3.29 \(\int (c g+d g x)^4 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\)

Optimal. Leaf size=188 \[ \frac {g^4 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d}-\frac {B g^4 n (b c-a d)^5 \log (a+b x)}{5 b^5 d}-\frac {B g^4 n x (b c-a d)^4}{5 b^4}-\frac {B g^4 n (c+d x)^2 (b c-a d)^3}{10 b^3 d}-\frac {B g^4 n (c+d x)^3 (b c-a d)^2}{15 b^2 d}-\frac {B g^4 n (c+d x)^4 (b c-a d)}{20 b d} \]

[Out]

-1/5*B*(-a*d+b*c)^4*g^4*n*x/b^4-1/10*B*(-a*d+b*c)^3*g^4*n*(d*x+c)^2/b^3/d-1/15*B*(-a*d+b*c)^2*g^4*n*(d*x+c)^3/
b^2/d-1/20*B*(-a*d+b*c)*g^4*n*(d*x+c)^4/b/d-1/5*B*(-a*d+b*c)^5*g^4*n*ln(b*x+a)/b^5/d+1/5*g^4*(d*x+c)^5*(A+B*ln
(e*((b*x+a)/(d*x+c))^n))/d

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Rubi [A]  time = 0.13, antiderivative size = 188, 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, 43} \[ \frac {g^4 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d}-\frac {B g^4 n x (b c-a d)^4}{5 b^4}-\frac {B g^4 n (c+d x)^2 (b c-a d)^3}{10 b^3 d}-\frac {B g^4 n (c+d x)^3 (b c-a d)^2}{15 b^2 d}-\frac {B g^4 n (b c-a d)^5 \log (a+b x)}{5 b^5 d}-\frac {B g^4 n (c+d x)^4 (b c-a d)}{20 b d} \]

Antiderivative was successfully verified.

[In]

Int[(c*g + d*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

-(B*(b*c - a*d)^4*g^4*n*x)/(5*b^4) - (B*(b*c - a*d)^3*g^4*n*(c + d*x)^2)/(10*b^3*d) - (B*(b*c - a*d)^2*g^4*n*(
c + d*x)^3)/(15*b^2*d) - (B*(b*c - a*d)*g^4*n*(c + d*x)^4)/(20*b*d) - (B*(b*c - a*d)^5*g^4*n*Log[a + b*x])/(5*
b^5*d) + (g^4*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

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 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 (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac {g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac {(B n) \int \frac {(b c-a d) g^5 (c+d x)^4}{a+b x} \, dx}{5 d g}\\ &=\frac {g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac {\left (B (b c-a d) g^4 n\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d}\\ &=\frac {g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac {\left (B (b c-a d) g^4 n\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{5 d}\\ &=-\frac {B (b c-a d)^4 g^4 n x}{5 b^4}-\frac {B (b c-a d)^3 g^4 n (c+d x)^2}{10 b^3 d}-\frac {B (b c-a d)^2 g^4 n (c+d x)^3}{15 b^2 d}-\frac {B (b c-a d) g^4 n (c+d x)^4}{20 b d}-\frac {B (b c-a d)^5 g^4 n \log (a+b x)}{5 b^5 d}+\frac {g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 146, normalized size = 0.78 \[ \frac {g^4 \left ((c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {B n (b c-a d) \left (4 b^3 (c+d x)^3 (b c-a d)+6 b^2 (c+d x)^2 (b c-a d)^2+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{12 b^5}\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*g + d*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

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fricas [B]  time = 1.04, size = 572, normalized size = 3.04 \[ \frac {12 \, A b^{5} d^{5} g^{4} x^{5} - 12 \, B b^{5} c^{5} g^{4} n \log \left (d x + c\right ) + 12 \, {\left (5 \, B a b^{4} c^{4} d - 10 \, B a^{2} b^{3} c^{3} d^{2} + 10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{4} n \log \left (b x + a\right ) + 3 \, {\left (20 \, A b^{5} c d^{4} g^{4} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \, {\left (30 \, A b^{5} c^{2} d^{3} g^{4} - {\left (4 \, B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{4} n\right )} x^{3} + 6 \, {\left (20 \, A b^{5} c^{3} d^{2} g^{4} - {\left (6 \, B b^{5} c^{3} d^{2} - 10 \, B a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{4} n\right )} x^{2} + 12 \, {\left (5 \, A b^{5} c^{4} d g^{4} - {\left (4 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} + B a^{4} b d^{5}\right )} g^{4} n\right )} x + 12 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B b^{5} c d^{4} g^{4} x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} x^{2} + 5 \, B b^{5} c^{4} d g^{4} x\right )} \log \relax (e) + 12 \, {\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B b^{5} c d^{4} g^{4} n x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} n x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} n x^{2} + 5 \, B b^{5} c^{4} d g^{4} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{60 \, b^{5} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*g*x+c*g)^4*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/60*(12*A*b^5*d^5*g^4*x^5 - 12*B*b^5*c^5*g^4*n*log(d*x + c) + 12*(5*B*a*b^4*c^4*d - 10*B*a^2*b^3*c^3*d^2 + 10
*B*a^3*b^2*c^2*d^3 - 5*B*a^4*b*c*d^4 + B*a^5*d^5)*g^4*n*log(b*x + a) + 3*(20*A*b^5*c*d^4*g^4 - (B*b^5*c*d^4 -
B*a*b^4*d^5)*g^4*n)*x^4 + 4*(30*A*b^5*c^2*d^3*g^4 - (4*B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^4*n)
*x^3 + 6*(20*A*b^5*c^3*d^2*g^4 - (6*B*b^5*c^3*d^2 - 10*B*a*b^4*c^2*d^3 + 5*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g^
4*n)*x^2 + 12*(5*A*b^5*c^4*d*g^4 - (4*B*b^5*c^4*d - 10*B*a*b^4*c^3*d^2 + 10*B*a^2*b^3*c^2*d^3 - 5*B*a^3*b^2*c*
d^4 + B*a^4*b*d^5)*g^4*n)*x + 12*(B*b^5*d^5*g^4*x^5 + 5*B*b^5*c*d^4*g^4*x^4 + 10*B*b^5*c^2*d^3*g^4*x^3 + 10*B*
b^5*c^3*d^2*g^4*x^2 + 5*B*b^5*c^4*d*g^4*x)*log(e) + 12*(B*b^5*d^5*g^4*n*x^5 + 5*B*b^5*c*d^4*g^4*n*x^4 + 10*B*b
^5*c^2*d^3*g^4*n*x^3 + 10*B*b^5*c^3*d^2*g^4*n*x^2 + 5*B*b^5*c^4*d*g^4*n*x)*log((b*x + a)/(d*x + c)))/(b^5*d)

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giac [B]  time = 5.99, size = 1862, normalized size = 9.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*g*x+c*g)^4*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

1/60*(12*(B*b^6*c^6*g^4*n - 6*B*a*b^5*c^5*d*g^4*n + 15*B*a^2*b^4*c^4*d^2*g^4*n - 20*B*a^3*b^3*c^3*d^3*g^4*n +
15*B*a^4*b^2*c^2*d^4*g^4*n - 6*B*a^5*b*c*d^5*g^4*n + B*a^6*d^6*g^4*n)*log((b*x + a)/(d*x + c))/(b^5*d - 5*(b*x
+ a)*b^4*d^2/(d*x + c) + 10*(b*x + a)^2*b^3*d^3/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^4/(d*x + c)^3 + 5*(b*x + a
)^4*b*d^5/(d*x + c)^4 - (b*x + a)^5*d^6/(d*x + c)^5) - (25*B*b^10*c^6*g^4*n - 150*B*a*b^9*c^5*d*g^4*n - 77*(b*
x + a)*B*b^9*c^6*d*g^4*n/(d*x + c) + 375*B*a^2*b^8*c^4*d^2*g^4*n + 462*(b*x + a)*B*a*b^8*c^5*d^2*g^4*n/(d*x +
c) + 94*(b*x + a)^2*B*b^8*c^6*d^2*g^4*n/(d*x + c)^2 - 500*B*a^3*b^7*c^3*d^3*g^4*n - 1155*(b*x + a)*B*a^2*b^7*c
^4*d^3*g^4*n/(d*x + c) - 564*(b*x + a)^2*B*a*b^7*c^5*d^3*g^4*n/(d*x + c)^2 - 54*(b*x + a)^3*B*b^7*c^6*d^3*g^4*
n/(d*x + c)^3 + 375*B*a^4*b^6*c^2*d^4*g^4*n + 1540*(b*x + a)*B*a^3*b^6*c^3*d^4*g^4*n/(d*x + c) + 1410*(b*x + a
)^2*B*a^2*b^6*c^4*d^4*g^4*n/(d*x + c)^2 + 324*(b*x + a)^3*B*a*b^6*c^5*d^4*g^4*n/(d*x + c)^3 + 12*(b*x + a)^4*B
*b^6*c^6*d^4*g^4*n/(d*x + c)^4 - 150*B*a^5*b^5*c*d^5*g^4*n - 1155*(b*x + a)*B*a^4*b^5*c^2*d^5*g^4*n/(d*x + c)
- 1880*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g^4*n/(d*x + c)^2 - 810*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g^4*n/(d*x + c)^3 -
72*(b*x + a)^4*B*a*b^5*c^5*d^5*g^4*n/(d*x + c)^4 + 25*B*a^6*b^4*d^6*g^4*n + 462*(b*x + a)*B*a^5*b^4*c*d^6*g^4
*n/(d*x + c) + 1410*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^4*n/(d*x + c)^2 + 1080*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^4*n
/(d*x + c)^3 + 180*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g^4*n/(d*x + c)^4 - 77*(b*x + a)*B*a^6*b^3*d^7*g^4*n/(d*x + c
) - 564*(b*x + a)^2*B*a^5*b^3*c*d^7*g^4*n/(d*x + c)^2 - 810*(b*x + a)^3*B*a^4*b^3*c^2*d^7*g^4*n/(d*x + c)^3 -
240*(b*x + a)^4*B*a^3*b^3*c^3*d^7*g^4*n/(d*x + c)^4 + 94*(b*x + a)^2*B*a^6*b^2*d^8*g^4*n/(d*x + c)^2 + 324*(b*
x + a)^3*B*a^5*b^2*c*d^8*g^4*n/(d*x + c)^3 + 180*(b*x + a)^4*B*a^4*b^2*c^2*d^8*g^4*n/(d*x + c)^4 - 54*(b*x + a
)^3*B*a^6*b*d^9*g^4*n/(d*x + c)^3 - 72*(b*x + a)^4*B*a^5*b*c*d^9*g^4*n/(d*x + c)^4 + 12*(b*x + a)^4*B*a^6*d^10
*g^4*n/(d*x + c)^4 - 12*A*b^10*c^6*g^4 - 12*B*b^10*c^6*g^4 + 72*A*a*b^9*c^5*d*g^4 + 72*B*a*b^9*c^5*d*g^4 - 180
*A*a^2*b^8*c^4*d^2*g^4 - 180*B*a^2*b^8*c^4*d^2*g^4 + 240*A*a^3*b^7*c^3*d^3*g^4 + 240*B*a^3*b^7*c^3*d^3*g^4 - 1
80*A*a^4*b^6*c^2*d^4*g^4 - 180*B*a^4*b^6*c^2*d^4*g^4 + 72*A*a^5*b^5*c*d^5*g^4 + 72*B*a^5*b^5*c*d^5*g^4 - 12*A*
a^6*b^4*d^6*g^4 - 12*B*a^6*b^4*d^6*g^4)/(b^9*d - 5*(b*x + a)*b^8*d^2/(d*x + c) + 10*(b*x + a)^2*b^7*d^3/(d*x +
c)^2 - 10*(b*x + a)^3*b^6*d^4/(d*x + c)^3 + 5*(b*x + a)^4*b^5*d^5/(d*x + c)^4 - (b*x + a)^5*b^4*d^6/(d*x + c)
^5) + 12*(B*b^6*c^6*g^4*n - 6*B*a*b^5*c^5*d*g^4*n + 15*B*a^2*b^4*c^4*d^2*g^4*n - 20*B*a^3*b^3*c^3*d^3*g^4*n +
15*B*a^4*b^2*c^2*d^4*g^4*n - 6*B*a^5*b*c*d^5*g^4*n + B*a^6*d^6*g^4*n)*log(b - (b*x + a)*d/(d*x + c))/(b^5*d) -
12*(B*b^6*c^6*g^4*n - 6*B*a*b^5*c^5*d*g^4*n + 15*B*a^2*b^4*c^4*d^2*g^4*n - 20*B*a^3*b^3*c^3*d^3*g^4*n + 15*B*
a^4*b^2*c^2*d^4*g^4*n - 6*B*a^5*b*c*d^5*g^4*n + B*a^6*d^6*g^4*n)*log((b*x + a)/(d*x + c))/(b^5*d))*(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 \[ \int \left (d g x +c g \right )^{4} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*g*x+c*g)^4*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

[Out]

int((d*g*x+c*g)^4*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [B]  time = 1.38, size = 676, normalized size = 3.60 \[ \frac {1}{5} \, B d^{4} g^{4} x^{5} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{5} \, A d^{4} g^{4} x^{5} + B c d^{3} g^{4} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c d^{3} g^{4} x^{4} + 2 \, B c^{2} d^{2} g^{4} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{2} d^{2} g^{4} x^{3} + 2 \, B c^{3} d g^{4} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{3} d g^{4} x^{2} + \frac {1}{60} \, B d^{4} g^{4} n {\left (\frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} - \frac {1}{6} \, B c d^{3} g^{4} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + B c^{2} d^{2} g^{4} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - 2 \, B c^{3} d g^{4} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B c^{4} g^{4} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B c^{4} g^{4} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c^{4} g^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*g*x+c*g)^4*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/5*B*d^4*g^4*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*d^4*g^4*x^5 + B*c*d^3*g^4*x^4*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n) + A*c*d^3*g^4*x^4 + 2*B*c^2*d^2*g^4*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*c
^2*d^2*g^4*x^3 + 2*B*c^3*d*g^4*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*c^3*d*g^4*x^2 + 1/60*B*d^4*g^4
*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*
b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/6*B*c*d^3*g^4*n*(6*a^4
*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 +
6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) + B*c^2*d^2*g^4*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2
*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) - 2*B*c^3*d*g^4*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*
x + c)/d^2 + (b*c - a*d)*x/(b*d)) + B*c^4*g^4*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*c^4*g^4*x*log(e*(b*x
/(d*x + c) + a/(d*x + c))^n) + A*c^4*g^4*x

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mupad [B]  time = 4.48, size = 1045, normalized size = 5.56 \[ x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {c\,d^2\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^3\,g^4}{b}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )}{2\,b\,d}+\frac {c^2\,d\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}\right )-x^3\,\left (\frac {\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {c\,d^2\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,b}+\frac {A\,a\,c\,d^3\,g^4}{3\,b}\right )+x^4\,\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{20\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,b}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,c^4\,g^4\,x+2\,B\,c^3\,d\,g^4\,x^2+2\,B\,c^2\,d^2\,g^4\,x^3+B\,c\,d^3\,g^4\,x^4+\frac {B\,d^4\,g^4\,x^5}{5}\right )+x\,\left (\frac {c^3\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+2\,B\,a\,d\,n-2\,B\,b\,c\,n\right )}{b}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {c\,d^2\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^3\,g^4}{b}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )}{b\,d}+\frac {2\,c^2\,d\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {d^3\,g^4\,\left (5\,A\,a\,d+25\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,b}-\frac {A\,d^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {c\,d^2\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^3\,g^4}{b}\right )}{b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (\frac {B\,n\,a^5\,d^4\,g^4}{5}-B\,n\,a^4\,b\,c\,d^3\,g^4+2\,B\,n\,a^3\,b^2\,c^2\,d^2\,g^4-2\,B\,n\,a^2\,b^3\,c^3\,d\,g^4+B\,n\,a\,b^4\,c^4\,g^4\right )}{b^5}+\frac {A\,d^4\,g^4\,x^5}{5}-\frac {B\,c^5\,g^4\,n\,\ln \left (c+d\,x\right )}{5\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*g + d*g*x)^4*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

x^2*(((5*a*d + 5*b*c)*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c)
)/(5*b))*(5*a*d + 5*b*c))/(5*b*d) - (c*d^2*g^4*(5*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d^3*g^4)/b
))/(10*b*d) - (a*c*((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*
b)))/(2*b*d) + (c^2*d*g^4*(5*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/b) - x^3*((((d^3*g^4*(5*A*a*d + 25*A*b*c +
B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b))*(5*a*d + 5*b*c))/(15*b*d) - (c*d^2*g^4*(5*A*a*d
+ 10*A*b*c + B*a*d*n - B*b*c*n))/(3*b) + (A*a*c*d^3*g^4)/(3*b)) + x^4*((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n
- B*b*c*n))/(20*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(20*b)) + log(e*((a + b*x)/(c + d*x))^n)*((B*d^4*g^4*x^5)/5
+ B*c^4*g^4*x + 2*B*c^3*d*g^4*x^2 + B*c*d^3*g^4*x^4 + 2*B*c^2*d^2*g^4*x^3) + x*((c^3*g^4*(10*A*a*d + 5*A*b*c +
2*B*a*d*n - 2*B*b*c*n))/b - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*
b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b))*(5*a*d + 5*b*c))/(5*b*d) - (c*d^2*g^4*(5*A*a*d + 10*A*b*c +
B*a*d*n - B*b*c*n))/b + (A*a*c*d^3*g^4)/b))/(5*b*d) - (a*c*((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n)
)/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b)))/(b*d) + (2*c^2*d*g^4*(5*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/b)
)/(5*b*d) + (a*c*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*
b))*(5*a*d + 5*b*c))/(5*b*d) - (c*d^2*g^4*(5*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d^3*g^4)/b))/(b
*d)) + (log(a + b*x)*((B*a^5*d^4*g^4*n)/5 + B*a*b^4*c^4*g^4*n - B*a^4*b*c*d^3*g^4*n - 2*B*a^2*b^3*c^3*d*g^4*n
+ 2*B*a^3*b^2*c^2*d^2*g^4*n))/b^5 + (A*d^4*g^4*x^5)/5 - (B*c^5*g^4*n*log(c + d*x))/(5*d)

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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((d*g*x+c*g)**4*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

Timed out

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