )} b^{4} c^{4} + 32 \, {\left (3 i \, A + i \, B\right )} a b^{3} c^{3} d + 36 \, {\left (-2 i \, A - i \, B\right )} a^{2} b^{2} c^{2} d^{2} - {\left (-12 i \, A - 13 i \, B\right )} a^{4} d^{4} + 12 \, {\left (-i \, B b^{4} c d^{3} + i \, B a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (i \, B b^{4} c^{2} d^{2} - 8 i \, B a b^{3} c d^{3} + 7 i \, B a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left ({\left (-12 i \, A - i \, B\right )} b^{4} c^{3} d + 6 \, {\left (6 i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 18 \, {\left (-2 i \, A - i \, B\right )} a^{2} b^{2} c d^{3} + {\left (12 i \, A + 13 i \, B\right )} a^{3} b d^{4}\right )} x + 12 \, {\left (-i \, B b^{4} d^{4} x^{4} - 4 i \, B a b^{3} d^{4} x^{3} - 6 i \, B a^{2} b^{2} d^{4} x^{2} - 3 i \, B b^{4} c^{4} + 8 i \, B a b^{3} c^{3} d - 6 i \, B a^{2} b^{2} c^{2} d^{2} + 4 \, {\left (-i \, B b^{4} c^{3} d + 3 i \, B a b^{3} c^{2} d^{2} - 3 i \, B a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

1/144*(9*(-4*I*A - I*B)*b^4*c^4 + 32*(3*I*A + I*B)*a*b^3*c^3*d + 36*(-2*I*A - I*B)*a^2*b^2*c^2*d^2 - (-12*I*A
- 13*I*B)*a^4*d^4 + 12*(-I*B*b^4*c*d^3 + I*B*a*b^3*d^4)*x^3 + 6*(I*B*b^4*c^2*d^2 - 8*I*B*a*b^3*c*d^3 + 7*I*B*a
^2*b^2*d^4)*x^2 + 4*((-12*I*A - I*B)*b^4*c^3*d + 6*(6*I*A + I*B)*a*b^3*c^2*d^2 + 18*(-2*I*A - I*B)*a^2*b^2*c*d
^3 + (12*I*A + 13*I*B)*a^3*b*d^4)*x + 12*(-I*B*b^4*d^4*x^4 - 4*I*B*a*b^3*d^4*x^3 - 6*I*B*a^2*b^2*d^4*x^2 - 3*I
*B*b^4*c^4 + 8*I*B*a*b^3*c^3*d - 6*I*B*a^2*b^2*c^2*d^2 + 4*(-I*B*b^4*c^3*d + 3*I*B*a*b^3*c^2*d^2 - 3*I*B*a^2*b
^2*c*d^3)*x)*log((b*x + a)*e/(d*x + c)))/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 +
4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3
*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*
x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (252) = 504\).
time = 8.47, size = 928, normalized size = 3.45 \begin {gather*} - \frac {B d^{4} i \log {\left (x + \frac {- \frac {B a^{4} d^{8} i}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{7} i}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{6} i}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{5} i}{\left (a d - b c\right )^{3}} + B a d^{5} i - \frac {B b^{4} c^{4} d^{4} i}{\left (a d - b c\right )^{3}} + B b c d^{4} i}{2 B b d^{5} i} \right )}}{12 b^{2} g^{5} \left (a d - b c\right )^{3}} + \frac {B d^{4} i \log {\left (x + \frac {\frac {B a^{4} d^{8} i}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{7} i}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{6} i}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{5} i}{\left (a d - b c\right )^{3}} + B a d^{5} i + \frac {B b^{4} c^{4} d^{4} i}{\left (a d - b c\right )^{3}} + B b c d^{4} i}{2 B b d^{5} i} \right )}}{12 b^{2} g^{5} \left (a d - b c\right )^{3}} + \frac {\left (- B a d i - 3 B b c i - 4 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{12 a^{4} b^{2} g^{5} + 48 a^{3} b^{3} g^{5} x + 72 a^{2} b^{4} g^{5} x^{2} + 48 a b^{5} g^{5} x^{3} + 12 b^{6} g^{5} x^{4}} + \frac {- 12 A a^{3} d^{3} i - 12 A a^{2} b c d^{2} i + 60 A a b^{2} c^{2} d i - 36 A b^{3} c^{3} i - 13 B a^{3} d^{3} i - 13 B a^{2} b c d^{2} i + 23 B a b^{2} c^{2} d i - 9 B b^{3} c^{3} i - 12 B b^{3} d^{3} i x^{3} + x^{2} \left (- 42 B a b^{2} d^{3} i + 6 B b^{3} c d^{2} i\right ) + x \left (- 48 A a^{2} b d^{3} i + 96 A a b^{2} c d^{2} i - 48 A b^{3} c^{2} d i - 52 B a^{2} b d^{3} i + 20 B a b^{2} c d^{2} i - 4 B b^{3} c^{2} d i\right )}{144 a^{6} b^{2} d^{2} g^{5} - 288 a^{5} b^{3} c d g^{5} + 144 a^{4} b^{4} c^{2} g^{5} + x^{4} \cdot \left (144 a^{2} b^{6} d^{2} g^{5} - 288 a b^{7} c d g^{5} + 144 b^{8} c^{2} g^{5}\right ) + x^{3} \cdot \left (576 a^{3} b^{5} d^{2} g^{5} - 1152 a^{2} b^{6} c d g^{5} + 576 a b^{7} c^{2} g^{5}\right ) + x^{2} \cdot \left (864 a^{4} b^{4} d^{2} g^{5} - 1728 a^{3} b^{5} c d g^{5} + 864 a^{2} b^{6} c^{2} g^{5}\right ) + x \left (576 a^{5} b^{3} d^{2} g^{5} - 1152 a^{4} b^{4} c d g^{5} + 576 a^{3} b^{5} c^{2} g^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)

[Out]

-B*d**4*i*log(x + (-B*a**4*d**8*i/(a*d - b*c)**3 + 4*B*a**3*b*c*d**7*i/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**
6*i/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**5*i/(a*d - b*c)**3 + B*a*d**5*i - B*b**4*c**4*d**4*i/(a*d - b*c)**3 +
B*b*c*d**4*i)/(2*B*b*d**5*i))/(12*b**2*g**5*(a*d - b*c)**3) + B*d**4*i*log(x + (B*a**4*d**8*i/(a*d - b*c)**3 -
4*B*a**3*b*c*d**7*i/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**6*i/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**5*i/(a*d -
b*c)**3 + B*a*d**5*i + B*b**4*c**4*d**4*i/(a*d - b*c)**3 + B*b*c*d**4*i)/(2*B*b*d**5*i))/(12*b**2*g**5*(a*d -
b*c)**3) + (-B*a*d*i - 3*B*b*c*i - 4*B*b*d*i*x)*log(e*(a + b*x)/(c + d*x))/(12*a**4*b**2*g**5 + 48*a**3*b**3*
g**5*x + 72*a**2*b**4*g**5*x**2 + 48*a*b**5*g**5*x**3 + 12*b**6*g**5*x**4) + (-12*A*a**3*d**3*i - 12*A*a**2*b*
c*d**2*i + 60*A*a*b**2*c**2*d*i - 36*A*b**3*c**3*i - 13*B*a**3*d**3*i - 13*B*a**2*b*c*d**2*i + 23*B*a*b**2*c**
2*d*i - 9*B*b**3*c**3*i - 12*B*b**3*d**3*i*x**3 + x**2*(-42*B*a*b**2*d**3*i + 6*B*b**3*c*d**2*i) + x*(-48*A*a*
*2*b*d**3*i + 96*A*a*b**2*c*d**2*i - 48*A*b**3*c**2*d*i - 52*B*a**2*b*d**3*i + 20*B*a*b**2*c*d**2*i - 4*B*b**3
*c**2*d*i))/(144*a**6*b**2*d**2*g**5 - 288*a**5*b**3*c*d*g**5 + 144*a**4*b**4*c**2*g**5 + x**4*(144*a**2*b**6*
d**2*g**5 - 288*a*b**7*c*d*g**5 + 144*b**8*c**2*g**5) + x**3*(576*a**3*b**5*d**2*g**5 - 1152*a**2*b**6*c*d*g**
5 + 576*a*b**7*c**2*g**5) + x**2*(864*a**4*b**4*d**2*g**5 - 1728*a**3*b**5*c*d*g**5 + 864*a**2*b**6*c**2*g**5)
+ x*(576*a**5*b**3*d**2*g**5 - 1152*a**4*b**4*c*d*g**5 + 576*a**3*b**5*c**2*g**5))

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Giac [A]
time = 5.17, size = 382, normalized size = 1.42 \begin {gather*} \frac {{\left (-36 i \, B b^{2} e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {96 i \, {\left (b x e + a e\right )} B b d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {72 i \, {\left (b x e + a e\right )}^{2} B d^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - 36 i \, A b^{2} e^{5} - 9 i \, B b^{2} e^{5} + \frac {96 i \, {\left (b x e + a e\right )} A b d e^{4}}{d x + c} + \frac {32 i \, {\left (b x e + a e\right )} B b d e^{4}}{d x + c} - \frac {72 i \, {\left (b x e + a e\right )}^{2} A d^{2} e^{3}}{{\left (d x + c\right )}^{2}} - \frac {36 i \, {\left (b x e + a e\right )}^{2} B d^{2} e^{3}}{{\left (d x + c\right )}^{2}}\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 )}}{144 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x e + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x e + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

1/144*(-36*I*B*b^2*e^5*log((b*x*e + a*e)/(d*x + c)) + 96*I*(b*x*e + a*e)*B*b*d*e^4*log((b*x*e + a*e)/(d*x + c)
)/(d*x + c) - 72*I*(b*x*e + a*e)^2*B*d^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 36*I*A*b^2*e^5 - 9*I*B
*b^2*e^5 + 96*I*(b*x*e + a*e)*A*b*d*e^4/(d*x + c) + 32*I*(b*x*e + a*e)*B*b*d*e^4/(d*x + c) - 72*I*(b*x*e + a*e
)^2*A*d^2*e^3/(d*x + c)^2 - 36*I*(b*x*e + a*e)^2*B*d^2*e^3/(d*x + c)^2)*(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)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x*e + a*e)^4*a*b*c*d*g^5/(d*
x + c)^4 + (b*x*e + a*e)^4*a^2*d^2*g^5/(d*x + c)^4)

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Mupad [B]
time = 6.46, size = 590, normalized size = 2.19 \begin {gather*} \frac {B\,d^4\,i\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{4\,b^2\,g^5}+\frac {B\,a\,d\,i}{12\,b^3\,g^5}+\frac {B\,d\,i\,x}{3\,b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i+9\,B\,b^3\,c^3\,i-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i+13\,B\,a^2\,b\,c\,d^2\,i}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+13\,B\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i+B\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i-5\,B\,a\,b^2\,c\,d^2\,i\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i-7\,B\,a\,b^2\,d^2\,i\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^5,x)

[Out]

(B*d^4*i*atanh((12*b^5*c^3*g^5 + 12*a^3*b^2*d^3*g^5 - 12*a*b^4*c^2*d*g^5 - 12*a^2*b^3*c*d^2*g^5)/(12*b^2*g^5*(
a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(6*b^2*g^5*(a*d - b*c)^3) - (log((e*
(a + b*x))/(c + d*x))*((B*c*i)/(4*b^2*g^5) + (B*a*d*i)/(12*b^3*g^5) + (B*d*i*x)/(3*b^2*g^5)))/(4*a^3*x + a^4/b
+ b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3) - ((12*A*a^3*d^3*i + 36*A*b^3*c^3*i + 13*B*a^3*d^3*i + 9*B*b^3*c^3*i -
60*A*a*b^2*c^2*d*i + 12*A*a^2*b*c*d^2*i - 23*B*a*b^2*c^2*d*i + 13*B*a^2*b*c*d^2*i)/(12*(a^2*d^2 + b^2*c^2 - 2
*a*b*c*d)) + (x*(12*A*a^2*b*d^3*i + 13*B*a^2*b*d^3*i + 12*A*b^3*c^2*d*i + B*b^3*c^2*d*i - 24*A*a*b^2*c*d^2*i -
5*B*a*b^2*c*d^2*i))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (d*x^2*(B*b^3*c*d*i - 7*B*a*b^2*d^2*i))/(2*(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)) + (B*b^3*d^3*i*x^3)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(12*a^4*b^2*g^5 + 12*b^6*g^5*x^4
+ 48*a^3*b^3*g^5*x + 48*a*b^5*g^5*x^3 + 72*a^2*b^4*g^5*x^2)

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