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

Optimal. Leaf size=387 \[ \frac{b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac{g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac{2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac{B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac{B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac{B g^2 i^3 n (b c-a d)^6 \log \left (\frac{a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac{B g^2 i^3 n (b c-a d)^6 \log (c+d x)}{60 b^4 d^3}-\frac{B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac{7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac{b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.699664, antiderivative size = 345, normalized size of antiderivative = 0.89, 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} \[ \frac{b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac{g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac{2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac{B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac{B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac{B g^2 i^3 n (b c-a d)^6 \log (a+b x)}{60 b^4 d^3}-\frac{B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac{7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac{b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (128 c+128 d x)^3 (a g+b g x)^2 \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)^2 g^2 (128 c+128 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}-\frac{b (b c-a d) g^2 (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{64 d^2}+\frac{b^2 g^2 (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16384 d^2}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{16384 d^2}-\frac{\left (b (b c-a d) g^2\right ) \int (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{64 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int (128 c+128 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}\\ &=\frac{524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac{1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{\left (b^2 B g^2 n\right ) \int \frac{4398046511104 (b c-a d) (c+d x)^5}{a+b x} \, dx}{12582912 d^3}+\frac{\left (b B (b c-a d) g^2 n\right ) \int \frac{34359738368 (b c-a d) (c+d x)^4}{a+b x} \, dx}{40960 d^3}-\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{268435456 (b c-a d) (c+d x)^3}{a+b x} \, dx}{512 d^3}\\ &=\frac{524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac{1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{\left (1048576 b^2 B (b c-a d) g^2 n\right ) \int \frac{(c+d x)^5}{a+b x} \, dx}{3 d^3}+\frac{\left (4194304 b B (b c-a d)^2 g^2 n\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^3}-\frac{\left (524288 B (b c-a d)^3 g^2 n\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^3}\\ &=\frac{524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac{1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{\left (1048576 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac{d (b c-a d)^4}{b^5}+\frac{(b c-a d)^5}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x)}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2}{b^3}+\frac{d (b c-a d) (c+d x)^3}{b^2}+\frac{d (c+d x)^4}{b}\right ) \, dx}{3 d^3}+\frac{\left (4194304 b B (b c-a d)^2 g^2 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^3}-\frac{\left (524288 B (b c-a d)^3 g^2 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}{d^3}\\ &=-\frac{524288 B (b c-a d)^5 g^2 n x}{15 b^3 d^2}-\frac{262144 B (b c-a d)^4 g^2 n (c+d x)^2}{15 b^2 d^3}-\frac{524288 B (b c-a d)^3 g^2 n (c+d x)^3}{45 b d^3}+\frac{1835008 B (b c-a d)^2 g^2 n (c+d x)^4}{15 d^3}-\frac{1048576 b B (b c-a d) g^2 n (c+d x)^5}{15 d^3}-\frac{524288 B (b c-a d)^6 g^2 n \log (a+b x)}{15 b^4 d^3}+\frac{524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac{1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}\\ \end{align*}

Mathematica [A]  time = 0.345287, size = 441, normalized size = 1.14 \[ \frac{g^2 i^3 \left (60 b^6 (c+d x)^6 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-144 b^5 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+90 b^4 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-15 B n (b c-a d)^3 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )+12 B n (b c-a d)^2 \left (6 b^2 (c+d x)^2 (b c-a d)^2+4 b^3 (c+d x)^3 (b c-a d)+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )-B n (b c-a d) \left (30 b^2 (c+d x)^2 (b c-a d)^3+20 b^3 (c+d x)^3 (b c-a d)^2+15 b^4 (c+d x)^4 (b c-a d)+60 b d x (b c-a d)^4+60 (b c-a d)^5 \log (a+b x)+12 b^5 (c+d x)^5\right )\right )}{360 b^4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.688, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \left ( dix+ci \right ) ^{3} \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)^2*(d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.53378, size = 2670, normalized size = 6.9 \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)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.15396, size = 2202, normalized size = 5.69 \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)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out