3.18 \(\int \frac{(c i+d i x)^2 (A+B \log (\frac{e (a+b x)}{c+d x}))}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=181 \[ -\frac{b i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^2}+\frac{d i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^2}-\frac{b B i^2 (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^2}+\frac{B d i^2 (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^2} \]

[Out]

(B*d*i^2*(c + d*x)^3)/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B*i^2*(c + d*x)^4)/(16*(b*c - a*d)^2*g^5*(a + b*x
)^4) + (d*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c
+ d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)

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Rubi [A]  time = 0.571379, antiderivative size = 325, normalized size of antiderivative = 1.8, 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} \[ -\frac{d^2 i^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^3 g^5 (a+b x)^2}-\frac{2 d i^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^3 g^5 (a+b x)^3}-\frac{i^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^3 g^5 (a+b x)^4}+\frac{B d^3 i^2}{12 b^3 g^5 (a+b x) (b c-a d)}+\frac{B d^4 i^2 \log (a+b x)}{12 b^3 g^5 (b c-a d)^2}-\frac{B d^4 i^2 \log (c+d x)}{12 b^3 g^5 (b c-a d)^2}-\frac{5 B d i^2 (b c-a d)}{36 b^3 g^5 (a+b x)^3}-\frac{B i^2 (b c-a d)^2}{16 b^3 g^5 (a+b x)^4}-\frac{B d^2 i^2}{24 b^3 g^5 (a+b x)^2} \]

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)^5,x]

[Out]

-(B*(b*c - a*d)^2*i^2)/(16*b^3*g^5*(a + b*x)^4) - (5*B*d*(b*c - a*d)*i^2)/(36*b^3*g^5*(a + b*x)^3) - (B*d^2*i^
2)/(24*b^3*g^5*(a + b*x)^2) + (B*d^3*i^2)/(12*b^3*(b*c - a*d)*g^5*(a + b*x)) + (B*d^4*i^2*Log[a + b*x])/(12*b^
3*(b*c - a*d)^2*g^5) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^3*g^5*(a + b*x)^4) - (2*d
*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B*Log[(e*(a + b
*x))/(c + d*x)]))/(2*b^3*g^5*(a + b*x)^2) - (B*d^4*i^2*Log[c + d*x])/(12*b^3*(b*c - a*d)^2*g^5)

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{(18 c+18 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac{324 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^5}+\frac{648 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^4}+\frac{324 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac{\left (324 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac{(648 d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{\left (324 (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac{81 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{216 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{162 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{\left (162 B d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{(216 B d (b c-a d)) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (81 B (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{81 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{216 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{162 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{\left (162 B d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (216 B d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (81 B (b c-a d)^3\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{81 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{216 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{162 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{\left (162 B d^2 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac{\left (216 B d (b c-a d)^2\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^5}+\frac{\left (81 B (b c-a d)^3\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^5}\\ &=-\frac{81 B (b c-a d)^2}{4 b^3 g^5 (a+b x)^4}-\frac{45 B d (b c-a d)}{b^3 g^5 (a+b x)^3}-\frac{27 B d^2}{2 b^3 g^5 (a+b x)^2}+\frac{27 B d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac{27 B d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{81 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{216 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{162 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}-\frac{27 B d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}\\ \end{align*}

Mathematica [B]  time = 0.38713, size = 454, normalized size = 2.51 \[ -\frac{i^2 \left (72 a^2 A b^2 d^4 x^2+48 a^3 A b d^4 x+12 a^4 A d^4+12 B (b c-a d)^2 \left (a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )+72 a^2 b^2 B d^4 x^2 \log (c+d x)+42 a^2 b^2 B d^4 x^2+48 a^3 b B d^4 x \log (c+d x)+28 a^3 b B d^4 x+12 a^4 B d^4 \log (c+d x)+7 a^4 B d^4-144 a A b^3 c^2 d^2 x-48 a A b^3 c^3 d-144 a A b^3 c d^3 x^2-48 a b^3 B c^2 d^2 x-16 a b^3 B c^3 d-48 a b^3 B c d^3 x^2+48 a b^3 B d^4 x^3 \log (c+d x)+12 a b^3 B d^4 x^3-12 B d^4 (a+b x)^4 \log (a+b x)+72 A b^4 c^2 d^2 x^2+96 A b^4 c^3 d x+36 A b^4 c^4+6 b^4 B c^2 d^2 x^2+20 b^4 B c^3 d x+9 b^4 B c^4-12 b^4 B c d^3 x^3+12 b^4 B d^4 x^4 \log (c+d x)\right )}{144 b^3 g^5 (a+b x)^4 (b c-a d)^2} \]

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)^5,x]

[Out]

-(i^2*(36*A*b^4*c^4 + 9*b^4*B*c^4 - 48*a*A*b^3*c^3*d - 16*a*b^3*B*c^3*d + 12*a^4*A*d^4 + 7*a^4*B*d^4 + 96*A*b^
4*c^3*d*x + 20*b^4*B*c^3*d*x - 144*a*A*b^3*c^2*d^2*x - 48*a*b^3*B*c^2*d^2*x + 48*a^3*A*b*d^4*x + 28*a^3*b*B*d^
4*x + 72*A*b^4*c^2*d^2*x^2 + 6*b^4*B*c^2*d^2*x^2 - 144*a*A*b^3*c*d^3*x^2 - 48*a*b^3*B*c*d^3*x^2 + 72*a^2*A*b^2
*d^4*x^2 + 42*a^2*b^2*B*d^4*x^2 - 12*b^4*B*c*d^3*x^3 + 12*a*b^3*B*d^4*x^3 - 12*B*d^4*(a + b*x)^4*Log[a + b*x]
+ 12*B*(b*c - a*d)^2*(a^2*d^2 + 2*a*b*d*(c + 2*d*x) + b^2*(3*c^2 + 8*c*d*x + 6*d^2*x^2))*Log[(e*(a + b*x))/(c
+ d*x)] + 12*a^4*B*d^4*Log[c + d*x] + 48*a^3*b*B*d^4*x*Log[c + d*x] + 72*a^2*b^2*B*d^4*x^2*Log[c + d*x] + 48*a
*b^3*B*d^4*x^3*Log[c + d*x] + 12*b^4*B*d^4*x^4*Log[c + d*x]))/(144*b^3*(b*c - a*d)^2*g^5*(a + b*x)^4)

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Maple [B]  time = 0.05, size = 828, normalized size = 4.6 \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)^5,x)

[Out]

1/3*e^3*d^2*i^2/(a*d-b*c)^3/g^5*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/3*e^3*d*i^2/(a*d-b*c)^3/g^5*A/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/4*e^4*d*i^2/(a*d-b*c)^3/g^5*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*
a+1/4*e^4*i^2/(a*d-b*c)^3/g^5*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+1/3*e^3*d^2*i^2/(a*d-b*c)^3/g^5*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*i^2/(a*d-b*c)^3/g^5*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^2*i^2/(a*d-b*c)^3/g^5*B/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/9*e^3*d*i^2/(a*d-b*c)^3/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/4*
e^4*d*i^2/(a*d-b*c)^3/g^5*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+1/4*e^4*
i^2/(a*d-b*c)^3/g^5*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-1/16*e^4*d*i
^2/(a*d-b*c)^3/g^5*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/16*e^4*i^2/(a*d-b*c)^3/g^5*B*b^2/(b*e/d+e/(d*
x+c)*a-e/d/(d*x+c)*b*c)^4*c

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Maxima [B]  time = 1.78848, size = 2994, normalized size = 16.54 \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)^5,x, algorithm="maxima")

[Out]

-1/144*B*d^2*i^2*(12*(6*b^2*x^2 + 4*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g
^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*
d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*
b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^1
0*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*
d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^
3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^
4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*
d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d
*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/72*B*c*d*i^2*
(12*(4*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^
3*b^3*g^5*x + a^4*b^2*g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 -
a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^
2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c
^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c
*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b
^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4
- 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x +
c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/48*B*c^2*i^2*((12*
b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(
b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^
4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d
+ 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*
g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*e*x/(d*x + c) + a*e/(d*x
+ c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*log(b*x + a)
/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5
*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/6*(4*b*x + a)*A*c*d*i^2/(b^6
*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(6*b^2*x^2 + 4*a*b*x +
a^2)*A*d^2*i^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) - 1/4*A*c^2
*i^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 0.813267, size = 1046, normalized size = 5.78 \begin{align*} \frac{12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{2} x^{3} - 6 \,{\left ({\left (12 \, A + B\right )} b^{4} c^{2} d^{2} - 8 \,{\left (3 \, A + B\right )} a b^{3} c d^{3} +{\left (12 \, A + 7 \, B\right )} a^{2} b^{2} d^{4}\right )} i^{2} x^{2} - 4 \,{\left ({\left (24 \, A + 5 \, B\right )} b^{4} c^{3} d - 12 \,{\left (3 \, A + B\right )} a b^{3} c^{2} d^{2} +{\left (12 \, A + 7 \, B\right )} a^{3} b d^{4}\right )} i^{2} x -{\left (9 \,{\left (4 \, A + B\right )} b^{4} c^{4} - 16 \,{\left (3 \, A + B\right )} a b^{3} c^{3} d +{\left (12 \, A + 7 \, B\right )} a^{4} d^{4}\right )} i^{2} + 12 \,{\left (B b^{4} d^{4} i^{2} x^{4} + 4 \, B a b^{3} d^{4} i^{2} x^{3} - 6 \,{\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3}\right )} i^{2} x^{2} - 4 \,{\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2}\right )} i^{2} x -{\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{144 \,{\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x +{\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \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)^5,x, algorithm="fricas")

[Out]

1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i^2*x^3 - 6*((12*A + B)*b^4*c^2*d^2 - 8*(3*A + B)*a*b^3*c*d^3 + (12*A +
7*B)*a^2*b^2*d^4)*i^2*x^2 - 4*((24*A + 5*B)*b^4*c^3*d - 12*(3*A + B)*a*b^3*c^2*d^2 + (12*A + 7*B)*a^3*b*d^4)*i
^2*x - (9*(4*A + B)*b^4*c^4 - 16*(3*A + B)*a*b^3*c^3*d + (12*A + 7*B)*a^4*d^4)*i^2 + 12*(B*b^4*d^4*i^2*x^4 + 4
*B*a*b^3*d^4*i^2*x^3 - 6*(B*b^4*c^2*d^2 - 2*B*a*b^3*c*d^3)*i^2*x^2 - 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2)*i^2
*x - (3*B*b^4*c^4 - 4*B*a*b^3*c^3*d)*i^2)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)
*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)
*g^5*x^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*g
^5)

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Sympy [B]  time = 48.2751, size = 928, normalized size = 5.13 \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)**5,x)

[Out]

-B*d**4*i**2*log(x + (-B*a**3*d**7*i**2/(a*d - b*c)**2 + 3*B*a**2*b*c*d**6*i**2/(a*d - b*c)**2 - 3*B*a*b**2*c*
*2*d**5*i**2/(a*d - b*c)**2 + B*a*d**5*i**2 + B*b**3*c**3*d**4*i**2/(a*d - b*c)**2 + B*b*c*d**4*i**2)/(2*B*b*d
**5*i**2))/(12*b**3*g**5*(a*d - b*c)**2) + B*d**4*i**2*log(x + (B*a**3*d**7*i**2/(a*d - b*c)**2 - 3*B*a**2*b*c
*d**6*i**2/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**5*i**2/(a*d - b*c)**2 + B*a*d**5*i**2 - B*b**3*c**3*d**4*i**2/(
a*d - b*c)**2 + B*b*c*d**4*i**2)/(2*B*b*d**5*i**2))/(12*b**3*g**5*(a*d - b*c)**2) - (12*A*a**3*d**3*i**2 + 12*
A*a**2*b*c*d**2*i**2 + 12*A*a*b**2*c**2*d*i**2 - 36*A*b**3*c**3*i**2 + 7*B*a**3*d**3*i**2 + 7*B*a**2*b*c*d**2*
i**2 + 7*B*a*b**2*c**2*d*i**2 - 9*B*b**3*c**3*i**2 + 12*B*b**3*d**3*i**2*x**3 + x**2*(72*A*a*b**2*d**3*i**2 -
72*A*b**3*c*d**2*i**2 + 42*B*a*b**2*d**3*i**2 - 6*B*b**3*c*d**2*i**2) + x*(48*A*a**2*b*d**3*i**2 + 48*A*a*b**2
*c*d**2*i**2 - 96*A*b**3*c**2*d*i**2 + 28*B*a**2*b*d**3*i**2 + 28*B*a*b**2*c*d**2*i**2 - 20*B*b**3*c**2*d*i**2
))/(144*a**5*b**3*d*g**5 - 144*a**4*b**4*c*g**5 + x**4*(144*a*b**7*d*g**5 - 144*b**8*c*g**5) + x**3*(576*a**2*
b**6*d*g**5 - 576*a*b**7*c*g**5) + x**2*(864*a**3*b**5*d*g**5 - 864*a**2*b**6*c*g**5) + x*(576*a**4*b**4*d*g**
5 - 576*a**3*b**5*c*g**5)) + (-B*a**2*d**2*i**2 - 2*B*a*b*c*d*i**2 - 4*B*a*b*d**2*i**2*x - 3*B*b**2*c**2*i**2
- 8*B*b**2*c*d*i**2*x - 6*B*b**2*d**2*i**2*x**2)*log(e*(a + b*x)/(c + d*x))/(12*a**4*b**3*g**5 + 48*a**3*b**4*
g**5*x + 72*a**2*b**5*g**5*x**2 + 48*a*b**6*g**5*x**3 + 12*b**7*g**5*x**4)

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Giac [B]  time = 1.715, size = 783, normalized size = 4.33 \begin{align*} -\frac{B d^{4} \log \left (b x + a\right )}{12 \,{\left (b^{5} c^{2} g^{5} - 2 \, a b^{4} c d g^{5} + a^{2} b^{3} d^{2} g^{5}\right )}} + \frac{B d^{4} \log \left (d x + c\right )}{12 \,{\left (b^{5} c^{2} g^{5} - 2 \, a b^{4} c d g^{5} + a^{2} b^{3} d^{2} g^{5}\right )}} + \frac{{\left (6 \, B b^{2} d^{2} x^{2} + 8 \, B b^{2} c d x + 4 \, B a b d^{2} x + 3 \, B b^{2} c^{2} + 2 \, B a b c d + B a^{2} d^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{12 \,{\left (b^{7} g^{5} x^{4} + 4 \, a b^{6} g^{5} x^{3} + 6 \, a^{2} b^{5} g^{5} x^{2} + 4 \, a^{3} b^{4} g^{5} x + a^{4} b^{3} g^{5}\right )}} - \frac{12 \, B b^{3} d^{3} x^{3} - 72 \, A b^{3} c d^{2} x^{2} - 78 \, B b^{3} c d^{2} x^{2} + 72 \, A a b^{2} d^{3} x^{2} + 114 \, B a b^{2} d^{3} x^{2} - 96 \, A b^{3} c^{2} d x - 116 \, B b^{3} c^{2} d x + 48 \, A a b^{2} c d^{2} x + 76 \, B a b^{2} c d^{2} x + 48 \, A a^{2} b d^{3} x + 76 \, B a^{2} b d^{3} x - 36 \, A b^{3} c^{3} - 45 \, B b^{3} c^{3} + 12 \, A a b^{2} c^{2} d + 19 \, B a b^{2} c^{2} d + 12 \, A a^{2} b c d^{2} + 19 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 19 \, B a^{3} d^{3}}{144 \,{\left (b^{8} c g^{5} x^{4} - a b^{7} d g^{5} x^{4} + 4 \, a b^{7} c g^{5} x^{3} - 4 \, a^{2} b^{6} d g^{5} x^{3} + 6 \, a^{2} b^{6} c g^{5} x^{2} - 6 \, a^{3} b^{5} d g^{5} x^{2} + 4 \, a^{3} b^{5} c g^{5} x - 4 \, a^{4} b^{4} d g^{5} x + a^{4} b^{4} c g^{5} - a^{5} b^{3} d g^{5}\right )}} \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)^5,x, algorithm="giac")

[Out]

-1/12*B*d^4*log(b*x + a)/(b^5*c^2*g^5 - 2*a*b^4*c*d*g^5 + a^2*b^3*d^2*g^5) + 1/12*B*d^4*log(d*x + c)/(b^5*c^2*
g^5 - 2*a*b^4*c*d*g^5 + a^2*b^3*d^2*g^5) + 1/12*(6*B*b^2*d^2*x^2 + 8*B*b^2*c*d*x + 4*B*a*b*d^2*x + 3*B*b^2*c^2
+ 2*B*a*b*c*d + B*a^2*d^2)*log((b*x + a)/(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^
3*b^4*g^5*x + a^4*b^3*g^5) - 1/144*(12*B*b^3*d^3*x^3 - 72*A*b^3*c*d^2*x^2 - 78*B*b^3*c*d^2*x^2 + 72*A*a*b^2*d^
3*x^2 + 114*B*a*b^2*d^3*x^2 - 96*A*b^3*c^2*d*x - 116*B*b^3*c^2*d*x + 48*A*a*b^2*c*d^2*x + 76*B*a*b^2*c*d^2*x +
48*A*a^2*b*d^3*x + 76*B*a^2*b*d^3*x - 36*A*b^3*c^3 - 45*B*b^3*c^3 + 12*A*a*b^2*c^2*d + 19*B*a*b^2*c^2*d + 12*
A*a^2*b*c*d^2 + 19*B*a^2*b*c*d^2 + 12*A*a^3*d^3 + 19*B*a^3*d^3)/(b^8*c*g^5*x^4 - a*b^7*d*g^5*x^4 + 4*a*b^7*c*g
^5*x^3 - 4*a^2*b^6*d*g^5*x^3 + 6*a^2*b^6*c*g^5*x^2 - 6*a^3*b^5*d*g^5*x^2 + 4*a^3*b^5*c*g^5*x - 4*a^4*b^4*d*g^5
*x + a^4*b^4*c*g^5 - a^5*b^3*d*g^5)