3.3.9 \(\int \frac {A+B \log (\frac {e (c+d x)^2}{(a+b x)^2})}{(a g+b g x)^5} \, dx\) [209]

Optimal. Leaf size=208 \[ \frac {B}{8 b g^5 (a+b x)^4}-\frac {B d}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4} \]

[Out]

1/8*B/b/g^5/(b*x+a)^4-1/6*B*d/b/(-a*d+b*c)/g^5/(b*x+a)^3+1/4*B*d^2/b/(-a*d+b*c)^2/g^5/(b*x+a)^2-1/2*B*d^3/b/(-
a*d+b*c)^3/g^5/(b*x+a)-1/2*B*d^4*ln(b*x+a)/b/(-a*d+b*c)^4/g^5+1/2*B*d^4*ln(d*x+c)/b/(-a*d+b*c)^4/g^5+1/4*(-A-B
*ln(e*(d*x+c)^2/(b*x+a)^2))/b/g^5/(b*x+a)^4

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Rubi [A]
time = 0.11, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2548, 21, 46} \begin {gather*} -\frac {B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A}{4 b g^5 (a+b x)^4}-\frac {B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}+\frac {B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}-\frac {B d^3}{2 b g^5 (a+b x) (b c-a d)^3}+\frac {B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}-\frac {B d}{6 b g^5 (a+b x)^3 (b c-a d)}+\frac {B}{8 b g^5 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^5,x]

[Out]

B/(8*b*g^5*(a + b*x)^4) - (B*d)/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2
) - (B*d^3)/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(2*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c +
d*x])/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(4*b*g^5*(a + b*x)^4)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps

\begin {align*} \int \frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(a g+b g x)^5} \, dx &=-\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {2 (-b c+a d)}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \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}{2 b g^5}\\ &=\frac {B}{8 b g^5 (a+b x)^4}-\frac {B d}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 162, normalized size = 0.78 \begin {gather*} \frac {\frac {B \left (3 (b c-a d)^4+4 d (-b c+a d)^3 (a+b x)+6 d^2 (b c-a d)^2 (a+b x)^2+12 d^3 (-b c+a d) (a+b x)^3-12 d^4 (a+b x)^4 \log (a+b x)+12 d^4 (a+b x)^4 \log (c+d x)\right )}{(b c-a d)^4}-6 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{24 b g^5 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^5,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(197)=394\).
time = 0.49, size = 582, normalized size = 2.80 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)

[Out]

-1/b*(1/4/(b*x+a)^4/g^5*A+1/4/g^5*B/(b*x+a)^4*ln(e*(a*d/(b*x+a)-b*c/(b*x+a)-d)^2/b^2)-1/8/g^5*B*a^4*d^4/(a*d-b
*c)^4/(b*x+a)^4+1/2/g^5*B*a^3*d^3/(a*d-b*c)^4*b*c/(b*x+a)^4-3/4/g^5*B*a^2*d^2/(a*d-b*c)^4*b^2*c^2/(b*x+a)^4+1/
2/g^5*B*a*d/(a*d-b*c)^4*b^3*c^3/(b*x+a)^4-1/6/g^5*B*a^3*d^4/(a*d-b*c)^4/(b*x+a)^3+1/2/g^5*B*a^2*d^3/(a*d-b*c)^
4*b*c/(b*x+a)^3-1/2/g^5*B*a*d^2/(a*d-b*c)^4*b^2*c^2/(b*x+a)^3-1/4/g^5*B*a^2*d^4/(a*d-b*c)^4/(b*x+a)^2+1/2/g^5*
B*a*d^3/(a*d-b*c)^4*b*c/(b*x+a)^2-1/2/g^5*B*a*d^4/(a*d-b*c)^4/(b*x+a)-1/2/g^5*B*a*d^5/(a*d-b*c)^5*ln(a*d/(b*x+
a)-b*c/(b*x+a)-d)-1/8/g^5*B*b^4*c^4/(a*d-b*c)^4/(b*x+a)^4+1/6/g^5*B*b^3*c^3/(a*d-b*c)^4*d/(b*x+a)^3-1/4/g^5*B*
b^2*c^2/(a*d-b*c)^4*d^2/(b*x+a)^2+1/2/g^5*B*b*c/(a*d-b*c)^4*d^3/(b*x+a)+1/2/g^5*B*b*c*d^4/(a*d-b*c)^5*ln(a*d/(
b*x+a)-b*c/(b*x+a)-d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (195) = 390\).
time = 0.33, size = 702, normalized size = 3.38 \begin {gather*} -\frac {1}{24} \, B {\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 {6 \, \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\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 {A}{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((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/24*B*((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) + 6*log(d^2*x^2*e/(b^2
*x^2 + 2*a*b*x + a^2) + 2*c*d*x*e/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2))/(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)) - 1/4*A/(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] Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (195) = 390\).
time = 0.36, size = 656, normalized size = 3.15 \begin {gather*} -\frac {3 \, {\left (2 \, A - B\right )} b^{4} c^{4} - 8 \, {\left (3 \, A - 2 \, B\right )} a b^{3} c^{3} d + 36 \, {\left (A - B\right )} a^{2} b^{2} c^{2} d^{2} - 24 \, {\left (A - 2 \, B\right )} a^{3} b c d^{3} + {\left (6 \, A - 25 \, B\right )} a^{4} d^{4} + 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 6 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{24 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/24*(3*(2*A - B)*b^4*c^4 - 8*(3*A - 2*B)*a*b^3*c^3*d + 36*(A - B)*a^2*b^2*c^2*d^2 - 24*(A - 2*B)*a^3*b*c*d^3
+ (6*A - 25*B)*a^4*d^4 + 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^
2*d^4)*x^2 + 4*(B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*x - 6*(B*b^4*d^4*x^4 +
4*B*a*b^3*d^4*x^3 + 6*B*a^2*b^2*d^4*x^2 + 4*B*a^3*b*d^4*x - B*b^4*c^4 + 4*B*a*b^3*c^3*d - 6*B*a^2*b^2*c^2*d^2
+ 4*B*a^3*b*c*d^3)*log((d^2*x^2 + 2*c*d*x + c^2)*e/(b^2*x^2 + 2*a*b*x + a^2)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a
^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 -
4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d
^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d
^4)*g^5*x + (a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (182) = 364\).
time = 3.38, size = 947, normalized size = 4.55 \begin {gather*} - \frac {B \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}}{4 a^{4} b g^{5} + 16 a^{3} b^{2} g^{5} x + 24 a^{2} b^{3} g^{5} x^{2} + 16 a b^{4} g^{5} x^{3} + 4 b^{5} g^{5} x^{4}} + \frac {B d^{4} \log {\left (x + \frac {- \frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} + \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} - \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} + \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{2 b g^{5} \left (a d - b c\right )^{4}} - \frac {B d^{4} \log {\left (x + \frac {\frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} - \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} + \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} - \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{2 b g^{5} \left (a d - b c\right )^{4}} + \frac {- 6 A a^{3} d^{3} + 18 A a^{2} b c d^{2} - 18 A a b^{2} c^{2} d + 6 A b^{3} c^{3} + 25 B a^{3} d^{3} - 23 B a^{2} b c d^{2} + 13 B a b^{2} c^{2} d - 3 B b^{3} c^{3} + 12 B b^{3} d^{3} x^{3} + x^{2} \cdot \left (42 B a b^{2} d^{3} - 6 B b^{3} c d^{2}\right ) + x \left (52 B a^{2} b d^{3} - 20 B a b^{2} c d^{2} + 4 B b^{3} c^{2} d\right )}{24 a^{7} b d^{3} g^{5} - 72 a^{6} b^{2} c d^{2} g^{5} + 72 a^{5} b^{3} c^{2} d g^{5} - 24 a^{4} b^{4} c^{3} g^{5} + x^{4} \cdot \left (24 a^{3} b^{5} d^{3} g^{5} - 72 a^{2} b^{6} c d^{2} g^{5} + 72 a b^{7} c^{2} d g^{5} - 24 b^{8} c^{3} g^{5}\right ) + x^{3} \cdot \left (96 a^{4} b^{4} d^{3} g^{5} - 288 a^{3} b^{5} c d^{2} g^{5} + 288 a^{2} b^{6} c^{2} d g^{5} - 96 a b^{7} c^{3} g^{5}\right ) + x^{2} \cdot \left (144 a^{5} b^{3} d^{3} g^{5} - 432 a^{4} b^{4} c d^{2} g^{5} + 432 a^{3} b^{5} c^{2} d g^{5} - 144 a^{2} b^{6} c^{3} g^{5}\right ) + x \left (96 a^{6} b^{2} d^{3} g^{5} - 288 a^{5} b^{3} c d^{2} g^{5} + 288 a^{4} b^{4} c^{2} d g^{5} - 96 a^{3} b^{5} c^{3} g^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(d*x+c)**2/(b*x+a)**2))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(c + d*x)**2/(a + b*x)**2)/(4*a**4*b*g**5 + 16*a**3*b**2*g**5*x + 24*a**2*b**3*g**5*x**2 + 16*a*b**4*
g**5*x**3 + 4*b**5*g**5*x**4) + B*d**4*log(x + (-B*a**5*d**9/(a*d - b*c)**4 + 5*B*a**4*b*c*d**8/(a*d - b*c)**4
- 10*B*a**3*b**2*c**2*d**7/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 - 5*B*a*b**4*c**4*d**5/(a
*d - b*c)**4 + B*a*d**5 + B*b**5*c**5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(2*b*g**5*(a*d - b*c)**4
) - B*d**4*log(x + (B*a**5*d**9/(a*d - b*c)**4 - 5*B*a**4*b*c*d**8/(a*d - b*c)**4 + 10*B*a**3*b**2*c**2*d**7/(
a*d - b*c)**4 - 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 + 5*B*a*b**4*c**4*d**5/(a*d - b*c)**4 + B*a*d**5 - B*b
**5*c**5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(2*b*g**5*(a*d - b*c)**4) + (-6*A*a**3*d**3 + 18*A*a*
*2*b*c*d**2 - 18*A*a*b**2*c**2*d + 6*A*b**3*c**3 + 25*B*a**3*d**3 - 23*B*a**2*b*c*d**2 + 13*B*a*b**2*c**2*d -
3*B*b**3*c**3 + 12*B*b**3*d**3*x**3 + x**2*(42*B*a*b**2*d**3 - 6*B*b**3*c*d**2) + x*(52*B*a**2*b*d**3 - 20*B*a
*b**2*c*d**2 + 4*B*b**3*c**2*d))/(24*a**7*b*d**3*g**5 - 72*a**6*b**2*c*d**2*g**5 + 72*a**5*b**3*c**2*d*g**5 -
24*a**4*b**4*c**3*g**5 + x**4*(24*a**3*b**5*d**3*g**5 - 72*a**2*b**6*c*d**2*g**5 + 72*a*b**7*c**2*d*g**5 - 24*
b**8*c**3*g**5) + x**3*(96*a**4*b**4*d**3*g**5 - 288*a**3*b**5*c*d**2*g**5 + 288*a**2*b**6*c**2*d*g**5 - 96*a*
b**7*c**3*g**5) + x**2*(144*a**5*b**3*d**3*g**5 - 432*a**4*b**4*c*d**2*g**5 + 432*a**3*b**5*c**2*d*g**5 - 144*
a**2*b**6*c**3*g**5) + x*(96*a**6*b**2*d**3*g**5 - 288*a**5*b**3*c*d**2*g**5 + 288*a**4*b**4*c**2*d*g**5 - 96*
a**3*b**5*c**3*g**5))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (195) = 390\).
time = 3.80, size = 416, normalized size = 2.00 \begin {gather*} \frac {B d^{4} \log \left (-\frac {b c g}{b g x + a g} + \frac {a d g}{b g x + a g} - d\right )}{2 \, {\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac {B d^{3}}{2 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} + \frac {B d^{2}}{4 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b g^{2}} - \frac {B \log \left (\frac {\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}{b^{2}}\right )}{4 \, {\left (b g x + a g\right )}^{4} b g} - \frac {B d}{6 \, {\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b g^{2}} - \frac {2 \, A b^{3} g^{3} + B b^{3} g^{3}}{8 \, {\left (b g x + a g\right )}^{4} b^{4} g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)^2/(b*x+a)^2))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

1/2*B*d^4*log(-b*c*g/(b*g*x + a*g) + a*d*g/(b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2
*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) - 1/2*B*d^3/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*
g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) + 1/4*B*d^2/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2
) - 1/4*B*log((b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b
*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2)/b^2)/((b*g*x + a*g)^4*b*g) - 1/6*B*d/((b*g*x + a*g)^3*(b
*c - a*d)*b*g^2) - 1/8*(2*A*b^3*g^3 + B*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)

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Mupad [B]
time = 7.90, size = 579, normalized size = 2.78 \begin {gather*} \frac {B\,d^4\,\mathrm {atanh}\left (\frac {-2\,a^4\,b\,d^4\,g^5+4\,a^3\,b^2\,c\,d^3\,g^5-4\,a\,b^4\,c^3\,d\,g^5+2\,b^5\,c^4\,g^5}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {\frac {6\,A\,a^3\,d^3-6\,A\,b^3\,c^3-25\,B\,a^3\,d^3+3\,B\,b^3\,c^3+18\,A\,a\,b^2\,c^2\,d-18\,A\,a^2\,b\,c\,d^2-13\,B\,a\,b^2\,c^2\,d+23\,B\,a^2\,b\,c\,d^2}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{2\,a^4\,b\,g^5+8\,a^3\,b^2\,g^5\,x+12\,a^2\,b^3\,g^5\,x^2+8\,a\,b^4\,g^5\,x^3+2\,b^5\,g^5\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(c + d*x)^2)/(a + b*x)^2))/(a*g + b*g*x)^5,x)

[Out]

(B*d^4*atanh((2*b^5*c^4*g^5 - 2*a^4*b*d^4*g^5 - 4*a*b^4*c^3*d*g^5 + 4*a^3*b^2*c*d^3*g^5)/(2*b*g^5*(a*d - b*c)^
4) - (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^4))/(b*g^5*(a*d - b*c)^4) - (B*
log((e*(c + d*x)^2)/(a + b*x)^2))/(4*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)) - ((6*A*
a^3*d^3 - 6*A*b^3*c^3 - 25*B*a^3*d^3 + 3*B*b^3*c^3 + 18*A*a*b^2*c^2*d - 18*A*a^2*b*c*d^2 - 13*B*a*b^2*c^2*d +
23*B*a^2*b*c*d^2)/(12*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d^2*x^2*(B*b^3*c - 7*B*a*b^2*d))
/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (d*x*(B*b^3*c^2 + 13*B*a^2*b*d^2 - 5*B*a*b^2*c*d))/
(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B*b^3*d^3*x^3)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d -
3*a^2*b*c*d^2))/(2*a^4*b*g^5 + 2*b^5*g^5*x^4 + 8*a^3*b^2*g^5*x + 8*a*b^4*g^5*x^3 + 12*a^2*b^3*g^5*x^2)

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