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

Optimal. Leaf size=283 \[ -\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2}+\frac {B g i^3 n (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B g i^3 n (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac {B g i^3 n x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 n (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 n (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 n (c+d x)^4 (b c-a d)}{20 d^2} \]

[Out]

1/20*B*(-a*d+b*c)^4*g*i^3*n*x/b^3/d+1/40*B*(-a*d+b*c)^3*g*i^3*n*(d*x+c)^2/b^2/d^2+1/60*B*(-a*d+b*c)^2*g*i^3*n*
(d*x+c)^3/b/d^2-1/20*B*(-a*d+b*c)*g*i^3*n*(d*x+c)^4/d^2-1/4*(-a*d+b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x
+c))^n))/d^2+1/5*b*g*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*n*ln((b*x+a)/
(d*x+c))/b^4/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*n*ln(d*x+c)/b^4/d^2

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Rubi [A]  time = 0.38, antiderivative size = 243, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 43} \[ -\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2}+\frac {B g i^3 n (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 n (b c-a d)^5 \log (a+b x)}{20 b^4 d^2}+\frac {B g i^3 n x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 n (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 n (c+d x)^4 (b c-a d)}{20 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^4*g*i^3*n*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2
*g*i^3*n*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^4)/(20*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log
[a + b*x])/(20*b^4*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^2) + (b*
g*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^2)

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]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (129 c+129 d x)^3 (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac {(-b c+a d) g (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {b g (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{129 d}\right ) \, dx\\ &=\frac {(b g) \int (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{129 d}+\frac {((-b c+a d) g) \int (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(b B g n) \int \frac {35723051649 (b c-a d) (c+d x)^4}{a+b x} \, dx}{83205 d^2}+\frac {(B (b c-a d) g n) \int \frac {276922881 (b c-a d) (c+d x)^3}{a+b x} \, dx}{516 d^2}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(2146689 b B (b c-a d) g n) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^2}+\frac {\left (2146689 B (b c-a d)^2 g n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{4 d^2}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(2146689 b B (b c-a d) g n) \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^2}+\frac {\left (2146689 B (b c-a d)^2 g n\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{4 d^2}\\ &=\frac {2146689 B (b c-a d)^4 g n x}{20 b^3 d}+\frac {2146689 B (b c-a d)^3 g n (c+d x)^2}{40 b^2 d^2}+\frac {715563 B (b c-a d)^2 g n (c+d x)^3}{20 b d^2}-\frac {2146689 B (b c-a d) g n (c+d x)^4}{20 d^2}+\frac {2146689 B (b c-a d)^5 g n \log (a+b x)}{20 b^4 d^2}-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 269, normalized size = 0.95 \[ \frac {g i^3 \left (24 b (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-30 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {5 B n (b c-a d)^2 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^4}-\frac {2 B n (b c-a d) \left (4 b^3 (c+d x)^3 (b c-a d)+6 b^2 (c+d x)^2 (b c-a d)^2+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{b^4}\right )}{120 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*((5*B*(b*c - a*d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b
*c - a*d)^3*Log[a + b*x]))/b^4 - (2*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^4 - 30*(b*c - a*d)*(c
+ d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(1
20*d^2)

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fricas [B]  time = 1.16, size = 721, normalized size = 2.55 \[ \frac {24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \, {\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} n \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} n \log \left (d x + c\right ) - 6 \, {\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g i^{3} n - 5 \, {\left (3 \, A b^{5} c d^{4} + A a b^{4} d^{5}\right )} g i^{3}\right )} x^{4} - 2 \, {\left ({\left (11 \, B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} - B a^{2} b^{3} d^{5}\right )} g i^{3} n - 60 \, {\left (A b^{5} c^{2} d^{3} + A a b^{4} c d^{4}\right )} g i^{3}\right )} x^{3} - 3 \, {\left ({\left (9 \, B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g i^{3} n - 20 \, {\left (A b^{5} c^{3} d^{2} + 3 \, A a b^{4} c^{2} d^{3}\right )} g i^{3}\right )} x^{2} + 6 \, {\left (20 \, A a b^{4} c^{3} d^{2} g i^{3} - {\left (B b^{5} c^{4} d + 5 \, B a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} n\right )} x + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \relax (e) + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} n x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} n x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} n x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} n x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*A*b^5*d^5*g*i^3*x^5 + 6*(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*i^3*n*log(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c^4*d)*g*i^3*n*log(d*x + c) - 6*((B*b^5*c*d^4 - B*a*b^4*d^5)*g
*i^3*n - 5*(3*A*b^5*c*d^4 + A*a*b^4*d^5)*g*i^3)*x^4 - 2*((11*B*b^5*c^2*d^3 - 10*B*a*b^4*c*d^4 - B*a^2*b^3*d^5)
*g*i^3*n - 60*(A*b^5*c^2*d^3 + A*a*b^4*c*d^4)*g*i^3)*x^3 - 3*((9*B*b^5*c^3*d^2 - 5*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*i^3*n - 20*(A*b^5*c^3*d^2 + 3*A*a*b^4*c^2*d^3)*g*i^3)*x^2 + 6*(20*A*a*b^4*c^3*d^2*
g*i^3 - (B*b^5*c^4*d + 5*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*i^3*n)*x
+ 6*(4*B*b^5*d^5*g*i^3*x^5 + 20*B*a*b^4*c^3*d^2*g*i^3*x + 5*(3*B*b^5*c*d^4 + B*a*b^4*d^5)*g*i^3*x^4 + 20*(B*b^
5*c^2*d^3 + B*a*b^4*c*d^4)*g*i^3*x^3 + 10*(B*b^5*c^3*d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*x^2)*log(e) + 6*(4*B*b^5*d
^5*g*i^3*n*x^5 + 20*B*a*b^4*c^3*d^2*g*i^3*n*x + 5*(3*B*b^5*c*d^4 + B*a*b^4*d^5)*g*i^3*n*x^4 + 20*(B*b^5*c^2*d^
3 + B*a*b^4*c*d^4)*g*i^3*n*x^3 + 10*(B*b^5*c^3*d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*n*x^2)*log((b*x + a)/(d*x + c)))
/(b^4*d^2)

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giac [B]  time = 5.67, size = 2374, normalized size = 8.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(6*(B*b^7*c^6*g*i*n - 6*B*a*b^6*c^5*d*g*i*n - 5*(b*x + a)*B*b^6*c^6*d*g*i*n/(d*x + c) + 15*B*a^2*b^5*c^4
*d^2*g*i*n + 30*(b*x + a)*B*a*b^5*c^5*d^2*g*i*n/(d*x + c) - 20*B*a^3*b^4*c^3*d^3*g*i*n - 75*(b*x + a)*B*a^2*b^
4*c^4*d^3*g*i*n/(d*x + c) + 15*B*a^4*b^3*c^2*d^4*g*i*n + 100*(b*x + a)*B*a^3*b^3*c^3*d^4*g*i*n/(d*x + c) - 6*B
*a^5*b^2*c*d^5*g*i*n - 75*(b*x + a)*B*a^4*b^2*c^2*d^5*g*i*n/(d*x + c) + B*a^6*b*d^6*g*i*n + 30*(b*x + a)*B*a^5
*b*c*d^6*g*i*n/(d*x + c) - 5*(b*x + a)*B*a^6*d^7*g*i*n/(d*x + c))*log((b*x + a)/(d*x + c))/(b^5*d^2 - 5*(b*x +
a)*b^4*d^3/(d*x + c) + 10*(b*x + a)^2*b^3*d^4/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^5/(d*x + c)^3 + 5*(b*x + a)^
4*b*d^6/(d*x + c)^4 - (b*x + a)^5*d^7/(d*x + c)^5) - (5*B*b^10*c^6*g*i*n - 30*B*a*b^9*c^5*d*g*i*n - 31*(b*x +
a)*B*b^9*c^6*d*g*i*n/(d*x + c) + 75*B*a^2*b^8*c^4*d^2*g*i*n + 186*(b*x + a)*B*a*b^8*c^5*d^2*g*i*n/(d*x + c) +
47*(b*x + a)^2*B*b^8*c^6*d^2*g*i*n/(d*x + c)^2 - 100*B*a^3*b^7*c^3*d^3*g*i*n - 465*(b*x + a)*B*a^2*b^7*c^4*d^3
*g*i*n/(d*x + c) - 282*(b*x + a)^2*B*a*b^7*c^5*d^3*g*i*n/(d*x + c)^2 - 27*(b*x + a)^3*B*b^7*c^6*d^3*g*i*n/(d*x
+ c)^3 + 75*B*a^4*b^6*c^2*d^4*g*i*n + 620*(b*x + a)*B*a^3*b^6*c^3*d^4*g*i*n/(d*x + c) + 705*(b*x + a)^2*B*a^2
*b^6*c^4*d^4*g*i*n/(d*x + c)^2 + 162*(b*x + a)^3*B*a*b^6*c^5*d^4*g*i*n/(d*x + c)^3 + 6*(b*x + a)^4*B*b^6*c^6*d
^4*g*i*n/(d*x + c)^4 - 30*B*a^5*b^5*c*d^5*g*i*n - 465*(b*x + a)*B*a^4*b^5*c^2*d^5*g*i*n/(d*x + c) - 940*(b*x +
a)^2*B*a^3*b^5*c^3*d^5*g*i*n/(d*x + c)^2 - 405*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g*i*n/(d*x + c)^3 - 36*(b*x + a)
^4*B*a*b^5*c^5*d^5*g*i*n/(d*x + c)^4 + 5*B*a^6*b^4*d^6*g*i*n + 186*(b*x + a)*B*a^5*b^4*c*d^6*g*i*n/(d*x + c) +
705*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g*i*n/(d*x + c)^2 + 540*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g*i*n/(d*x + c)^3 + 9
0*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g*i*n/(d*x + c)^4 - 31*(b*x + a)*B*a^6*b^3*d^7*g*i*n/(d*x + c) - 282*(b*x + a)
^2*B*a^5*b^3*c*d^7*g*i*n/(d*x + c)^2 - 405*(b*x + a)^3*B*a^4*b^3*c^2*d^7*g*i*n/(d*x + c)^3 - 120*(b*x + a)^4*B
*a^3*b^3*c^3*d^7*g*i*n/(d*x + c)^4 + 47*(b*x + a)^2*B*a^6*b^2*d^8*g*i*n/(d*x + c)^2 + 162*(b*x + a)^3*B*a^5*b^
2*c*d^8*g*i*n/(d*x + c)^3 + 90*(b*x + a)^4*B*a^4*b^2*c^2*d^8*g*i*n/(d*x + c)^4 - 27*(b*x + a)^3*B*a^6*b*d^9*g*
i*n/(d*x + c)^3 - 36*(b*x + a)^4*B*a^5*b*c*d^9*g*i*n/(d*x + c)^4 + 6*(b*x + a)^4*B*a^6*d^10*g*i*n/(d*x + c)^4
- 6*A*b^10*c^6*g*i - 6*B*b^10*c^6*g*i + 36*A*a*b^9*c^5*d*g*i + 36*B*a*b^9*c^5*d*g*i + 30*(b*x + a)*A*b^9*c^6*d
*g*i/(d*x + c) + 30*(b*x + a)*B*b^9*c^6*d*g*i/(d*x + c) - 90*A*a^2*b^8*c^4*d^2*g*i - 90*B*a^2*b^8*c^4*d^2*g*i
- 180*(b*x + a)*A*a*b^8*c^5*d^2*g*i/(d*x + c) - 180*(b*x + a)*B*a*b^8*c^5*d^2*g*i/(d*x + c) + 120*A*a^3*b^7*c^
3*d^3*g*i + 120*B*a^3*b^7*c^3*d^3*g*i + 450*(b*x + a)*A*a^2*b^7*c^4*d^3*g*i/(d*x + c) + 450*(b*x + a)*B*a^2*b^
7*c^4*d^3*g*i/(d*x + c) - 90*A*a^4*b^6*c^2*d^4*g*i - 90*B*a^4*b^6*c^2*d^4*g*i - 600*(b*x + a)*A*a^3*b^6*c^3*d^
4*g*i/(d*x + c) - 600*(b*x + a)*B*a^3*b^6*c^3*d^4*g*i/(d*x + c) + 36*A*a^5*b^5*c*d^5*g*i + 36*B*a^5*b^5*c*d^5*
g*i + 450*(b*x + a)*A*a^4*b^5*c^2*d^5*g*i/(d*x + c) + 450*(b*x + a)*B*a^4*b^5*c^2*d^5*g*i/(d*x + c) - 6*A*a^6*
b^4*d^6*g*i - 6*B*a^6*b^4*d^6*g*i - 180*(b*x + a)*A*a^5*b^4*c*d^6*g*i/(d*x + c) - 180*(b*x + a)*B*a^5*b^4*c*d^
6*g*i/(d*x + c) + 30*(b*x + a)*A*a^6*b^3*d^7*g*i/(d*x + c) + 30*(b*x + a)*B*a^6*b^3*d^7*g*i/(d*x + c))/(b^8*d^
2 - 5*(b*x + a)*b^7*d^3/(d*x + c) + 10*(b*x + a)^2*b^6*d^4/(d*x + c)^2 - 10*(b*x + a)^3*b^5*d^5/(d*x + c)^3 +
5*(b*x + a)^4*b^4*d^6/(d*x + c)^4 - (b*x + a)^5*b^3*d^7/(d*x + c)^5) + 6*(B*b^6*c^6*g*i*n - 6*B*a*b^5*c^5*d*g*
i*n + 15*B*a^2*b^4*c^4*d^2*g*i*n - 20*B*a^3*b^3*c^3*d^3*g*i*n + 15*B*a^4*b^2*c^2*d^4*g*i*n - 6*B*a^5*b*c*d^5*g
*i*n + B*a^6*d^6*g*i*n)*log(b - (b*x + a)*d/(d*x + c))/(b^4*d^2) - 6*(B*b^6*c^6*g*i*n - 6*B*a*b^5*c^5*d*g*i*n
+ 15*B*a^2*b^4*c^4*d^2*g*i*n - 20*B*a^3*b^3*c^3*d^3*g*i*n + 15*B*a^4*b^2*c^2*d^4*g*i*n - 6*B*a^5*b*c*d^5*g*i*n
+ B*a^6*d^6*g*i*n)*log((b*x + a)/(d*x + c))/(b^4*d^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.40, size = 1118, normalized size = 3.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*b*d^3*g*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b*d^3*g*i^3*x^5 + 3/4*B*b*c*d^2*g*i^3*x^4
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a*d^3*g*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A
*b*c*d^2*g*i^3*x^4 + 1/4*A*a*d^3*g*i^3*x^4 + B*b*c^2*d*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a*
c*d^2*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*b*c^2*d*g*i^3*x^3 + A*a*c*d^2*g*i^3*x^3 + 1/2*B*b*c
^3*g*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a*c^2*d*g*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x +
c))^n) + 1/2*A*b*c^3*g*i^3*x^2 + 3/2*A*a*c^2*d*g*i^3*x^2 + 1/60*B*b*d^3*g*i^3*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)) - 1/8*B*b*c*d^2*g*i^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)) - 1/24*B*a*d^3*g*i^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*b*c^2*d*g*i^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)) + 1/2*B*a*
c*d^2*g*i^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)) - 1/2*B*b*c^3*g*i^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)) -
3/2*B*a*c^2*d*g*i^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*c^3*g*i^3*n*(a
*log(b*x + a)/b - c*log(d*x + c)/d) + B*a*c^3*g*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*c^3*g*i^3*x

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mupad [B]  time = 5.40, size = 1234, normalized size = 4.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((a*c*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d +
20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2
*B*a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/(b*d) - ((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((
d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*
i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4*b) + A*a*c*d
^2*g*i^3))/(20*b*d) - (a*c*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 2
0*b*c))/20))/(b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/b))/(20*
b*d) + (c^2*g*i^3*(12*A*a^2*d^2 + 2*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 16*A*a*b*c*d - 2*B*a*b*c*d*n))/(
2*b*d)) - x^3*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*
a*d + 20*b*c))/20))/(60*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c
*d + 2*B*a*b*c*d*n))/(12*b) + (A*a*c*d^2*g*i^3)/3) + x^2*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((d^2*g*i^3*(
10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^
2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/
(40*b*d) - (a*c*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20)
)/(2*b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/(2*b)) + log(e*((
a + b*x)/(c + d*x))^n)*((B*c^2*g*i^3*x^2*(3*a*d + b*c))/2 + (B*d^2*g*i^3*x^4*(a*d + 3*b*c))/4 + B*a*c^3*g*i^3*
x + (B*b*d^3*g*i^3*x^5)/5 + B*c*d*g*i^3*x^3*(a*d + b*c)) + x^4*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*
b*c*n))/20 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/80) + (log(c + d*x)*(B*b*c^5*g*i^3*n - 5*B*a*c^4*d*g*i^3*n))/(20*
d^2) - (log(a + b*x)*(B*a^5*d^3*g*i^3*n - 10*B*a^2*b^3*c^3*g*i^3*n - 5*B*a^4*b*c*d^2*g*i^3*n + 10*B*a^3*b^2*c^
2*d*g*i^3*n))/(20*b^4) + (A*b*d^3*g*i^3*x^5)/5

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

[Out]

Timed out

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