3.126 \(\int \frac{(c i+d i x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n))}{(a g+b g x)^6} \, dx\)
Optimal. Leaf size=293 \[ -\frac{b^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac{d^2 i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b d i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac{b^2 B i^2 n (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac{B d^2 i^2 n (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b B d i^2 n (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]
[Out]
-(B*d^2*i^2*n*(c + d*x)^3)/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*x)^4)/(8*(b*c - a*d)^3*g^6*
(a + b*x)^4) - (b^2*B*i^2*n*(c + d*x)^5)/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/
(c + d*x))^n]))/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
))/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)
________________________________________________________________________________________
Rubi [A] time = 0.720867, antiderivative size = 375, normalized size of antiderivative =
1.28, 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, 44} \[ -\frac{d^2 i^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^6 (a+b x)^3}-\frac{d i^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^6 (a+b x)^4}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3 g^6 (a+b x)^5}-\frac{B d^4 i^2 n}{30 b^3 g^6 (a+b x) (b c-a d)^2}+\frac{B d^3 i^2 n}{60 b^3 g^6 (a+b x)^2 (b c-a d)}-\frac{B d^5 i^2 n \log (a+b x)}{30 b^3 g^6 (b c-a d)^3}+\frac{B d^5 i^2 n \log (c+d x)}{30 b^3 g^6 (b c-a d)^3}-\frac{3 B d i^2 n (b c-a d)}{40 b^3 g^6 (a+b x)^4}-\frac{B i^2 n (b c-a d)^2}{25 b^3 g^6 (a+b x)^5}-\frac{B d^2 i^2 n}{90 b^3 g^6 (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^6,x]
[Out]
-(B*(b*c - a*d)^2*i^2*n)/(25*b^3*g^6*(a + b*x)^5) - (3*B*d*(b*c - a*d)*i^2*n)/(40*b^3*g^6*(a + b*x)^4) - (B*d^
2*i^2*n)/(90*b^3*g^6*(a + b*x)^3) + (B*d^3*i^2*n)/(60*b^3*(b*c - a*d)*g^6*(a + b*x)^2) - (B*d^4*i^2*n)/(30*b^3
*(b*c - a*d)^2*g^6*(a + b*x)) - (B*d^5*i^2*n*Log[a + b*x])/(30*b^3*(b*c - a*d)^3*g^6) - ((b*c - a*d)^2*i^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^3*g^6*(a + b*x)^5) - (d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(2*b^3*g^6*(a + b*x)^4) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*g^6*(a + b*x)^3)
+ (B*d^5*i^2*n*Log[c + d*x])/(30*b^3*(b*c - a*d)^3*g^6)
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 44
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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{(126 c+126 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx &=\int \left (\frac{15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^6 (a+b x)^6}+\frac{31752 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^6 (a+b x)^5}+\frac{15876 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac{\left (15876 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^6}+\frac{(31752 d (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^6}+\frac{\left (15876 (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^6} \, dx}{b^2 g^6}\\ &=-\frac{15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{7938 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^4}-\frac{5292 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac{\left (5292 B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac{(7938 B d (b c-a d) n) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (15876 B (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{7938 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^4}-\frac{5292 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac{\left (5292 B d^2 (b c-a d) n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (7938 B d (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (15876 B (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{7938 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^4}-\frac{5292 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac{\left (5292 B d^2 (b c-a d) n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac{\left (7938 B d (b c-a d)^2 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac{\left (15876 B (b c-a d)^3 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^6}-\frac{b d}{(b c-a d)^2 (a+b x)^5}+\frac{b d^2}{(b c-a d)^3 (a+b x)^4}-\frac{b d^3}{(b c-a d)^4 (a+b x)^3}+\frac{b d^4}{(b c-a d)^5 (a+b x)^2}-\frac{b d^5}{(b c-a d)^6 (a+b x)}+\frac{d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^3 g^6}\\ &=-\frac{15876 B (b c-a d)^2 n}{25 b^3 g^6 (a+b x)^5}-\frac{11907 B d (b c-a d) n}{10 b^3 g^6 (a+b x)^4}-\frac{882 B d^2 n}{5 b^3 g^6 (a+b x)^3}+\frac{1323 B d^3 n}{5 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{2646 B d^4 n}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{2646 B d^5 n \log (a+b x)}{5 b^3 (b c-a d)^3 g^6}-\frac{15876 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{7938 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^4}-\frac{5292 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^6 (a+b x)^3}+\frac{2646 B d^5 n \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}\\ \end{align*}
Mathematica [A] time = 1.05588, size = 357, normalized size = 1.22 \[ \frac{i^2 \left (-\frac{360 a^2 A d^2}{(a+b x)^5}-\frac{60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5}-\frac{72 a^2 B d^2 n}{(a+b x)^5}-\frac{360 A b^2 c^2}{(a+b x)^5}-\frac{900 A b c d}{(a+b x)^4}+\frac{720 a A b c d}{(a+b x)^5}-\frac{600 A d^2}{(a+b x)^3}+\frac{900 a A d^2}{(a+b x)^4}-\frac{72 b^2 B c^2 n}{(a+b x)^5}-\frac{60 B d^4 n}{(a+b x) (b c-a d)^2}+\frac{30 B d^3 n}{(a+b x)^2 (b c-a d)}-\frac{60 B d^5 n \log (a+b x)}{(b c-a d)^3}+\frac{60 B d^5 n \log (c+d x)}{(b c-a d)^3}-\frac{135 b B c d n}{(a+b x)^4}+\frac{144 a b B c d n}{(a+b x)^5}-\frac{20 B d^2 n}{(a+b x)^3}+\frac{135 a B d^2 n}{(a+b x)^4}\right )}{1800 b^3 g^6} \]
Antiderivative was successfully verified.
[In]
Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^6,x]
[Out]
(i^2*((-360*A*b^2*c^2)/(a + b*x)^5 + (720*a*A*b*c*d)/(a + b*x)^5 - (360*a^2*A*d^2)/(a + b*x)^5 - (72*b^2*B*c^2
*n)/(a + b*x)^5 + (144*a*b*B*c*d*n)/(a + b*x)^5 - (72*a^2*B*d^2*n)/(a + b*x)^5 - (900*A*b*c*d)/(a + b*x)^4 + (
900*a*A*d^2)/(a + b*x)^4 - (135*b*B*c*d*n)/(a + b*x)^4 + (135*a*B*d^2*n)/(a + b*x)^4 - (600*A*d^2)/(a + b*x)^3
- (20*B*d^2*n)/(a + b*x)^3 + (30*B*d^3*n)/((b*c - a*d)*(a + b*x)^2) - (60*B*d^4*n)/((b*c - a*d)^2*(a + b*x))
- (60*B*d^5*n*Log[a + b*x])/(b*c - a*d)^3 - (60*B*(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*
d^2*x^2))*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^5 + (60*B*d^5*n*Log[c + d*x])/(b*c - a*d)^3))/(1800*b^3*g^
6)
________________________________________________________________________________________
Maple [F] time = 0.771, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{6}} \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((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x)
[Out]
int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x)
________________________________________________________________________________________
Maxima [B] time = 2.28833, size = 4128, normalized size = 14.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="maxima")
[Out]
-1/300*B*c^2*i^2*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 13
7*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3
*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2
*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6
*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*
b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*
x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c
^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 -
5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x
+ c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6))
- 1/1800*B*d^2*i^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4
+ 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*
d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4
*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*
x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 -
4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*
c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5
*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 -
4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 +
a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a
^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 +
a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 -
a^5*b^3*d^5)*g^6)) - 1/600*B*c*d*i^2*n*((27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c
*d^3 + 77*a^5*d^4 - 60*(5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3
- 10*(10*b^5*c^3*d - 67*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^
3*d + 232*a^2*b^3*c^2*d^2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d
^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*
c*d^3 + a^5*b^6*d^4)*g^6*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b
^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x
^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^
4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x
+ a)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6
) + 60*(5*b*c*d^4 - a*d^5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 +
5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6)) - 1/10*(5*b*x + a)*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^7*
g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(1
0*b^2*x^2 + 5*a*b*x + a^2)*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 1
0*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/10*(5*b*x + a)*A*c*d*i^2/(b^7*g^6*
x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(10*b^
2*x^2 + 5*a*b*x + a^2)*A*d^2*i^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*
a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/5*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^6*x^5 + 5*a*b^5*g^6
*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6) - 1/5*A*c^2*i^2/(b^6*g^6*x^5 + 5
*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6)
________________________________________________________________________________________
Fricas [B] time = 0.634855, size = 2213, normalized size = 7.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="fricas")
[Out]
-1/1800*(60*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^2*n*x^4 - 30*(B*b^5*c^2*d^3 - 10*B*a*b^4*c*d^4 + 9*B*a^2*b^3*d^5)*i^
2*n*x^3 + (72*B*b^5*c^5 - 225*B*a*b^4*c^4*d + 200*B*a^2*b^3*c^3*d^2 - 47*B*a^5*d^5)*i^2*n + 60*(6*A*b^5*c^5 -
15*A*a*b^4*c^4*d + 10*A*a^2*b^3*c^3*d^2 - A*a^5*d^5)*i^2 + 10*((2*B*b^5*c^3*d^2 - 15*B*a*b^4*c^2*d^3 + 60*B*a^
2*b^3*c*d^4 - 47*B*a^3*b^2*d^5)*i^2*n + 60*(A*b^5*c^3*d^2 - 3*A*a*b^4*c^2*d^3 + 3*A*a^2*b^3*c*d^4 - A*a^3*b^2*
d^5)*i^2)*x^2 + 5*((27*B*b^5*c^4*d - 100*B*a*b^4*c^3*d^2 + 120*B*a^2*b^3*c^2*d^3 - 47*B*a^4*b*d^5)*i^2*n + 60*
(3*A*b^5*c^4*d - 8*A*a*b^4*c^3*d^2 + 6*A*a^2*b^3*c^2*d^3 - A*a^4*b*d^5)*i^2)*x + 60*(10*(B*b^5*c^3*d^2 - 3*B*a
*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*i^2*x^2 + 5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3
*c^2*d^3 - B*a^4*b*d^5)*i^2*x + (6*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - B*a^5*d^5)*i^2)*log(e
) + 60*(B*b^5*d^5*i^2*n*x^5 + 5*B*a*b^4*d^5*i^2*n*x^4 + 10*B*a^2*b^3*d^5*i^2*n*x^3 + 10*(B*b^5*c^3*d^2 - 3*B*a
*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4)*i^2*n*x^2 + 5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3*c^2*d^3)*i^2*
n*x + (6*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*i^2*n)*log((b*x + a)/(d*x + c)))/((b^11*c^3 - 3*
a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^6*x^5 + 5*(a*b^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^
4*b^7*d^3)*g^6*x^4 + 10*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d^2 - a^5*b^6*d^3)*g^6*x^3 + 10*(a^3*b^8*
c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^6*x^2 + 5*(a^4*b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*
c*d^2 - a^7*b^4*d^3)*g^6*x + (a^5*b^6*c^3 - 3*a^6*b^5*c^2*d + 3*a^7*b^4*c*d^2 - a^8*b^3*d^3)*g^6)
________________________________________________________________________________________
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((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**6,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.33954, size = 1443, normalized size = 4.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="giac")
[Out]
1/30*B*d^5*n*log(b*x + a)/(b^6*c^3*g^6 - 3*a*b^5*c^2*d*g^6 + 3*a^2*b^4*c*d^2*g^6 - a^3*b^3*d^3*g^6) - 1/30*B*d
^5*n*log(d*x + c)/(b^6*c^3*g^6 - 3*a*b^5*c^2*d*g^6 + 3*a^2*b^4*c*d^2*g^6 - a^3*b^3*d^3*g^6) + 1/30*(10*B*b^2*d
^2*n*x^2 + 15*B*b^2*c*d*n*x + 5*B*a*b*d^2*n*x + 6*B*b^2*c^2*n + 3*B*a*b*c*d*n + B*a^2*d^2*n)*log((b*x + a)/(d*
x + c))/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g
^6) + 1/1800*(60*B*b^4*d^4*n*x^4 - 30*B*b^4*c*d^3*n*x^3 + 270*B*a*b^3*d^4*n*x^3 + 20*B*b^4*c^2*d^2*n*x^2 - 130
*B*a*b^3*c*d^3*n*x^2 + 470*B*a^2*b^2*d^4*n*x^2 + 135*B*b^4*c^3*d*n*x - 365*B*a*b^3*c^2*d^2*n*x + 235*B*a^2*b^2
*c*d^3*n*x + 235*B*a^3*b*d^4*n*x + 600*A*b^4*c^2*d^2*x^2 + 600*B*b^4*c^2*d^2*x^2 - 1200*A*a*b^3*c*d^3*x^2 - 12
00*B*a*b^3*c*d^3*x^2 + 600*A*a^2*b^2*d^4*x^2 + 600*B*a^2*b^2*d^4*x^2 + 72*B*b^4*c^4*n - 153*B*a*b^3*c^3*d*n +
47*B*a^2*b^2*c^2*d^2*n + 47*B*a^3*b*c*d^3*n + 47*B*a^4*d^4*n + 900*A*b^4*c^3*d*x + 900*B*b^4*c^3*d*x - 1500*A*
a*b^3*c^2*d^2*x - 1500*B*a*b^3*c^2*d^2*x + 300*A*a^2*b^2*c*d^3*x + 300*B*a^2*b^2*c*d^3*x + 300*A*a^3*b*d^4*x +
300*B*a^3*b*d^4*x + 360*A*b^4*c^4 + 360*B*b^4*c^4 - 540*A*a*b^3*c^3*d - 540*B*a*b^3*c^3*d + 60*A*a^2*b^2*c^2*
d^2 + 60*B*a^2*b^2*c^2*d^2 + 60*A*a^3*b*c*d^3 + 60*B*a^3*b*c*d^3 + 60*A*a^4*d^4 + 60*B*a^4*d^4)/(b^10*c^2*g^6*
x^5 - 2*a*b^9*c*d*g^6*x^5 + a^2*b^8*d^2*g^6*x^5 + 5*a*b^9*c^2*g^6*x^4 - 10*a^2*b^8*c*d*g^6*x^4 + 5*a^3*b^7*d^2
*g^6*x^4 + 10*a^2*b^8*c^2*g^6*x^3 - 20*a^3*b^7*c*d*g^6*x^3 + 10*a^4*b^6*d^2*g^6*x^3 + 10*a^3*b^7*c^2*g^6*x^2 -
20*a^4*b^6*c*d*g^6*x^2 + 10*a^5*b^5*d^2*g^6*x^2 + 5*a^4*b^6*c^2*g^6*x - 10*a^5*b^5*c*d*g^6*x + 5*a^6*b^4*d^2*
g^6*x + a^5*b^5*c^2*g^6 - 2*a^6*b^4*c*d*g^6 + a^7*b^3*d^2*g^6)