3.117 \(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n)) \, dx\)

Optimal. Leaf size=442 \[ -\frac{3 b^2 g^3 i^2 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4}+\frac{b^3 g^3 i^2 (c+d x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4}-\frac{g^3 i^2 (c+d x)^3 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^4}+\frac{3 b g^3 i^2 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4}+\frac{B g^3 i^2 n x (b c-a d)^5}{60 b^2 d^3}-\frac{b^2 B g^3 i^2 n (c+d x)^5 (b c-a d)}{30 d^4}+\frac{B g^3 i^2 n (b c-a d)^6 \log \left (\frac{a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac{B g^3 i^2 n (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}+\frac{B g^3 i^2 n (c+d x)^2 (b c-a d)^4}{120 b d^4}-\frac{19 B g^3 i^2 n (c+d x)^3 (b c-a d)^3}{180 d^4}+\frac{13 b B g^3 i^2 n (c+d x)^4 (b c-a d)^2}{120 d^4} \]

[Out]

(B*(b*c - a*d)^5*g^3*i^2*n*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*n*(c + d*x)^2)/(120*b*d^4) - (19*B*(b*c
- a*d)^3*g^3*i^2*n*(c + d*x)^3)/(180*d^4) + (13*b*B*(b*c - a*d)^2*g^3*i^2*n*(c + d*x)^4)/(120*d^4) - (b^2*B*(b
*c - a*d)*g^3*i^2*n*(c + d*x)^5)/(30*d^4) - ((b*c - a*d)^3*g^3*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*
x))^n]))/(3*d^4) + (3*b*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^4) - (3
*b^2*(b*c - a*d)*g^3*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^4) + (b^3*g^3*i^2*(c + d*x)^
6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^4) + (B*(b*c - a*d)^6*g^3*i^2*n*Log[(a + b*x)/(c + d*x)])/(60*b
^3*d^4) + (B*(b*c - a*d)^6*g^3*i^2*n*Log[c + d*x])/(60*b^3*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.685045, antiderivative size = 345, normalized size of antiderivative = 0.78, number of steps used = 14, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {2528, 2525, 12, 43} \[ \frac{d^2 g^3 i^2 (a+b x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3}+\frac{g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^3}+\frac{2 d g^3 i^2 (a+b x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3}-\frac{B g^3 i^2 n x (b c-a d)^5}{60 b^2 d^3}+\frac{B g^3 i^2 n (a+b x)^2 (b c-a d)^4}{120 b^3 d^2}+\frac{B g^3 i^2 n (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac{B g^3 i^2 n (a+b x)^3 (b c-a d)^3}{180 b^3 d}-\frac{7 B g^3 i^2 n (a+b x)^4 (b c-a d)^2}{120 b^3}-\frac{B d g^3 i^2 n (a+b x)^5 (b c-a d)}{30 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

-(B*(b*c - a*d)^5*g^3*i^2*n*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*n*(a + b*x)^2)/(120*b^3*d^2) - (B*(b*c
- a*d)^3*g^3*i^2*n*(a + b*x)^3)/(180*b^3*d) - (7*B*(b*c - a*d)^2*g^3*i^2*n*(a + b*x)^4)/(120*b^3) - (B*d*(b*c
- a*d)*g^3*i^2*n*(a + b*x)^5)/(30*b^3) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]))/(4*b^3) + (2*d*(b*c - a*d)*g^3*i^2*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^3) + (d^2*g^
3*i^2*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^3) + (B*(b*c - a*d)^6*g^3*i^2*n*Log[c + d*x])/(
60*b^3*d^4)

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]

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 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])

Rubi steps

\begin{align*} \int (117 c+117 d x)^2 (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 &=\int \left (\frac{13689 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{27378 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac{13689 d^2 (a g+b g x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}\right ) \, dx\\ &=\frac{\left (13689 (b c-a d)^2\right ) \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}{b^2}+\frac{\left (13689 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g^2}+\frac{(27378 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}\\ &=\frac{13689 (b c-a d)^2 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^3}+\frac{27378 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{4563 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3}-\frac{\left (4563 B d^2 n\right ) \int \frac{(b c-a d) g^6 (a+b x)^5}{c+d x} \, dx}{2 b^3 g^3}-\frac{(27378 B d (b c-a d) n) \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b^3 g^2}-\frac{\left (13689 B (b c-a d)^2 n\right ) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b^3 g}\\ &=\frac{13689 (b c-a d)^2 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^3}+\frac{27378 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{4563 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3}-\frac{\left (4563 B d^2 (b c-a d) g^3 n\right ) \int \frac{(a+b x)^5}{c+d x} \, dx}{2 b^3}-\frac{\left (27378 B d (b c-a d)^2 g^3 n\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b^3}-\frac{\left (13689 B (b c-a d)^3 g^3 n\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b^3}\\ &=\frac{13689 (b c-a d)^2 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^3}+\frac{27378 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{4563 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3}-\frac{\left (4563 B d^2 (b c-a d) g^3 n\right ) \int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{2 b^3}-\frac{\left (27378 B d (b c-a d)^2 g^3 n\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac{\left (13689 B (b c-a d)^3 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^3}\\ &=-\frac{4563 B (b c-a d)^5 g^3 n x}{20 b^2 d^3}+\frac{4563 B (b c-a d)^4 g^3 n (a+b x)^2}{40 b^3 d^2}-\frac{1521 B (b c-a d)^3 g^3 n (a+b x)^3}{20 b^3 d}-\frac{31941 B (b c-a d)^2 g^3 n (a+b x)^4}{40 b^3}-\frac{4563 B d (b c-a d) g^3 n (a+b x)^5}{10 b^3}+\frac{13689 (b c-a d)^2 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^3}+\frac{27378 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{4563 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3}+\frac{4563 B (b c-a d)^6 g^3 n \log (c+d x)}{20 b^3 d^4}\\ \end{align*}

Mathematica [A]  time = 0.39755, size = 441, normalized size = 1. \[ \frac{g^3 i^2 \left (60 d^6 (a+b x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+144 d^5 (a+b x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+90 d^4 (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-15 B n (b c-a d)^3 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+12 B n (b c-a d)^2 \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )-B n (b c-a d) \left (20 d^3 (a+b x)^3 (b c-a d)^2+15 d^4 (a+b x)^4 (a d-b c)+30 d^2 (a+b x)^2 (a d-b c)^3+60 b d x (b c-a d)^4-60 (b c-a d)^5 \log (c+d x)+12 d^5 (a+b x)^5\right )\right )}{360 b^3 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

(g^3*i^2*(90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^5*(b*c - a*d)*(a + b
*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 60*d^6*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 15*
B*(b*c - a*d)^3*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]) + 12*B*(b*c - a*d)^2*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*
c - a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) - B*(b*c - a*d)*n*(60*b*d*(b*c - a*d
)^4*x + 30*d^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*
x)^4 + 12*d^5*(a + b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x])))/(360*b^3*d^4)

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Maple [F]  time = 0.631, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{2} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

________________________________________________________________________________________

Maxima [B]  time = 1.5242, size = 2670, normalized size = 6.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/6*B*b^3*d^2*g^3*i^2*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A*b^3*d^2*g^3*i^2*x^6 + 2/5*B*b^3*c*d*g
^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*B*a*b^2*d^2*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^3*i^2*x^5 + 1/4*B*b^3*c^2*g^3*i^2*x^4*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n) + 3/2*B*a*b^2*c*d*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*B*a^2*b*d^2
*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^3*c^2*g^3*i^2*x^4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4
+ 3/4*A*a^2*b*d^2*g^3*i^2*x^4 + B*a*b^2*c^2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*B*a^2*b*c*d
*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*B*a^3*d^2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2*b*c*d*g^3*i^2*x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*B*a^2*b*c^2*g
^3*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^3*c*d*g^3*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^
n) + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d*g^3*i^2*x^2 - 1/360*B*b^3*d^2*g^3*i^2*n*(60*a^6*log(b*x + a)/b^6
- 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3
*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/30*B*b^3*c
*d*g^3*i^2*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)) + 1/20*B*a*b^2*d
^2*g^3*i^2*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)) - 1/24*B*b^3*c^2
*g^3*i^2*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)) - 1/4*B*a*b^2*c*d*g^3*i^2*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)) - 1/8*B*a^2*b*d^2*g^3*i^2*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*c^2*g^3*i^2*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)) + B*a^2*b*c*d*g^3*i^2*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)) + 1/6*B*a^3*d^2*g^3*i^2*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*c^2*g^3*i^2*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*c*d*g^3*i^2*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*c^2*g^3*i^2*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^3*c^2*g^3*i^2*x
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^3*c^2*g^3*i^2*x

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Fricas [B]  time = 1.10175, size = 2202, normalized size = 4.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/360*(60*A*b^6*d^6*g^3*i^2*x^6 + 6*(15*B*a^4*b^2*c^2*d^4 - 6*B*a^5*b*c*d^5 + B*a^6*d^6)*g^3*i^2*n*log(b*x + a
) + 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2 - 20*B*a^3*b^3*c^3*d^3)*g^3*i^2*n*log(d*x + c) - 12*
((B*b^6*c*d^5 - B*a*b^5*d^6)*g^3*i^2*n - 6*(2*A*b^6*c*d^5 + 3*A*a*b^5*d^6)*g^3*i^2)*x^5 - 3*((7*B*b^6*c^2*d^4
+ 6*B*a*b^5*c*d^5 - 13*B*a^2*b^4*d^6)*g^3*i^2*n - 30*(A*b^6*c^2*d^4 + 6*A*a*b^5*c*d^5 + 3*A*a^2*b^4*d^6)*g^3*i
^2)*x^4 - 2*((B*b^6*c^3*d^3 + 39*B*a*b^5*c^2*d^4 - 21*B*a^2*b^4*c*d^5 - 19*B*a^3*b^3*d^6)*g^3*i^2*n - 60*(3*A*
a*b^5*c^2*d^4 + 6*A*a^2*b^4*c*d^5 + A*a^3*b^3*d^6)*g^3*i^2)*x^3 + 3*((B*b^6*c^4*d^2 - 6*B*a*b^5*c^3*d^3 - 30*B
*a^2*b^4*c^2*d^4 + 34*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^3*i^2*n + 60*(3*A*a^2*b^4*c^2*d^4 + 2*A*a^3*b^3*c*d^5
)*g^3*i^2)*x^2 + 6*(60*A*a^3*b^3*c^2*d^4*g^3*i^2 - (B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 15*B*a^2*b^4*c^3*d^3 - 5
*B*a^3*b^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^3*i^2*n)*x + 6*(10*B*b^6*d^6*g^3*i^2*x^6 + 60*B*a^3*b^
3*c^2*d^4*g^3*i^2*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*x^5 + 15*(B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + 3
*B*a^2*b^4*d^6)*g^3*i^2*x^4 + 20*(3*B*a*b^5*c^2*d^4 + 6*B*a^2*b^4*c*d^5 + B*a^3*b^3*d^6)*g^3*i^2*x^3 + 30*(3*B
*a^2*b^4*c^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*x^2)*log(e) + 6*(10*B*b^6*d^6*g^3*i^2*n*x^6 + 60*B*a^3*b^3*c^2*d
^4*g^3*i^2*n*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*n*x^5 + 15*(B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + 3*B*
a^2*b^4*d^6)*g^3*i^2*n*x^4 + 20*(3*B*a*b^5*c^2*d^4 + 6*B*a^2*b^4*c*d^5 + B*a^3*b^3*d^6)*g^3*i^2*n*x^3 + 30*(3*
B*a^2*b^4*c^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*n*x^2)*log((b*x + a)/(d*x + c)))/(b^3*d^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

Timed out