3.19 \(\int \frac{(c i+d i x)^2 (A+B \log (\frac{e (a+b x)}{c+d x}))}{(a g+b g x)^6} \, dx\)

Optimal. Leaf size=281 \[ -\frac{b^2 i^2 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac{d^2 i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b d i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac{b^2 B i^2 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac{B d^2 i^2 (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b B d i^2 (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]

[Out]

-(B*d^2*i^2*(c + d*x)^3)/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*(c + d*x)^4)/(8*(b*c - a*d)^3*g^6*(a +
b*x)^4) - (b^2*B*i^2*(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)]))/(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)]))/(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)]))/(5*(b*c -
a*d)^3*g^6*(a + b*x)^5)

________________________________________________________________________________________

Rubi [A]  time = 0.678321, antiderivative size = 359, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^2 i^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^3 g^6 (a+b x)^3}-\frac{d i^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^3 g^6 (a+b x)^4}-\frac{i^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^3 g^6 (a+b x)^5}-\frac{B d^4 i^2}{30 b^3 g^6 (a+b x) (b c-a d)^2}+\frac{B d^3 i^2}{60 b^3 g^6 (a+b x)^2 (b c-a d)}-\frac{B d^5 i^2 \log (a+b x)}{30 b^3 g^6 (b c-a d)^3}+\frac{B d^5 i^2 \log (c+d x)}{30 b^3 g^6 (b c-a d)^3}-\frac{3 B d i^2 (b c-a d)}{40 b^3 g^6 (a+b x)^4}-\frac{B i^2 (b c-a d)^2}{25 b^3 g^6 (a+b x)^5}-\frac{B d^2 i^2}{90 b^3 g^6 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

-(B*(b*c - a*d)^2*i^2)/(25*b^3*g^6*(a + b*x)^5) - (3*B*d*(b*c - a*d)*i^2)/(40*b^3*g^6*(a + b*x)^4) - (B*d^2*i^
2)/(90*b^3*g^6*(a + b*x)^3) + (B*d^3*i^2)/(60*b^3*(b*c - a*d)*g^6*(a + b*x)^2) - (B*d^4*i^2)/(30*b^3*(b*c - a*
d)^2*g^6*(a + b*x)) - (B*d^5*i^2*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)]))/(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)]))/(2*b
^3*g^6*(a + b*x)^4) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3*g^6*(a + b*x)^3) + (B*d^5*i^2*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{(19 c+19 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx &=\int \left (\frac{361 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^6 (a+b x)^6}+\frac{722 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^6 (a+b x)^5}+\frac{361 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac{\left (361 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^6}+\frac{(722 d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^2 g^6}+\frac{\left (361 (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{b^2 g^6}\\ &=-\frac{361 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{361 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^3 g^6 (a+b x)^4}-\frac{361 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^6 (a+b x)^3}+\frac{\left (361 B d^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}+\frac{(361 B d (b c-a d)) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{2 b^3 g^6}+\frac{\left (361 B (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{361 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{361 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^3 g^6 (a+b x)^4}-\frac{361 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^6 (a+b x)^3}+\frac{\left (361 B d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}+\frac{\left (361 B d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{2 b^3 g^6}+\frac{\left (361 B (b c-a d)^3\right ) \int \frac{1}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{361 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{361 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^3 g^6 (a+b x)^4}-\frac{361 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^6 (a+b x)^3}+\frac{\left (361 B d^2 (b c-a d)\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}{3 b^3 g^6}+\frac{\left (361 B d (b c-a d)^2\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}{2 b^3 g^6}+\frac{\left (361 B (b c-a d)^3\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{361 B (b c-a d)^2}{25 b^3 g^6 (a+b x)^5}-\frac{1083 B d (b c-a d)}{40 b^3 g^6 (a+b x)^4}-\frac{361 B d^2}{90 b^3 g^6 (a+b x)^3}+\frac{361 B d^3}{60 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{361 B d^4}{30 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{361 B d^5 \log (a+b x)}{30 b^3 (b c-a d)^3 g^6}-\frac{361 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^3 g^6 (a+b x)^5}-\frac{361 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^3 g^6 (a+b x)^4}-\frac{361 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^6 (a+b x)^3}+\frac{361 B d^5 \log (c+d x)}{30 b^3 (b c-a d)^3 g^6}\\ \end{align*}

Mathematica [A]  time = 0.888242, size = 344, normalized size = 1.22 \[ \frac{i^2 \left (-\frac{360 a^2 A d^2}{(a+b x)^5}-\frac{60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5}-\frac{72 a^2 B d^2}{(a+b x)^5}-\frac{360 A b^2 c^2}{(a+b x)^5}-\frac{900 A b c d}{(a+b x)^4}+\frac{720 a A b c d}{(a+b x)^5}-\frac{600 A d^2}{(a+b x)^3}+\frac{900 a A d^2}{(a+b x)^4}-\frac{72 b^2 B c^2}{(a+b x)^5}-\frac{60 B d^4}{(a+b x) (b c-a d)^2}+\frac{30 B d^3}{(a+b x)^2 (b c-a d)}-\frac{60 B d^5 \log (a+b x)}{(b c-a d)^3}+\frac{60 B d^5 \log (c+d x)}{(b c-a d)^3}-\frac{135 b B c d}{(a+b x)^4}+\frac{144 a b B c d}{(a+b x)^5}-\frac{20 B d^2}{(a+b x)^3}+\frac{135 a B d^2}{(a+b x)^4}\right )}{1800 b^3 g^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

(i^2*((-360*A*b^2*c^2)/(a + b*x)^5 - (72*b^2*B*c^2)/(a + b*x)^5 + (720*a*A*b*c*d)/(a + b*x)^5 + (144*a*b*B*c*d
)/(a + b*x)^5 - (360*a^2*A*d^2)/(a + b*x)^5 - (72*a^2*B*d^2)/(a + b*x)^5 - (900*A*b*c*d)/(a + b*x)^4 - (135*b*
B*c*d)/(a + b*x)^4 + (900*a*A*d^2)/(a + b*x)^4 + (135*a*B*d^2)/(a + b*x)^4 - (600*A*d^2)/(a + b*x)^3 - (20*B*d
^2)/(a + b*x)^3 + (30*B*d^3)/((b*c - a*d)*(a + b*x)^2) - (60*B*d^4)/((b*c - a*d)^2*(a + b*x)) - (60*B*d^5*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)])/(a + b*x)^5 + (60*B*d^5*Log[c + d*x])/(b*c - a*d)^3))/(1800*b^3*g^6)

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Maple [B]  time = 0.054, size = 1262, normalized size = 4.5 \begin{align*} \text{result too large to display} \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)))/(b*g*x+a*g)^6,x)

[Out]

1/3*e^3*d^3*i^2/(a*d-b*c)^4/g^6*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/3*e^3*d^2*i^2/(a*d-b*c)^4/g^6*A/(b
*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/2*e^4*d^2*i^2/(a*d-b*c)^4/g^6*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c
)^4*a+1/2*e^4*d*i^2/(a*d-b*c)^4/g^6*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+1/5*e^5*d*i^2/(a*d-b*c)^4/g^
6*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*a-1/5*e^5*i^2/(a*d-b*c)^4/g^6*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^5*c+1/3*e^3*d^3*i^2/(a*d-b*c)^4/g^6*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*
x+c))*a-1/3*e^3*d^2*i^2/(a*d-b*c)^4/g^6*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)*b*c+1/9*e^3*d^3*i^2/(a*d-b*c)^4/g^6*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/9*e^3*d^2*i^2/(a*d-b*c)^4/g^
6*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/2*e^4*d^2*i^2/(a*d-b*c)^4/g^6*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+1/2*e^4*d*i^2/(a*d-b*c)^4/g^6*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-1/8*e^4*d^2*i^2/(a*d-b*c)^4/g^6*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b
*c)^4*a+1/8*e^4*d*i^2/(a*d-b*c)^4/g^6*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+1/5*e^5*d*i^2/(a*d-b*c)^4/
g^6*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/5*e^5*i^2/(a*d-b*c)^4/g^6*
B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+1/25*e^5*d*i^2/(a*d-b*c)^4/g^6*B
*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*a-1/25*e^5*i^2/(a*d-b*c)^4/g^6*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)^5*c

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Maxima [B]  time = 2.69166, size = 4089, normalized size = 14.55 \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)))/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/1800*B*d^2*i^2*(60*(10*b^2*x^2 + 5*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(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) + (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 - 14
0*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*(60*(5*b
*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(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) + (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)) - 1/300*B*c^2*i^2*((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 + 137*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*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(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) + 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)) - 1/10*(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*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)

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Fricas [B]  time = 0.796824, size = 1661, normalized size = 5.91 \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)))/(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*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*
x^3 + 10*(2*(30*A + B)*b^5*c^3*d^2 - 15*(12*A + B)*a*b^4*c^2*d^3 + 60*(3*A + B)*a^2*b^3*c*d^4 - (60*A + 47*B)*
a^3*b^2*d^5)*i^2*x^2 + 5*(9*(20*A + 3*B)*b^5*c^4*d - 20*(24*A + 5*B)*a*b^4*c^3*d^2 + 120*(3*A + B)*a^2*b^3*c^2
*d^3 - (60*A + 47*B)*a^4*b*d^5)*i^2*x + (72*(5*A + B)*b^5*c^5 - 225*(4*A + B)*a*b^4*c^4*d + 200*(3*A + B)*a^2*
b^3*c^3*d^2 - (60*A + 47*B)*a^5*d^5)*i^2 + 60*(B*b^5*d^5*i^2*x^5 + 5*B*a*b^4*d^5*i^2*x^4 + 10*B*a^2*b^3*d^5*i^
2*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*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*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)*log((b*e*x + a
*e)/(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)

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Sympy [B]  time = 88.4711, size = 1300, normalized size = 4.63 \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*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**6,x)

[Out]

-B*d**5*i**2*log(x + (-B*a**4*d**9*i**2/(a*d - b*c)**3 + 4*B*a**3*b*c*d**8*i**2/(a*d - b*c)**3 - 6*B*a**2*b**2
*c**2*d**7*i**2/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**6*i**2/(a*d - b*c)**3 + B*a*d**6*i**2 - B*b**4*c**4*d**5*i
**2/(a*d - b*c)**3 + B*b*c*d**5*i**2)/(2*B*b*d**6*i**2))/(30*b**3*g**6*(a*d - b*c)**3) + B*d**5*i**2*log(x + (
B*a**4*d**9*i**2/(a*d - b*c)**3 - 4*B*a**3*b*c*d**8*i**2/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**7*i**2/(a*d -
b*c)**3 - 4*B*a*b**3*c**3*d**6*i**2/(a*d - b*c)**3 + B*a*d**6*i**2 + B*b**4*c**4*d**5*i**2/(a*d - b*c)**3 + B*
b*c*d**5*i**2)/(2*B*b*d**6*i**2))/(30*b**3*g**6*(a*d - b*c)**3) - (60*A*a**4*d**4*i**2 + 60*A*a**3*b*c*d**3*i*
*2 + 60*A*a**2*b**2*c**2*d**2*i**2 - 540*A*a*b**3*c**3*d*i**2 + 360*A*b**4*c**4*i**2 + 47*B*a**4*d**4*i**2 + 4
7*B*a**3*b*c*d**3*i**2 + 47*B*a**2*b**2*c**2*d**2*i**2 - 153*B*a*b**3*c**3*d*i**2 + 72*B*b**4*c**4*i**2 + 60*B
*b**4*d**4*i**2*x**4 + x**3*(270*B*a*b**3*d**4*i**2 - 30*B*b**4*c*d**3*i**2) + x**2*(600*A*a**2*b**2*d**4*i**2
- 1200*A*a*b**3*c*d**3*i**2 + 600*A*b**4*c**2*d**2*i**2 + 470*B*a**2*b**2*d**4*i**2 - 130*B*a*b**3*c*d**3*i**
2 + 20*B*b**4*c**2*d**2*i**2) + x*(300*A*a**3*b*d**4*i**2 + 300*A*a**2*b**2*c*d**3*i**2 - 1500*A*a*b**3*c**2*d
**2*i**2 + 900*A*b**4*c**3*d*i**2 + 235*B*a**3*b*d**4*i**2 + 235*B*a**2*b**2*c*d**3*i**2 - 365*B*a*b**3*c**2*d
**2*i**2 + 135*B*b**4*c**3*d*i**2))/(1800*a**7*b**3*d**2*g**6 - 3600*a**6*b**4*c*d*g**6 + 1800*a**5*b**5*c**2*
g**6 + x**5*(1800*a**2*b**8*d**2*g**6 - 3600*a*b**9*c*d*g**6 + 1800*b**10*c**2*g**6) + x**4*(9000*a**3*b**7*d*
*2*g**6 - 18000*a**2*b**8*c*d*g**6 + 9000*a*b**9*c**2*g**6) + x**3*(18000*a**4*b**6*d**2*g**6 - 36000*a**3*b**
7*c*d*g**6 + 18000*a**2*b**8*c**2*g**6) + x**2*(18000*a**5*b**5*d**2*g**6 - 36000*a**4*b**6*c*d*g**6 + 18000*a
**3*b**7*c**2*g**6) + x*(9000*a**6*b**4*d**2*g**6 - 18000*a**5*b**5*c*d*g**6 + 9000*a**4*b**6*c**2*g**6)) + (-
B*a**2*d**2*i**2 - 3*B*a*b*c*d*i**2 - 5*B*a*b*d**2*i**2*x - 6*B*b**2*c**2*i**2 - 15*B*b**2*c*d*i**2*x - 10*B*b
**2*d**2*i**2*x**2)*log(e*(a + b*x)/(c + d*x))/(30*a**5*b**3*g**6 + 150*a**4*b**4*g**6*x + 300*a**3*b**5*g**6*
x**2 + 300*a**2*b**6*g**6*x**3 + 150*a*b**7*g**6*x**4 + 30*b**8*g**6*x**5)

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Giac [B]  time = 1.41879, size = 1211, normalized size = 4.31 \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)))/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

1/30*B*d^5*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
*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*x
^2 + 15*B*b^2*c*d*x + 5*B*a*b*d^2*x + 6*B*b^2*c^2 + 3*B*a*b*c*d + B*a^2*d^2)*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*x^4 - 30*B*b^4*c*d^3*x^3 + 270*B*a*b^3*d^4*x^3 + 600*A*b^4*c^2*d^2*x^2 + 620*B*b^4*c^2*d^2*x^2 - 12
00*A*a*b^3*c*d^3*x^2 - 1330*B*a*b^3*c*d^3*x^2 + 600*A*a^2*b^2*d^4*x^2 + 1070*B*a^2*b^2*d^4*x^2 + 900*A*b^4*c^3
*d*x + 1035*B*b^4*c^3*d*x - 1500*A*a*b^3*c^2*d^2*x - 1865*B*a*b^3*c^2*d^2*x + 300*A*a^2*b^2*c*d^3*x + 535*B*a^
2*b^2*c*d^3*x + 300*A*a^3*b*d^4*x + 535*B*a^3*b*d^4*x + 360*A*b^4*c^4 + 432*B*b^4*c^4 - 540*A*a*b^3*c^3*d - 69
3*B*a*b^3*c^3*d + 60*A*a^2*b^2*c^2*d^2 + 107*B*a^2*b^2*c^2*d^2 + 60*A*a^3*b*c*d^3 + 107*B*a^3*b*c*d^3 + 60*A*a
^4*d^4 + 107*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)