3.1.32 \(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{c i+d i x} \, dx\) [32]

Optimal. Leaf size=198 \[ \frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i}-\frac {(b c-a d) g^2 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^2 i}-\frac {(b c-a d)^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3 i}-\frac {B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i} \]

[Out]

1/2*g^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i-1/2*(-a*d+b*c)*g^2*(b*x+a)*(2*A+B+2*B*ln(e*(b*x+a)/(d*x+c)))
/d^2/i-1/2*(-a*d+b*c)^2*g^2*ln((-a*d+b*c)/b/(d*x+c))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/d^3/i-B*(-a*d+b*c)^2*
g^2*polylog(2,d*(b*x+a)/b/(d*x+c))/d^3/i

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Rubi [A]
time = 0.16, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 2384, 2354, 2438} \begin {gather*} -\frac {B g^2 (b c-a d)^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {g^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{2 d^3 i}-\frac {g^2 (a+b x) (b c-a d) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{2 d^2 i}+\frac {g^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{32 c+32 d x} \, dx &=\int \left (-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{32 d^2}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 (32 c+32 d x)}+\frac {b g (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{32 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{32 d}-\frac {\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{32 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 c+32 d x} \, dx}{d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac {B \int \frac {(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{64 d}-\frac {\left (b B (b c-a d) g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{32 d^2}-\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (32 c+32 d x)}{e (a+b x)} \, dx}{32 d^3}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac {\left (B (b c-a d) g^2\right ) \int \frac {a+b x}{c+d x} \, dx}{64 d}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {1}{c+d x} \, dx}{32 d^2}-\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (32 c+32 d x)}{a+b x} \, dx}{32 d^3 e}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {B (b c-a d)^2 g^2 \log (c+d x)}{32 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac {\left (B (b c-a d) g^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{64 d}-\frac {\left (B (b c-a d)^2 g^2\right ) \int \left (\frac {b e \log (32 c+32 d x)}{a+b x}-\frac {d e \log (32 c+32 d x)}{c+d x}\right ) \, dx}{32 d^3 e}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {b B (b c-a d) g^2 x}{64 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac {\left (b B (b c-a d)^2 g^2\right ) \int \frac {\log (32 c+32 d x)}{a+b x} \, dx}{32 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {\log (32 c+32 d x)}{c+d x} \, dx}{32 d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {b B (b c-a d) g^2 x}{64 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}-\frac {B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \text {Subst}\left (\int \frac {32 \log (x)}{x} \, dx,x,32 c+32 d x\right )}{1024 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {32 d (a+b x)}{-32 b c+32 a d}\right )}{32 c+32 d x} \, dx}{d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {b B (b c-a d) g^2 x}{64 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}-\frac {B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,32 c+32 d x\right )}{32 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-32 b c+32 a d}\right )}{x} \, dx,x,32 c+32 d x\right )}{32 d^3}\\ &=-\frac {A b (b c-a d) g^2 x}{32 d^2}-\frac {b B (b c-a d) g^2 x}{64 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{32 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac {3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}+\frac {B (b c-a d)^2 g^2 \log ^2(32 (c+d x))}{64 d^3}-\frac {B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac {B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{32 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 254, normalized size = 1.28 \begin {gather*} \frac {g^2 \left (-2 A b d (b c-a d) x+2 B d (-b c+a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 B (b c-a d)^2 \log (c+d x)-B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (i (c+d x))-B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (i (c+d x))\right ) \log (i (c+d x))+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^3 i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*(-2*A*b*d*(b*c - a*d)*x + 2*B*d*(-(b*c) + a*d)*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + d^2*(a + b*x)^2*(
A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*B*(b*c - a*d)^2*Log[c + d*x] - B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*L
og[c + d*x]) + 2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[i*(c + d*x)] - B*(b*c - a*d)^2*((2*Log
[(d*(a + b*x))/(-(b*c) + a*d)] - Log[i*(c + d*x)])*Log[i*(c + d*x)] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])
))/(2*d^3*i)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1434\) vs. \(2(192)=384\).
time = 1.44, size = 1435, normalized size = 7.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1435\)
default	\(\text {Expression too large to display}\)	\(1435\)
risch	\(\text {Expression too large to display}\)	\(1998\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^2*e*(a*d-b*c)*(-1/2*A*g^2*e/i*b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a+1/2*A/d*g^2*e/i*b^3/(b*e-(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c+2*A*g^2/i*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a-2*A/d*g^2/i*b^2/(b*e-(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*d)*c+A*g^2/e/i*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a-A/d*g^2/e/i*ln(b*e-(b*e/d+(
a*d-b*c)*e/d/(d*x+c))*d)*b*c+3/2*B*g^2/e/i*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a-3/2*B/d*g^2/e/i*ln(b*e-(b
*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*b*c+1/2*B*g^2/i*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a-1/2*B/d*g^2/i*b^2/(b*
e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c-B*d*g^2/i*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))
/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a+B*g^2/i*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d
*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c+1/2*B*d^2*g^2/e/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d
-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a-1/2*B*d*g^2/e/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))
*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*b*c+2*B*d*g^2/e/i*ln(b*e/d+(a*d-b*c)*
e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a-2*B*g^2/e/i*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*b*c+B*g^2/e/i*dilog(-(-b*
e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)*a-B/d*g^2/e/i*dilog(-(-b*e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)*b*c+B
*g^2/e/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(-b*e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)*a-B/d*g^2/e/i*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))*ln(-(-b*e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)*b*c)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (182) = 364\).
time = 0.35, size = 425, normalized size = 2.15 \begin {gather*} 2 \, A a b g^{2} {\left (-\frac {i \, x}{d} + \frac {i \, c \log \left (d x + c\right )}{d^{2}}\right )} - \frac {1}{2} \, A b^{2} g^{2} {\left (\frac {2 i \, c^{2} \log \left (d x + c\right )}{d^{3}} + \frac {i \, {\left (d x^{2} - 2 \, c x\right )}}{d^{2}}\right )} - \frac {i \, A a^{2} g^{2} \log \left (i \, d x + i \, c\right )}{d} - \frac {i \, {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{3}} - \frac {i \, {\left (5 \, b^{2} c^{2} g^{2} - 8 \, a b c d g^{2} + 2 \, a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )}{2 \, d^{3}} + \frac {-i \, B b^{2} d^{2} g^{2} x^{2} + {\left (i \, b^{2} c^{2} g^{2} - 2 i \, a b c d g^{2} + i \, a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )^{2} + {\left (3 i \, b^{2} c d g^{2} - 5 i \, a b d^{2} g^{2}\right )} B x + {\left (-i \, B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (-i \, b^{2} c d g^{2} + 2 i \, a b d^{2} g^{2}\right )} B x + {\left (2 i \, a b c d g^{2} - 3 i \, a^{2} d^{2} g^{2}\right )} B\right )} \log \left (b x + a\right ) + {\left (i \, B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (i \, b^{2} c d g^{2} - 2 i \, a b d^{2} g^{2}\right )} B x\right )} \log \left (d x + c\right )}{2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*b*g^2*(-I*x/d + I*c*log(d*x + c)/d^2) - 1/2*A*b^2*g^2*(2*I*c^2*log(d*x + c)/d^3 + I*(d*x^2 - 2*c*x)/d^2)
- I*A*a^2*g^2*log(I*d*x + I*c)/d - I*(b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*(log(b*x + a)*log((b*d*x + a
*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/d^3 - 1/2*I*(5*b^2*c^2*g^2 - 8*a*b*c*d*g^2 + 2*a^2
*d^2*g^2)*B*log(d*x + c)/d^3 + 1/2*(-I*B*b^2*d^2*g^2*x^2 + (I*b^2*c^2*g^2 - 2*I*a*b*c*d*g^2 + I*a^2*d^2*g^2)*B
*log(d*x + c)^2 + (3*I*b^2*c*d*g^2 - 5*I*a*b*d^2*g^2)*B*x + (-I*B*b^2*d^2*g^2*x^2 - 2*(-I*b^2*c*d*g^2 + 2*I*a*
b*d^2*g^2)*B*x + (2*I*a*b*c*d*g^2 - 3*I*a^2*d^2*g^2)*B)*log(b*x + a) + (I*B*b^2*d^2*g^2*x^2 - 2*(I*b^2*c*d*g^2
- 2*I*a*b*d^2*g^2)*B*x)*log(d*x + c))/d^3

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-I*A*b^2*g^2*x^2 - 2*I*A*a*b*g^2*x - I*A*a^2*g^2 + (-I*B*b^2*g^2*x^2 - 2*I*B*a*b*g^2*x - I*B*a^2*g^2
)*log((b*x + a)*e/(d*x + c)))/(d*x + c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {g^{2} \left (\int \frac {A a^{2}}{c + d x}\, dx + \int \frac {A b^{2} x^{2}}{c + d x}\, dx + \int \frac {B a^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 A a b x}{c + d x}\, dx + \int \frac {B b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 B a b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g**2*(Integral(A*a**2/(c + d*x), x) + Integral(A*b**2*x**2/(c + d*x), x) + Integral(B*a**2*log(a*e/(c + d*x) +
b*e*x/(c + d*x))/(c + d*x), x) + Integral(2*A*a*b*x/(c + d*x), x) + Integral(B*b**2*x**2*log(a*e/(c + d*x) +
b*e*x/(c + d*x))/(c + d*x), x) + Integral(2*B*a*b*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x))/i

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3913 vs. \(2 (182) = 364\).
time = 67.51, size = 3913, normalized size = 19.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(-2*I*B*b^9*c^5*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 10*I*B*a*b^8*c^4*d*g^2*e^5*log(-b*e + (b*
x*e + a*e)*d/(d*x + c)) - 20*I*B*a^2*b^7*c^3*d^2*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 20*I*B*a^3*b^
6*c^2*d^3*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 10*I*B*a^4*b^5*c*d^4*g^2*e^5*log(-b*e + (b*x*e + a*e
)*d/(d*x + c)) + 2*I*B*a^5*b^4*d^5*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 8*I*(b*x*e + a*e)*B*b^8*c^5
*d*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 40*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*e^4*log(-b
*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 80*I*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*e^4*log(-b*e + (b*x*e + a
*e)*d/(d*x + c))/(d*x + c) - 80*I*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c) + 40*I*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 8*I
*(b*x*e + a*e)*B*a^5*b^3*d^6*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 12*I*(b*x*e + a*e)^2*B*
b^7*c^5*d^2*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 60*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g
^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*I*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^2*e^3*log
(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 120*I*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^2*e^3*log(-b*e + (b
*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 60*I*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^2*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^2 + 12*I*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*
x + c)^2 + 8*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 40*I*
(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 80*I*(b*x*e + a*e)
^3*B*a^2*b^4*c^3*d^5*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 80*I*(b*x*e + a*e)^3*B*a^3*b^
3*c^2*d^6*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 40*I*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^2
*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 8*I*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*e^2*log(-b*e + (b
*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 2*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^4 + 10*I*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c
)^4 - 20*I*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 20*I*(b
*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 10*I*(b*x*e + a*e)^4
*B*a^4*b*c*d^8*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 2*I*(b*x*e + a*e)^4*B*a^5*d^9*g^2*e*l
og(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 8*I*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^2*e^2*log((b*x*e + a*e)
/(d*x + c))/(d*x + c)^3 + 40*I*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^
3 - 80*I*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 80*I*(b*x*e + a*
e)^3*B*a^3*b^3*c^2*d^6*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 40*I*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7
*g^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 8*I*(b*x*e + a*e)^3*B*a^5*b*d^8*g^2*e^2*log((b*x*e + a*e)/
(d*x + c))/(d*x + c)^3 + 2*I*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 10
*I*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 20*I*(b*x*e + a*e)^4*B*a^2
*b^3*c^3*d^6*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 20*I*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^2*e*log
((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 10*I*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^2*e*log((b*x*e + a*e)/(d*x + c))/
(d*x + c)^4 - 2*I*(b*x*e + a*e)^4*B*a^5*d^9*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 2*I*A*b^9*c^5*g^2
*e^5 - I*B*b^9*c^5*g^2*e^5 + 10*I*A*a*b^8*c^4*d*g^2*e^5 + 5*I*B*a*b^8*c^4*d*g^2*e^5 - 20*I*A*a^2*b^7*c^3*d^2*g
^2*e^5 - 10*I*B*a^2*b^7*c^3*d^2*g^2*e^5 + 20*I*A*a^3*b^6*c^2*d^3*g^2*e^5 + 10*I*B*a^3*b^6*c^2*d^3*g^2*e^5 - 10
*I*A*a^4*b^5*c*d^4*g^2*e^5 - 5*I*B*a^4*b^5*c*d^4*g^2*e^5 + 2*I*A*a^5*b^4*d^5*g^2*e^5 + I*B*a^5*b^4*d^5*g^2*e^5
+ 8*I*(b*x*e + a*e)*A*b^8*c^5*d*g^2*e^4/(d*x + c) + 2*I*(b*x*e + a*e)*B*b^8*c^5*d*g^2*e^4/(d*x + c) - 40*I*(b
*x*e + a*e)*A*a*b^7*c^4*d^2*g^2*e^4/(d*x + c) - 10*I*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^2*e^4/(d*x + c) + 80*I*(b
*x*e + a*e)*A*a^2*b^6*c^3*d^3*g^2*e^4/(d*x + c) + 20*I*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^2*e^4/(d*x + c) - 80*
I*(b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g^2*e^4/(d*x + c) - 20*I*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^2*e^4/(d*x + c) +
40*I*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g^2*e^4/(d*x + c) + 10*I*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^2*e^4/(d*x + c) -
8*I*(b*x*e + a*e)*A*a^5*b^3*d^6*g^2*e^4/(d*x + c) - 2*I*(b*x*e + a*e)*B*a^5*b^3*d^6*g^2*e^4/(d*x + c) - 12*I*
(b*x*e + a*e)^2*A*b^7*c^5*d^2*g^2*e^3/(d*x + c)^2 + I*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g^2*e^3/(d*x + c)^2 + 60*I
*(b*x*e + a*e)^2*A*a*b^6*c^4*d^3*g^2*e^3/(d*x + c)^2 - 5*I*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^2*e^3/(d*x + c)^2
- 120*I*(b*x*e + a*e)^2*A*a^2*b^5*c^3*d^4*g^2*...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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