3.2 \(\int (a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\)

Optimal. Leaf size=156 \[ \frac {g^3 (a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b}+\frac {B g^3 n (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B g^3 n x (b c-a d)^3}{4 d^3}+\frac {B g^3 n (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac {B g^3 n (a+b x)^3 (b c-a d)}{12 b d} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 156, 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} \[ \frac {g^3 (a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b}-\frac {B g^3 n x (b c-a d)^3}{4 d^3}+\frac {B g^3 n (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac {B g^3 n (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B g^3 n (a+b x)^3 (b c-a d)}{12 b d} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

-(B*(b*c - a*d)^3*g^3*n*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*n*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*n*(a +
b*x)^3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b) + (B*(b*c - a*d)^4*g^3*n*Lo
g[c + d*x])/(4*b*d^4)

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

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Mathematica [A]  time = 0.11, size = 124, normalized size = 0.79 \[ \frac {g^3 \left ((a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {B n (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{6 d^4}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

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fricas [B]  time = 1.13, size = 426, normalized size = 2.73 \[ \frac {6 \, A b^{4} d^{4} g^{3} x^{4} + 6 \, B a^{4} d^{4} g^{3} n \log \left (b x + a\right ) + 6 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} n \log \left (d x + c\right ) + 2 \, {\left (12 \, A a b^{3} d^{4} g^{3} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3} n\right )} x^{3} + 3 \, {\left (12 \, A a^{2} b^{2} d^{4} g^{3} + {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \, B a^{2} b^{2} d^{4}\right )} g^{3} n\right )} x^{2} + 6 \, {\left (4 \, A a^{3} b d^{4} g^{3} - {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - 3 \, B a^{3} b d^{4}\right )} g^{3} n\right )} x + 6 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \relax (e) + 6 \, {\left (B b^{4} d^{4} g^{3} n x^{4} + 4 \, B a b^{3} d^{4} g^{3} n x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} n x^{2} + 4 \, B a^{3} b d^{4} g^{3} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/24*(6*A*b^4*d^4*g^3*x^4 + 6*B*a^4*d^4*g^3*n*log(b*x + a) + 6*(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)*g^3*n*log(d*x + c) + 2*(12*A*a*b^3*d^4*g^3 - (B*b^4*c*d^3 - B*a*b^3*d^4)*g^3*n)*x^3 + 3
*(12*A*a^2*b^2*d^4*g^3 + (B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 + 3*B*a^2*b^2*d^4)*g^3*n)*x^2 + 6*(4*A*a^3*b*d^4*g^3
- (B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 6*B*a^2*b^2*c*d^3 - 3*B*a^3*b*d^4)*g^3*n)*x + 6*(B*b^4*d^4*g^3*x^4 + 4*B
*a*b^3*d^4*g^3*x^3 + 6*B*a^2*b^2*d^4*g^3*x^2 + 4*B*a^3*b*d^4*g^3*x)*log(e) + 6*(B*b^4*d^4*g^3*n*x^4 + 4*B*a*b^
3*d^4*g^3*n*x^3 + 6*B*a^2*b^2*d^4*g^3*n*x^2 + 4*B*a^3*b*d^4*g^3*n*x)*log((b*x + a)/(d*x + c)))/(b*d^4)

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giac [B]  time = 4.41, size = 2986, normalized size = 19.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

-1/24*(6*(B*b^8*c^5*g^3*n - 5*B*a*b^7*c^4*d*g^3*n - 4*(b*x + a)*B*b^7*c^5*d*g^3*n/(d*x + c) + 10*B*a^2*b^6*c^3
*d^2*g^3*n + 20*(b*x + a)*B*a*b^6*c^4*d^2*g^3*n/(d*x + c) + 6*(b*x + a)^2*B*b^6*c^5*d^2*g^3*n/(d*x + c)^2 - 10
*B*a^3*b^5*c^2*d^3*g^3*n - 40*(b*x + a)*B*a^2*b^5*c^3*d^3*g^3*n/(d*x + c) - 30*(b*x + a)^2*B*a*b^5*c^4*d^3*g^3
*n/(d*x + c)^2 - 4*(b*x + a)^3*B*b^5*c^5*d^3*g^3*n/(d*x + c)^3 + 5*B*a^4*b^4*c*d^4*g^3*n + 40*(b*x + a)*B*a^3*
b^4*c^2*d^4*g^3*n/(d*x + c) + 60*(b*x + a)^2*B*a^2*b^4*c^3*d^4*g^3*n/(d*x + c)^2 + 20*(b*x + a)^3*B*a*b^4*c^4*
d^4*g^3*n/(d*x + c)^3 - B*a^5*b^3*d^5*g^3*n - 20*(b*x + a)*B*a^4*b^3*c*d^5*g^3*n/(d*x + c) - 60*(b*x + a)^2*B*
a^3*b^3*c^2*d^5*g^3*n/(d*x + c)^2 - 40*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g^3*n/(d*x + c)^3 + 4*(b*x + a)*B*a^5*b^2
*d^6*g^3*n/(d*x + c) + 30*(b*x + a)^2*B*a^4*b^2*c*d^6*g^3*n/(d*x + c)^2 + 40*(b*x + a)^3*B*a^3*b^2*c^2*d^6*g^3
*n/(d*x + c)^3 - 6*(b*x + a)^2*B*a^5*b*d^7*g^3*n/(d*x + c)^2 - 20*(b*x + a)^3*B*a^4*b*c*d^7*g^3*n/(d*x + c)^3
+ 4*(b*x + a)^3*B*a^5*d^8*g^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b^4*d^4 - 4*(b*x + a)*b^3*d^5/(d*x + c)
+ 6*(b*x + a)^2*b^2*d^6/(d*x + c)^2 - 4*(b*x + a)^3*b*d^7/(d*x + c)^3 + (b*x + a)^4*d^8/(d*x + c)^4) + (11*B*
b^8*c^5*g^3*n - 55*B*a*b^7*c^4*d*g^3*n - 38*(b*x + a)*B*b^7*c^5*d*g^3*n/(d*x + c) + 110*B*a^2*b^6*c^3*d^2*g^3*
n + 190*(b*x + a)*B*a*b^6*c^4*d^2*g^3*n/(d*x + c) + 45*(b*x + a)^2*B*b^6*c^5*d^2*g^3*n/(d*x + c)^2 - 110*B*a^3
*b^5*c^2*d^3*g^3*n - 380*(b*x + a)*B*a^2*b^5*c^3*d^3*g^3*n/(d*x + c) - 225*(b*x + a)^2*B*a*b^5*c^4*d^3*g^3*n/(
d*x + c)^2 - 18*(b*x + a)^3*B*b^5*c^5*d^3*g^3*n/(d*x + c)^3 + 55*B*a^4*b^4*c*d^4*g^3*n + 380*(b*x + a)*B*a^3*b
^4*c^2*d^4*g^3*n/(d*x + c) + 450*(b*x + a)^2*B*a^2*b^4*c^3*d^4*g^3*n/(d*x + c)^2 + 90*(b*x + a)^3*B*a*b^4*c^4*
d^4*g^3*n/(d*x + c)^3 - 11*B*a^5*b^3*d^5*g^3*n - 190*(b*x + a)*B*a^4*b^3*c*d^5*g^3*n/(d*x + c) - 450*(b*x + a)
^2*B*a^3*b^3*c^2*d^5*g^3*n/(d*x + c)^2 - 180*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g^3*n/(d*x + c)^3 + 38*(b*x + a)*B*
a^5*b^2*d^6*g^3*n/(d*x + c) + 225*(b*x + a)^2*B*a^4*b^2*c*d^6*g^3*n/(d*x + c)^2 + 180*(b*x + a)^3*B*a^3*b^2*c^
2*d^6*g^3*n/(d*x + c)^3 - 45*(b*x + a)^2*B*a^5*b*d^7*g^3*n/(d*x + c)^2 - 90*(b*x + a)^3*B*a^4*b*c*d^7*g^3*n/(d
*x + c)^3 + 18*(b*x + a)^3*B*a^5*d^8*g^3*n/(d*x + c)^3 + 6*A*b^8*c^5*g^3 + 6*B*b^8*c^5*g^3 - 30*A*a*b^7*c^4*d*
g^3 - 30*B*a*b^7*c^4*d*g^3 - 24*(b*x + a)*A*b^7*c^5*d*g^3/(d*x + c) - 24*(b*x + a)*B*b^7*c^5*d*g^3/(d*x + c) +
60*A*a^2*b^6*c^3*d^2*g^3 + 60*B*a^2*b^6*c^3*d^2*g^3 + 120*(b*x + a)*A*a*b^6*c^4*d^2*g^3/(d*x + c) + 120*(b*x
+ a)*B*a*b^6*c^4*d^2*g^3/(d*x + c) + 36*(b*x + a)^2*A*b^6*c^5*d^2*g^3/(d*x + c)^2 + 36*(b*x + a)^2*B*b^6*c^5*d
^2*g^3/(d*x + c)^2 - 60*A*a^3*b^5*c^2*d^3*g^3 - 60*B*a^3*b^5*c^2*d^3*g^3 - 240*(b*x + a)*A*a^2*b^5*c^3*d^3*g^3
/(d*x + c) - 240*(b*x + a)*B*a^2*b^5*c^3*d^3*g^3/(d*x + c) - 180*(b*x + a)^2*A*a*b^5*c^4*d^3*g^3/(d*x + c)^2 -
180*(b*x + a)^2*B*a*b^5*c^4*d^3*g^3/(d*x + c)^2 - 24*(b*x + a)^3*A*b^5*c^5*d^3*g^3/(d*x + c)^3 - 24*(b*x + a)
^3*B*b^5*c^5*d^3*g^3/(d*x + c)^3 + 30*A*a^4*b^4*c*d^4*g^3 + 30*B*a^4*b^4*c*d^4*g^3 + 240*(b*x + a)*A*a^3*b^4*c
^2*d^4*g^3/(d*x + c) + 240*(b*x + a)*B*a^3*b^4*c^2*d^4*g^3/(d*x + c) + 360*(b*x + a)^2*A*a^2*b^4*c^3*d^4*g^3/(
d*x + c)^2 + 360*(b*x + a)^2*B*a^2*b^4*c^3*d^4*g^3/(d*x + c)^2 + 120*(b*x + a)^3*A*a*b^4*c^4*d^4*g^3/(d*x + c)
^3 + 120*(b*x + a)^3*B*a*b^4*c^4*d^4*g^3/(d*x + c)^3 - 6*A*a^5*b^3*d^5*g^3 - 6*B*a^5*b^3*d^5*g^3 - 120*(b*x +
a)*A*a^4*b^3*c*d^5*g^3/(d*x + c) - 120*(b*x + a)*B*a^4*b^3*c*d^5*g^3/(d*x + c) - 360*(b*x + a)^2*A*a^3*b^3*c^2
*d^5*g^3/(d*x + c)^2 - 360*(b*x + a)^2*B*a^3*b^3*c^2*d^5*g^3/(d*x + c)^2 - 240*(b*x + a)^3*A*a^2*b^3*c^3*d^5*g
^3/(d*x + c)^3 - 240*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g^3/(d*x + c)^3 + 24*(b*x + a)*A*a^5*b^2*d^6*g^3/(d*x + c)
+ 24*(b*x + a)*B*a^5*b^2*d^6*g^3/(d*x + c) + 180*(b*x + a)^2*A*a^4*b^2*c*d^6*g^3/(d*x + c)^2 + 180*(b*x + a)^2
*B*a^4*b^2*c*d^6*g^3/(d*x + c)^2 + 240*(b*x + a)^3*A*a^3*b^2*c^2*d^6*g^3/(d*x + c)^3 + 240*(b*x + a)^3*B*a^3*b
^2*c^2*d^6*g^3/(d*x + c)^3 - 36*(b*x + a)^2*A*a^5*b*d^7*g^3/(d*x + c)^2 - 36*(b*x + a)^2*B*a^5*b*d^7*g^3/(d*x
+ c)^2 - 120*(b*x + a)^3*A*a^4*b*c*d^7*g^3/(d*x + c)^3 - 120*(b*x + a)^3*B*a^4*b*c*d^7*g^3/(d*x + c)^3 + 24*(b
*x + a)^3*A*a^5*d^8*g^3/(d*x + c)^3 + 24*(b*x + a)^3*B*a^5*d^8*g^3/(d*x + c)^3)/(b^4*d^4 - 4*(b*x + a)*b^3*d^5
/(d*x + c) + 6*(b*x + a)^2*b^2*d^6/(d*x + c)^2 - 4*(b*x + a)^3*b*d^7/(d*x + c)^3 + (b*x + a)^4*d^8/(d*x + c)^4
) + 6*(B*b^5*c^5*g^3*n - 5*B*a*b^4*c^4*d*g^3*n + 10*B*a^2*b^3*c^3*d^2*g^3*n - 10*B*a^3*b^2*c^2*d^3*g^3*n + 5*B
*a^4*b*c*d^4*g^3*n - B*a^5*d^5*g^3*n)*log(-b + (b*x + a)*d/(d*x + c))/(b*d^4) - 6*(B*b^5*c^5*g^3*n - 5*B*a*b^4
*c^4*d*g^3*n + 10*B*a^2*b^3*c^3*d^2*g^3*n - 10*B*a^3*b^2*c^2*d^3*g^3*n + 5*B*a^4*b*c*d^4*g^3*n - B*a^5*d^5*g^3
*n)*log((b*x + a)/(d*x + c))/(b*d^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

[Out]

int((b*g*x+a*g)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [B]  time = 1.41, size = 479, normalized size = 3.07 \[ \frac {1}{4} \, B b^{3} g^{3} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{4} \, A b^{3} g^{3} x^{4} + B a b^{2} g^{3} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a b^{2} g^{3} x^{3} + \frac {3}{2} \, B a^{2} b g^{3} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {3}{2} \, A a^{2} b g^{3} x^{2} - \frac {1}{24} \, B b^{3} g^{3} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{2} \, B a b^{2} g^{3} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {3}{2} \, B a^{2} b g^{3} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B a^{3} g^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B a^{3} g^{3} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a^{3} g^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/4*B*b^3*g^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^3*g^3*x^4 + B*a*b^2*g^3*x^3*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n) + A*a*b^2*g^3*x^3 + 3/2*B*a^2*b*g^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A
*a^2*b*g^3*x^2 - 1/24*B*b^3*g^3*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)) + 1/2*B*a*b^2*g^3*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)) - 3/2*B*a^
2*b*g^3*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*a^3*g^3*n*(a*log(b*x + a)/b
- c*log(d*x + c)/d) + B*a^3*g^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^3*g^3*x

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mupad [B]  time = 4.40, size = 588, normalized size = 3.77 \[ x^3\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,d}\right )-x^2\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,d}+\frac {A\,a\,b^2\,c\,g^3}{2\,d}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a^3\,g^3\,x+\frac {3\,B\,a^2\,b\,g^3\,x^2}{2}+B\,a\,b^2\,g^3\,x^3+\frac {B\,b^3\,g^3\,x^4}{4}\right )+x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{d}\right )}{4\,b\,d}+\frac {a^2\,g^3\,\left (8\,A\,a\,d+12\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{b\,d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,n\,a^3\,c\,d^3\,g^3+6\,B\,n\,a^2\,b\,c^2\,d^2\,g^3-4\,B\,n\,a\,b^2\,c^3\,d\,g^3+B\,n\,b^3\,c^4\,g^3\right )}{4\,d^4}+\frac {A\,b^3\,g^3\,x^4}{4}+\frac {B\,a^4\,g^3\,n\,\ln \left (a+b\,x\right )}{4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

x^3*((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(12*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(12*d)) - x^2*(((
(b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d))*(4*a*d + 4*b*c))
/(8*b*d) - (a*b*g^3*(6*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(2*d) + (A*a*b^2*c*g^3)/(2*d)) + log(e*((a + b*x)
/(c + d*x))^n)*((B*b^3*g^3*x^4)/4 + B*a^3*g^3*x + (3*B*a^2*b*g^3*x^2)/2 + B*a*b^2*g^3*x^3) + x*(((4*a*d + 4*b*
c)*((((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d))*(4*a*d + 4
*b*c))/(4*b*d) - (a*b*g^3*(6*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^2*c*g^3)/d))/(4*b*d) + (a^2*g^3*
(8*A*a*d + 12*A*b*c + 3*B*a*d*n - 3*B*b*c*n))/(2*d) - (a*c*((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))
/(4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d)))/(b*d)) + (log(c + d*x)*(B*b^3*c^4*g^3*n - 4*B*a^3*c*d^3*g^3*n - 4
*B*a*b^2*c^3*d*g^3*n + 6*B*a^2*b*c^2*d^2*g^3*n))/(4*d^4) + (A*b^3*g^3*x^4)/4 + (B*a^4*g^3*n*log(a + b*x))/(4*b
)

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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((b*g*x+a*g)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

Timed out

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