3.1.40 \(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^2} \, dx\) [40]

Optimal. Leaf size=260 \[ -\frac {2 B (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {(2 A+B) (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i^2 (c+d x)}+\frac {b (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}+\frac {2 b B (b c-a d) g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2562, 2384, 45, 2393, 2332, 2354, 2438} \begin {gather*} \frac {2 b B g^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {b g^2 (b c-a d) \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+B\right )}{d^3 i^2}+\frac {g^2 (2 A+B) (a+b x) (b c-a d)}{d^2 i^2 (c+d x)}+\frac {g^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (c+d x)}+\frac {2 B g^2 (a+b x) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}-\frac {2 B g^2 (a+b x) (b c-a d)}{d^2 i^2 (c+d x)} \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)^2,x]

[Out]

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

Rule 45

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])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 )}{(40 c+40 d x)^2} \, dx &=\int \left (\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1600 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 )}{1600 d^2 (c+d x)^2}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{800 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{1600 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{800 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 )}{(c+d x)^2} \, dx}{1600 d^2}\\ &=\frac {A b^2 g^2 x}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac {\left (b^2 B g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{1600 d^2}+\frac {\left (b B (b c-a d) 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 (c+d x)}{e (a+b x)} \, dx}{800 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1600 d^3}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \frac {1}{c+d x} \, dx}{1600 d^2}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1600 d^3}+\frac {\left (b B (b c-a d) 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 (c+d x)}{a+b x} \, dx}{800 d^3 e}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 \log (c+d x)}{1600 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1600 d^3}+\frac {\left (b B (b c-a d) g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{800 d^3 e}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 \log (c+d x)}{800 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}+\frac {\left (b^2 B (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{800 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{800 d^2}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac {b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{800 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{800 d^2}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac {b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac {b B (b c-a d) g^2 \log ^2(c+d x)}{1600 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{800 d^3}\\ &=\frac {A b^2 g^2 x}{1600 d^2}+\frac {B (b c-a d)^2 g^2}{1600 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 \log (a+b x)}{1600 d^3}+\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1600 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 )}{1600 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 \log (c+d x)}{800 d^3}+\frac {b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{800 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{800 d^3}-\frac {b B (b c-a d) g^2 \log ^2(c+d x)}{1600 d^3}+\frac {b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{800 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 239, normalized size = 0.92 \begin {gather*} \frac {g^2 \left (A b^2 d x+\frac {B (b c-a d)^2}{c+d x}+b B (b c-a d) \log (a+b x)+b B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-2 b B (b c-a d) \log (c+d x)-2 b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+b B (b c-a d) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3 i^2} \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)^2,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(260)=520\).
time = 1.38, size = 559, normalized size = 2.15

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {g^{2} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}+\frac {2 g^{2} A b \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d e \,i^{2}}+\frac {g^{2} A \,b^{2}}{d \,i^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g^{2} B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}-\frac {g^{2} B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}+\frac {2 g^{2} B b \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d e \,i^{2}}+\frac {2 g^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d e \,i^{2}}+\frac {g^{2} B b \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d e \,i^{2}}+\frac {g^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\)	\(559\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {g^{2} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}+\frac {2 g^{2} A b \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d e \,i^{2}}+\frac {g^{2} A \,b^{2}}{d \,i^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g^{2} B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}-\frac {g^{2} B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2}}+\frac {2 g^{2} B b \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d e \,i^{2}}+\frac {2 g^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d e \,i^{2}}+\frac {g^{2} B b \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d e \,i^{2}}+\frac {g^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\)	\(559\)
risch	\(\text {Expression too large to display}\)	\(2042\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (247) = 494\).
time = 0.38, size = 725, normalized size = 2.79 \begin {gather*} A b^{2} {\left (\frac {c^{2}}{d^{4} x + c d^{3}} - \frac {x}{d^{2}} + \frac {2 \, c \log \left (d x + c\right )}{d^{3}}\right )} g^{2} - B a^{2} g^{2} {\left (\frac {b \log \left (b x + a\right )}{b c d - a d^{2}} - \frac {b \log \left (d x + c\right )}{b c d - a d^{2}} - \frac {\log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} x + c d} + \frac {1}{d^{2} x + c d}\right )} - 2 \, A a b g^{2} {\left (\frac {c}{d^{3} x + c d^{2}} + \frac {\log \left (d x + c\right )}{d^{2}}\right )} + \frac {A a^{2} g^{2}}{d^{2} x + c d} + \frac {{\left (4 \, b^{3} c^{2} g^{2} - 7 \, a b^{2} c d g^{2} + 2 \, a^{2} b d^{2} g^{2}\right )} B \log \left (d x + c\right )}{b c d^{3} - a d^{4}} - \frac {{\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (b^{3} c^{2} d g^{2} - a b^{2} c d^{2} g^{2}\right )} B x + {\left ({\left (b^{3} c^{2} d g^{2} - 2 \, a b^{2} c d^{2} g^{2} + a^{2} b d^{3} g^{2}\right )} B x + {\left (b^{3} c^{3} g^{2} - 2 \, a b^{2} c^{2} d g^{2} + a^{2} b c d^{2} g^{2}\right )} B\right )} \log \left (d x + c\right )^{2} + {\left ({\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (2 \, b^{3} c^{2} d g^{2} - 2 \, a b^{2} c d^{2} g^{2} - a^{2} b d^{3} g^{2}\right )} B x + {\left (2 \, a b^{2} c^{2} d g^{2} - 3 \, a^{2} b c d^{2} g^{2}\right )} B\right )} \log \left (b x + a\right ) - {\left ({\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (b^{3} c^{2} d g^{2} - a b^{2} c d^{2} g^{2}\right )} B x - {\left (b^{3} c^{3} g^{2} - 3 \, a b^{2} c^{2} d g^{2} + 2 \, a^{2} b c d^{2} g^{2}\right )} B\right )} \log \left (d x + c\right )}{b c^{2} d^{3} - a c d^{4} + {\left (b c d^{4} - a d^{5}\right )} x} + \frac {2 \, {\left (b^{2} c g^{2} - a b d 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}} \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)^2,x, algorithm="maxima")

[Out]

A*b^2*(c^2/(d^4*x + c*d^3) - x/d^2 + 2*c*log(d*x + c)/d^3)*g^2 - B*a^2*g^2*(b*log(b*x + a)/(b*c*d - a*d^2) - b
*log(d*x + c)/(b*c*d - a*d^2) - log(b*x*e/(d*x + c) + a*e/(d*x + c))/(d^2*x + c*d) + 1/(d^2*x + c*d)) - 2*A*a*
b*g^2*(c/(d^3*x + c*d^2) + log(d*x + c)/d^2) + A*a^2*g^2/(d^2*x + c*d) + (4*b^3*c^2*g^2 - 7*a*b^2*c*d*g^2 + 2*
a^2*b*d^2*g^2)*B*log(d*x + c)/(b*c*d^3 - a*d^4) - ((b^3*c*d^2*g^2 - a*b^2*d^3*g^2)*B*x^2 + (b^3*c^2*d*g^2 - a*
b^2*c*d^2*g^2)*B*x + ((b^3*c^2*d*g^2 - 2*a*b^2*c*d^2*g^2 + a^2*b*d^3*g^2)*B*x + (b^3*c^3*g^2 - 2*a*b^2*c^2*d*g
^2 + a^2*b*c*d^2*g^2)*B)*log(d*x + c)^2 + ((b^3*c*d^2*g^2 - a*b^2*d^3*g^2)*B*x^2 + (2*b^3*c^2*d*g^2 - 2*a*b^2*
c*d^2*g^2 - a^2*b*d^3*g^2)*B*x + (2*a*b^2*c^2*d*g^2 - 3*a^2*b*c*d^2*g^2)*B)*log(b*x + a) - ((b^3*c*d^2*g^2 - a
*b^2*d^3*g^2)*B*x^2 + (b^3*c^2*d*g^2 - a*b^2*c*d^2*g^2)*B*x - (b^3*c^3*g^2 - 3*a*b^2*c^2*d*g^2 + 2*a^2*b*c*d^2
*g^2)*B)*log(d*x + c))/(b*c^2*d^3 - a*c*d^4 + (b*c*d^4 - a*d^5)*x) + 2*(b^2*c*g^2 - a*b*d*g^2)*(log(b*x + a)*l
og((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2452 vs. \(2 (247) = 494\).
time = 66.13, size = 2452, normalized size = 9.43 \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)^2,x, algorithm="giac")

[Out]

-1/6*(2*B*b^7*c^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*B*a*b^6*c^3*d*g^2*e^4*log(-b*e + (b*x*e +
a*e)*d/(d*x + c)) + 12*B*a^2*b^5*c^2*d^2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*B*a^3*b^4*c*d^3*g^2
*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 2*B*a^4*b^3*d^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6
*(b*x*e + a*e)*B*b^6*c^4*d*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a*b^5*
c^3*d^2*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 36*(b*x*e + a*e)*B*a^2*b^4*c^2*d^3*g^2*e^3*l
og(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a^3*b^3*c*d^4*g^2*e^3*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c) - 6*(b*x*e + a*e)*B*a^4*b^2*d^5*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c) + 6*(b*x*e + a*e)^2*B*b^5*c^4*d^2*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e
+ a*e)^2*B*a*b^4*c^3*d^3*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 36*(b*x*e + a*e)^2*B*a^2*
b^3*c^2*d^4*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e + a*e)^2*B*a^3*b^2*c*d^5*g^2
*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 6*(b*x*e + a*e)^2*B*a^4*b*d^6*g^2*e^2*log(-b*e + (b*x
*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 2*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c
))/(d*x + c)^3 + 8*(b*x*e + a*e)^3*B*a*b^3*c^3*d^4*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 1
2*(b*x*e + a*e)^3*B*a^2*b^2*c^2*d^5*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 8*(b*x*e + a*e)^
3*B*a^3*b*c*d^6*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 2*(b*x*e + a*e)^3*B*a^4*d^7*g^2*e*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 2*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g^2*e*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^3 - 8*(b*x*e + a*e)^3*B*a*b^3*c^3*d^4*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 12*(b*
x*e + a*e)^3*B*a^2*b^2*c^2*d^5*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 8*(b*x*e + a*e)^3*B*a^3*b*c*d^
6*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 2*(b*x*e + a*e)^3*B*a^4*d^7*g^2*e*log((b*x*e + a*e)/(d*x +
c))/(d*x + c)^3 + 2*A*b^7*c^4*g^2*e^4 + 3*B*b^7*c^4*g^2*e^4 - 8*A*a*b^6*c^3*d*g^2*e^4 - 12*B*a*b^6*c^3*d*g^2*e
^4 + 12*A*a^2*b^5*c^2*d^2*g^2*e^4 + 18*B*a^2*b^5*c^2*d^2*g^2*e^4 - 8*A*a^3*b^4*c*d^3*g^2*e^4 - 12*B*a^3*b^4*c*
d^3*g^2*e^4 + 2*A*a^4*b^3*d^4*g^2*e^4 + 3*B*a^4*b^3*d^4*g^2*e^4 - 6*(b*x*e + a*e)*A*b^6*c^4*d*g^2*e^3/(d*x + c
) - 7*(b*x*e + a*e)*B*b^6*c^4*d*g^2*e^3/(d*x + c) + 24*(b*x*e + a*e)*A*a*b^5*c^3*d^2*g^2*e^3/(d*x + c) + 28*(b
*x*e + a*e)*B*a*b^5*c^3*d^2*g^2*e^3/(d*x + c) - 36*(b*x*e + a*e)*A*a^2*b^4*c^2*d^3*g^2*e^3/(d*x + c) - 42*(b*x
*e + a*e)*B*a^2*b^4*c^2*d^3*g^2*e^3/(d*x + c) + 24*(b*x*e + a*e)*A*a^3*b^3*c*d^4*g^2*e^3/(d*x + c) + 28*(b*x*e
+ a*e)*B*a^3*b^3*c*d^4*g^2*e^3/(d*x + c) - 6*(b*x*e + a*e)*A*a^4*b^2*d^5*g^2*e^3/(d*x + c) - 7*(b*x*e + a*e)*
B*a^4*b^2*d^5*g^2*e^3/(d*x + c) + 6*(b*x*e + a*e)^2*A*b^5*c^4*d^2*g^2*e^2/(d*x + c)^2 + 4*(b*x*e + a*e)^2*B*b^
5*c^4*d^2*g^2*e^2/(d*x + c)^2 - 24*(b*x*e + a*e)^2*A*a*b^4*c^3*d^3*g^2*e^2/(d*x + c)^2 - 16*(b*x*e + a*e)^2*B*
a*b^4*c^3*d^3*g^2*e^2/(d*x + c)^2 + 36*(b*x*e + a*e)^2*A*a^2*b^3*c^2*d^4*g^2*e^2/(d*x + c)^2 + 24*(b*x*e + a*e
)^2*B*a^2*b^3*c^2*d^4*g^2*e^2/(d*x + c)^2 - 24*(b*x*e + a*e)^2*A*a^3*b^2*c*d^5*g^2*e^2/(d*x + c)^2 - 16*(b*x*e
+ a*e)^2*B*a^3*b^2*c*d^5*g^2*e^2/(d*x + c)^2 + 6*(b*x*e + a*e)^2*A*a^4*b*d^6*g^2*e^2/(d*x + c)^2 + 4*(b*x*e +
a*e)^2*B*a^4*b*d^6*g^2*e^2/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)
))^2/(b^4*d^3*e^3 - 3*(b*x*e + a*e)*b^3*d^4*e^2/(d*x + c) + 3*(b*x*e + a*e)^2*b^2*d^5*e/(d*x + c)^2 - (b*x*e +
a*e)^3*b*d^6/(d*x + c)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \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 )}{{\left (c\,i+d\,i\,x\right )}^2} \,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)^2,x)

[Out]

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

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