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

Optimal. Leaf size=206 \[ -\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 b g^5 (a+b x)^4}+\frac {B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac {B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac {B d^3}{4 b g^5 (a+b x) (b c-a d)^3}-\frac {B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac {B d}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac {B}{16 b g^5 (a+b x)^4} \]

[Out]

-1/16*B/b/g^5/(b*x+a)^4+1/12*B*d/b/(-a*d+b*c)/g^5/(b*x+a)^3-1/8*B*d^2/b/(-a*d+b*c)^2/g^5/(b*x+a)^2+1/4*B*d^3/b
/(-a*d+b*c)^3/g^5/(b*x+a)+1/4*B*d^4*ln(b*x+a)/b/(-a*d+b*c)^4/g^5+1/4*(-A-B*ln(e*(b*x+a)/(d*x+c)))/b/g^5/(b*x+a
)^4-1/4*B*d^4*ln(d*x+c)/b/(-a*d+b*c)^4/g^5

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Rubi [A]  time = 0.16, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 b g^5 (a+b x)^4}+\frac {B d^3}{4 b g^5 (a+b x) (b c-a d)^3}-\frac {B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac {B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac {B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac {B d}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac {B}{16 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-B/(16*b*g^5*(a + b*x)^4) + (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x
)^2) + (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(
e*(a + b*x))/(c + d*x)])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(4*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)}{c+d x}\right )}{(a g+b g x)^5} \, dx &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {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)}{c+d x}\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}{4 b g^5}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\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}{4 b g^5}\\ &=-\frac {B}{16 b g^5 (a+b x)^4}+\frac {B d}{12 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}-\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 158, normalized size = 0.77 \[ \frac {\frac {B \left (\frac {12 d^3 (b c-a d)}{a+b x}-\frac {6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac {4 d (b c-a d)^3}{(a+b x)^3}-\frac {3 (b c-a d)^4}{(a+b x)^4}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a+b x)^4}}{4 b g^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 3.02, size = 629, normalized size = 3.05 \[ -\frac {3 \, {\left (4 \, A + B\right )} b^{4} c^{4} - 16 \, {\left (3 \, A + B\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A + B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \, {\left (A + B\right )} a^{3} b c d^{3} + {\left (12 \, A + 25 \, B\right )} a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{48 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(3*(4*A + B)*b^4*c^4 - 16*(3*A + B)*a*b^3*c^3*d + 36*(2*A + B)*a^2*b^2*c^2*d^2 - 48*(A + B)*a^3*b*c*d^3
+ (12*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 - 12*(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*e*x + a*e)/(d*x + c)))/((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 = 2.04, size = 528, normalized size = 2.56 \[ -\frac {{\left (12 \, B b^{3} e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {48 \, {\left (b x e + a e\right )} B b^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {72 \, {\left (b x e + a e\right )}^{2} B b d^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b x e + a e\right )}^{3} B d^{3} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + 12 \, A b^{3} e^{5} + 3 \, B b^{3} e^{5} - \frac {48 \, {\left (b x e + a e\right )} A b^{2} d e^{4}}{d x + c} - \frac {16 \, {\left (b x e + a e\right )} B b^{2} d e^{4}}{d x + c} + \frac {72 \, {\left (b x e + a e\right )}^{2} A b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} + \frac {36 \, {\left (b x e + a e\right )}^{2} B b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b x e + a e\right )}^{3} A d^{3} e^{2}}{{\left (d x + c\right )}^{3}} - \frac {48 \, {\left (b x e + a e\right )}^{3} B d^{3} e^{2}}{{\left (d x + c\right )}^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{48 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b x e + a e\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b x e + a e\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x e + a e\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(12*B*b^3*e^5*log((b*x*e + a*e)/(d*x + c)) - 48*(b*x*e + a*e)*B*b^2*d*e^4*log((b*x*e + a*e)/(d*x + c))/(
d*x + c) + 72*(b*x*e + a*e)^2*B*b*d^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 48*(b*x*e + a*e)^3*B*d^3*
e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 12*A*b^3*e^5 + 3*B*b^3*e^5 - 48*(b*x*e + a*e)*A*b^2*d*e^4/(d*x
+ c) - 16*(b*x*e + a*e)*B*b^2*d*e^4/(d*x + c) + 72*(b*x*e + a*e)^2*A*b*d^2*e^3/(d*x + c)^2 + 36*(b*x*e + a*e)^
2*B*b*d^2*e^3/(d*x + c)^2 - 48*(b*x*e + a*e)^3*A*d^3*e^2/(d*x + c)^3 - 48*(b*x*e + a*e)^3*B*d^3*e^2/(d*x + c)^
3)*(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^3*c^3*g^5/(d*x +
c)^4 - 3*(b*x*e + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x*e + a*e)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x*e
+ a*e)^4*a^3*d^3*g^5/(d*x + c)^4)

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maple [B]  time = 0.05, size = 1607, normalized size = 7.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.54, size = 647, normalized size = 3.14 \[ \frac {1}{48} \, B {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac {12 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*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) - 12*log(b*e*x/(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)) - 1/4*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 = 6.17, size = 577, normalized size = 2.80 \[ -\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3-3\,B\,b^3\,c^3+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d-23\,B\,a^2\,b\,c\,d^2}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b\,g^5+16\,a^3\,b^2\,g^5\,x+24\,a^2\,b^3\,g^5\,x^2+16\,a\,b^4\,g^5\,x^3+4\,b^5\,g^5\,x^4}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {B\,d^4\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4\,g^5+8\,a^3\,b^2\,c\,d^3\,g^5-8\,a\,b^4\,c^3\,d\,g^5+4\,b^5\,c^4\,g^5}{4\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((12*A*a^3*d^3 - 12*A*b^3*c^3 + 25*B*a^3*d^3 - 3*B*b^3*c^3 + 36*A*a*b^2*c^2*d - 36*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))/(4*a^4*b*g^5 + 4*b^5*g^5*x^4 + 16*a^3*b^2*g^5*x + 16*a*b^4*g^5*x^3 + 24*a^2*b^3*g^5
*x^2) - (B*log((e*(a + b*x))/(c + d*x)))/(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((4*b^5*c^4*g^5 - 4*a^4*b*d^4*g^5 - 8*a*b^4*c^3*d*g^5 + 8*a^3*b^2*c*d^3*g^5)/(4*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))/(2*b*g^5*(a*d - b*c)^4)

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sympy [B]  time = 5.93, size = 944, normalized size = 4.58 \[ - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \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 )}}{4 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 )}}{4 b g^{5} \left (a d - b c\right )^{4}} + \frac {- 12 A a^{3} d^{3} + 36 A a^{2} b c d^{2} - 36 A a b^{2} c^{2} d + 12 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 )}{48 a^{7} b d^{3} g^{5} - 144 a^{6} b^{2} c d^{2} g^{5} + 144 a^{5} b^{3} c^{2} d g^{5} - 48 a^{4} b^{4} c^{3} g^{5} + x^{4} \left (48 a^{3} b^{5} d^{3} g^{5} - 144 a^{2} b^{6} c d^{2} g^{5} + 144 a b^{7} c^{2} d g^{5} - 48 b^{8} c^{3} g^{5}\right ) + x^{3} \left (192 a^{4} b^{4} d^{3} g^{5} - 576 a^{3} b^{5} c d^{2} g^{5} + 576 a^{2} b^{6} c^{2} d g^{5} - 192 a b^{7} c^{3} g^{5}\right ) + x^{2} \left (288 a^{5} b^{3} d^{3} g^{5} - 864 a^{4} b^{4} c d^{2} g^{5} + 864 a^{3} b^{5} c^{2} d g^{5} - 288 a^{2} b^{6} c^{3} g^{5}\right ) + x \left (192 a^{6} b^{2} d^{3} g^{5} - 576 a^{5} b^{3} c d^{2} g^{5} + 576 a^{4} b^{4} c^{2} d g^{5} - 192 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)/(d*x+c)))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(a + b*x)/(c + d*x))/(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))/(4*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))/(4*b*g**5*(a*d - b*c)**4) + (-12*A*a**3*d**3 + 36*A*a**2*b*
c*d**2 - 36*A*a*b**2*c**2*d + 12*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))/(48*a**7*b*d**3*g**5 - 144*a**6*b**2*c*d**2*g**5 + 144*a**5*b**3*c**2*d*g**5 -
48*a**4*b**4*c**3*g**5 + x**4*(48*a**3*b**5*d**3*g**5 - 144*a**2*b**6*c*d**2*g**5 + 144*a*b**7*c**2*d*g**5 - 4
8*b**8*c**3*g**5) + x**3*(192*a**4*b**4*d**3*g**5 - 576*a**3*b**5*c*d**2*g**5 + 576*a**2*b**6*c**2*d*g**5 - 19
2*a*b**7*c**3*g**5) + x**2*(288*a**5*b**3*d**3*g**5 - 864*a**4*b**4*c*d**2*g**5 + 864*a**3*b**5*c**2*d*g**5 -
288*a**2*b**6*c**3*g**5) + x*(192*a**6*b**2*d**3*g**5 - 576*a**5*b**3*c*d**2*g**5 + 576*a**4*b**4*c**2*d*g**5
- 192*a**3*b**5*c**3*g**5))

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