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

Optimal. Leaf size=215 \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d g^5 (c+d x)^4}+\frac {b^4 B n \log (a+b x)}{4 d g^5 (b c-a d)^4}-\frac {b^4 B n \log (c+d x)}{4 d g^5 (b c-a d)^4}+\frac {b^3 B n}{4 d g^5 (c+d x) (b c-a d)^3}+\frac {b^2 B n}{8 d g^5 (c+d x)^2 (b c-a d)^2}+\frac {b B n}{12 d g^5 (c+d x)^3 (b c-a d)}+\frac {B n}{16 d g^5 (c+d x)^4} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d g^5 (c+d x)^4}+\frac {b^3 B n}{4 d g^5 (c+d x) (b c-a d)^3}+\frac {b^2 B n}{8 d g^5 (c+d x)^2 (b c-a d)^2}+\frac {b^4 B n \log (a+b x)}{4 d g^5 (b c-a d)^4}-\frac {b^4 B n \log (c+d x)}{4 d g^5 (b c-a d)^4}+\frac {b B n}{12 d g^5 (c+d x)^3 (b c-a d)}+\frac {B n}{16 d g^5 (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*n)/(16*d*g^5*(c + d*x)^4) + (b*B*n)/(12*d*(b*c - a*d)*g^5*(c + d*x)^3) + (b^2*B*n)/(8*d*(b*c - a*d)^2*g^5*(
c + d*x)^2) + (b^3*B*n)/(4*d*(b*c - a*d)^3*g^5*(c + d*x)) + (b^4*B*n*Log[a + b*x])/(4*d*(b*c - a*d)^4*g^5) - (
A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*d*g^5*(c + d*x)^4) - (b^4*B*n*Log[c + d*x])/(4*d*(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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B n) \int \frac {b c-a d}{g^4 (a+b x) (c+d x)^5} \, dx}{4 d g}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x)^5} \, dx}{4 d g^5}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b^5}{(b c-a d)^5 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^5}-\frac {b d}{(b c-a d)^2 (c+d x)^4}-\frac {b^2 d}{(b c-a d)^3 (c+d x)^3}-\frac {b^3 d}{(b c-a d)^4 (c+d x)^2}-\frac {b^4 d}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 d g^5}\\ &=\frac {B n}{16 d g^5 (c+d x)^4}+\frac {b B n}{12 d (b c-a d) g^5 (c+d x)^3}+\frac {b^2 B n}{8 d (b c-a d)^2 g^5 (c+d x)^2}+\frac {b^3 B n}{4 d (b c-a d)^3 g^5 (c+d x)}+\frac {b^4 B n \log (a+b x)}{4 d (b c-a d)^4 g^5}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}-\frac {b^4 B n \log (c+d x)}{4 d (b c-a d)^4 g^5}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 162, normalized size = 0.75 \[ \frac {\frac {B n \left (12 b^4 \log (a+b x)+\frac {12 b^3 (b c-a d)}{c+d x}+\frac {6 b^2 (b c-a d)^2}{(c+d x)^2}+\frac {4 b (b c-a d)^3}{(c+d x)^3}+\frac {3 (b c-a d)^4}{(c+d x)^4}-12 b^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(c+d x)^4}}{4 d g^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.09, size = 735, normalized size = 3.42 \[ -\frac {12 \, A b^{4} c^{4} - 48 \, A a b^{3} c^{3} d + 72 \, A a^{2} b^{2} c^{2} d^{2} - 48 \, A a^{3} b c d^{3} + 12 \, A a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n x^{3} - 6 \, {\left (7 \, B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} n x^{2} - 4 \, {\left (13 \, B b^{4} c^{3} d - 18 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} n x - {\left (25 \, B b^{4} c^{4} - 48 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 16 \, B a^{3} b c d^{3} + 3 \, B a^{4} d^{4}\right )} n + 12 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} \log \relax (e) - 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B b^{4} c d^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} n x^{2} + 4 \, B b^{4} c^{3} d n x + {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{48 \, {\left ({\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} g^{5} x^{4} + 4 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} g^{5} x^{3} + 6 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} g^{5} x^{2} + 4 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} g^{5} x + {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(12*A*b^4*c^4 - 48*A*a*b^3*c^3*d + 72*A*a^2*b^2*c^2*d^2 - 48*A*a^3*b*c*d^3 + 12*A*a^4*d^4 - 12*(B*b^4*c*
d^3 - B*a*b^3*d^4)*n*x^3 - 6*(7*B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*n*x^2 - 4*(13*B*b^4*c^3*d - 1
8*B*a*b^3*c^2*d^2 + 6*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*n*x - (25*B*b^4*c^4 - 48*B*a*b^3*c^3*d + 36*B*a^2*b^2*c^2
*d^2 - 16*B*a^3*b*c*d^3 + 3*B*a^4*d^4)*n + 12*(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 + B*a^4*d^4)*log(e) - 12*(B*b^4*d^4*n*x^4 + 4*B*b^4*c*d^3*n*x^3 + 6*B*b^4*c^2*d^2*n*x^2 + 4*B*b^4*c^3*d*n
*x + (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 - B*a^4*d^4)*n)*log((b*x + a)/(d*x + c)))/((b^4*
c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7 - 4*a^3*b*c*d^8 + a^4*d^9)*g^5*x^4 + 4*(b^4*c^5*d^4 - 4*a*b^3*c^
4*d^5 + 6*a^2*b^2*c^3*d^6 - 4*a^3*b*c^2*d^7 + a^4*c*d^8)*g^5*x^3 + 6*(b^4*c^6*d^3 - 4*a*b^3*c^5*d^4 + 6*a^2*b^
2*c^4*d^5 - 4*a^3*b*c^3*d^6 + a^4*c^2*d^7)*g^5*x^2 + 4*(b^4*c^7*d^2 - 4*a*b^3*c^6*d^3 + 6*a^2*b^2*c^5*d^4 - 4*
a^3*b*c^4*d^5 + a^4*c^3*d^6)*g^5*x + (b^4*c^8*d - 4*a*b^3*c^7*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5*d^4 + a^4*
c^4*d^5)*g^5)

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giac [B]  time = 11.42, size = 676, normalized size = 3.14 \[ \frac {1}{48} \, {\left (12 \, {\left (\frac {4 \, {\left (b x + a\right )} B b^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}} - \frac {6 \, {\left (b x + a\right )}^{2} B b^{2} d n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (b x + a\right )}^{3} B b d^{2} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{4} B d^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {3 \, {\left (B d^{3} n - 4 \, A d^{3} - 4 \, B d^{3}\right )} {\left (b x + a\right )}^{4}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}} - \frac {16 \, {\left (B b d^{2} n - 3 \, A b d^{2} - 3 \, B b d^{2}\right )} {\left (b x + a\right )}^{3}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} + \frac {36 \, {\left (B b^{2} d n - 2 \, A b^{2} d - 2 \, B b^{2} d\right )} {\left (b x + a\right )}^{2}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} - \frac {48 \, {\left (B b^{3} n - A b^{3} - B b^{3}\right )} {\left (b x + a\right )}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (d g x +c g \right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.06, size = 652, normalized size = 3.03 \[ \frac {1}{48} \, B n {\left (\frac {12 \, b^{3} d^{3} x^{3} + 25 \, b^{3} c^{3} - 23 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3} + 6 \, {\left (7 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (13 \, b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{{\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} g^{5} x^{4} + 4 \, {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} g^{5} x^{3} + 6 \, {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} g^{5} x^{2} + 4 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} g^{5} x + {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} g^{5}} + \frac {12 \, b^{4} \log \left (b x + a\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}} - \frac {12 \, b^{4} \log \left (d x + c\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} - \frac {A}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*B*n*((12*b^3*d^3*x^3 + 25*b^3*c^3 - 23*a*b^2*c^2*d + 13*a^2*b*c*d^2 - 3*a^3*d^3 + 6*(7*b^3*c*d^2 - a*b^2*
d^3)*x^2 + 4*(13*b^3*c^2*d - 5*a*b^2*c*d^2 + a^2*b*d^3)*x)/((b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a
^3*d^8)*g^5*x^4 + 4*(b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*g^5*x^3 + 6*(b^3*c^5*d^3 - 3
*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3*c^2*d^6)*g^5*x^2 + 4*(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 -
a^3*c^3*d^5)*g^5*x + (b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^4)*g^5) + 12*b^4*log(b*x + a)
/((b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*g^5) - 12*b^4*log(d*x + c)/((b^4
*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*g^5)) - 1/4*B*log(e*(b*x/(d*x + c) + a
/(d*x + c))^n)/(d^5*g^5*x^4 + 4*c*d^4*g^5*x^3 + 6*c^2*d^3*g^5*x^2 + 4*c^3*d^2*g^5*x + c^4*d*g^5) - 1/4*A/(d^5*
g^5*x^4 + 4*c*d^4*g^5*x^3 + 6*c^2*d^3*g^5*x^2 + 4*c^3*d^2*g^5*x + c^4*d*g^5)

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mupad [B]  time = 4.99, size = 603, normalized size = 2.80 \[ \frac {B\,b^4\,n\,\mathrm {atanh}\left (\frac {4\,a^4\,d^5\,g^5-8\,a^3\,b\,c\,d^4\,g^5+8\,a\,b^3\,c^3\,d^2\,g^5-4\,b^4\,c^4\,d\,g^5}{4\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{4\,d\,\left (c^4\,g^5+4\,c^3\,d\,g^5\,x+6\,c^2\,d^2\,g^5\,x^2+4\,c\,d^3\,g^5\,x^3+d^4\,g^5\,x^4\right )}-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3-3\,B\,a^3\,d^3\,n+25\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2-23\,B\,a\,b^2\,c^2\,d\,n+13\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,x\,\left (B\,n\,a^2\,d^3-5\,B\,n\,a\,b\,c\,d^2+13\,B\,n\,b^2\,c^2\,d\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {b^2\,x^2\,\left (B\,a\,d^3\,n-7\,B\,b\,c\,d^2\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,c^4\,d\,g^5+16\,c^3\,d^2\,g^5\,x+24\,c^2\,d^3\,g^5\,x^2+16\,c\,d^4\,g^5\,x^3+4\,d^5\,g^5\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b^4*n*atanh((4*a^4*d^5*g^5 - 4*b^4*c^4*d*g^5 - 8*a^3*b*c*d^4*g^5 + 8*a*b^3*c^3*d^2*g^5)/(4*d*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*d*g^5*(a*d - b*c)^4) -
(B*log(e*((a + b*x)/(c + d*x))^n))/(4*d*(c^4*g^5 + d^4*g^5*x^4 + 4*c*d^3*g^5*x^3 + 6*c^2*d^2*g^5*x^2 + 4*c^3*
d*g^5*x)) - ((12*A*a^3*d^3 - 12*A*b^3*c^3 - 3*B*a^3*d^3*n + 25*B*b^3*c^3*n + 36*A*a*b^2*c^2*d - 36*A*a^2*b*c*d
^2 - 23*B*a*b^2*c^2*d*n + 13*B*a^2*b*c*d^2*n)/(12*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*x*
(B*a^2*d^3*n + 13*B*b^2*c^2*d*n - 5*B*a*b*c*d^2*n))/(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) -
(b^2*x^2*(B*a*d^3*n - 7*B*b*c*d^2*n))/(2*(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*n*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*c^4*d*g^5 + 4*d^5*g^5*x^4 + 16*c^3*d^2*g^5*x + 16*
c*d^4*g^5*x^3 + 24*c^2*d^3*g^5*x^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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