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

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

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

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Rubi [A]  time = 0.14, 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 = {2525, 12, 44} \[ -\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{4 b g^5 (a+b x)^4}+\frac {B d^3}{2 b g^5 (a+b x) (b c-a d)^3}-\frac {B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac {B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}-\frac {B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}+\frac {B d}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac {B}{8 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*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) - (A + B*Log[(e*
(a + b*x)^2)/(c + d*x)^2])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^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 44

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] && L
tQ[m + n + 2, 0])

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 162, normalized size = 0.78 \[ -\frac {6 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac {B \left (12 d^4 (a+b x)^4 \log (c+d x)+12 d^3 (a+b x)^3 (a d-b c)+6 d^2 (a+b x)^2 (b c-a d)^2+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )}{(b c-a d)^4}}{24 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(6*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (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)/(b*g^5*(a + b*x)^4)

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fricas [B]  time = 0.95, size = 654, normalized size = 3.14 \[ -\frac {3 \, {\left (2 \, A + B\right )} b^{4} c^{4} - 8 \, {\left (3 \, A + 2 \, B\right )} a b^{3} c^{3} d + 36 \, {\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 24 \, {\left (A + 2 \, B\right )} a^{3} b c d^{3} + {\left (6 \, A + 25 \, B\right )} a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 6 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{24 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^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((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^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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giac [B]  time = 0.77, size = 419, normalized size = 2.01 \[ -\frac {B d^{4} \log \left (-\frac {b c g}{b g x + a g} + \frac {a d g}{b g x + a g} - d\right )}{2 \, {\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} + \frac {B d^{3}}{2 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {B d^{2}}{4 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b g^{2}} - \frac {B \log \left (\frac {b^{2}}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{4 \, {\left (b g x + a g\right )}^{4} b g} + \frac {B d}{6 \, {\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b g^{2}} - \frac {2 \, A b^{3} g^{3} + 3 \, B b^{3} g^{3}}{8 \, {\left (b g x + a g\right )}^{4} b^{4} g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^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/(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*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 + 3*B*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)

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maple [B]  time = 0.22, size = 833, normalized size = 4.00 \[ \frac {B \,b^{3} d^{4} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{4 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a^{4} d^{4}-4 a^{3} c \,d^{3} b +6 a^{2} c^{2} d^{2} b^{2}-4 a \,c^{3} d \,b^{3}+c^{4} b^{4}\right ) g^{5}}-\frac {A \,b^{3} d^{4}}{4 \left (a d -b c \right )^{4} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} g^{5}}+\frac {B \,b^{2} d^{4} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a^{3} d^{3}-3 a^{2} c \,d^{2} b +3 a \,c^{2} d \,b^{2}-b^{3} c^{3}\right ) \left (d x +c \right ) g^{5}}+\frac {A \,b^{2} d^{4}}{\left (a d -b c \right )^{4} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} g^{5}}-\frac {B \,b^{2} d^{4}}{2 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a^{3} d^{3}-3 a^{2} c \,d^{2} b +3 a \,c^{2} d \,b^{2}-b^{3} c^{3}\right ) \left (d x +c \right ) g^{5}}+\frac {3 B b \,d^{4} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{2 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right )^{2} g^{5}}-\frac {3 A b \,d^{4}}{2 \left (a d -b c \right )^{4} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{5}}-\frac {7 B b \,d^{4}}{4 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right )^{2} g^{5}}+\frac {B \,d^{4} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a d -b c \right ) \left (d x +c \right )^{3} g^{5}}+\frac {A \,d^{4}}{\left (a d -b c \right )^{4} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) g^{5}}-\frac {13 B \,d^{4}}{6 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (a d -b c \right ) \left (d x +c \right )^{3} g^{5}}-\frac {25 B \,d^{4}}{24 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{4} \left (d x +c \right )^{4} b \,g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4/g^5*A*b^2/(a*d-b*c)^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3-1/4*d^4/g^5*A*b^3/(a*d-b*c)^4/(1/(d*x+c)*a*d-1/(d*
x+c)*b*c+b)^4+d^4/g^5*A/(a*d-b*c)^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)-3/2*d^4/g^5*A*b/(a*d-b*c)^4/(1/(d*x+c)*a*d
-1/(d*x+c)*b*c+b)^2-25/24*d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*B/b/(d*x+c)^4+1/4*d^4/g^5/(1/(d*x+c)*a*d-1
/(d*x+c)*b*c+b)^4*b^3*B/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln((1/(d*x+c)*a*d-1/(d
*x+c)*b*c+b)^2/d^2*e)-1/2*d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*B*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d
-b^3*c^3)/(d*x+c)-7/4*d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2-13/6
*d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*B/(a*d-b*c)/(d*x+c)^3+d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*B/(
a*d-b*c)/(d*x+c)^3*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)+3/2*d^4/g^5/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^4*b
*B/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)+d^4/g^5/(1/(d*x+c)*a*d-1/
(d*x+c)*b*c+b)^4*B*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b
)^2/d^2*e)

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maxima [B]  time = 1.62, size = 699, normalized size = 3.36 \[ \frac {1}{24} \, B {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac {6 \, \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^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(b^2*e*x^2/(d^2*
x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^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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mupad [B]  time = 6.51, size = 579, normalized size = 2.78 \[ -\frac {\frac {6\,A\,a^3\,d^3-6\,A\,b^3\,c^3+25\,B\,a^3\,d^3-3\,B\,b^3\,c^3+18\,A\,a\,b^2\,c^2\,d-18\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d-23\,B\,a^2\,b\,c\,d^2}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{2\,a^4\,b\,g^5+8\,a^3\,b^2\,g^5\,x+12\,a^2\,b^3\,g^5\,x^2+8\,a\,b^4\,g^5\,x^3+2\,b^5\,g^5\,x^4}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {B\,d^4\,\mathrm {atanh}\left (\frac {-2\,a^4\,b\,d^4\,g^5+4\,a^3\,b^2\,c\,d^3\,g^5-4\,a\,b^4\,c^3\,d\,g^5+2\,b^5\,c^4\,g^5}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{b\,g^5\,{\left (a\,d-b\,c\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((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
) - (B*log((e*(a + b*x)^2)/(c + d*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)) -
(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)

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sympy [B]  time = 5.81, size = 947, normalized size = 4.55 \[ - \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d 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} \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} \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} \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} \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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*log(e*(a + b*x)**2/(c + d*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 - 2
4*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 - 14
4*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 - 9
6*a**3*b**5*c**3*g**5))

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