3.1.48 \(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^3} \, dx\) [48]

Optimal. Leaf size=251 \[ \frac {B g^2 (a+b x)^2}{4 d i^3 (c+d x)^2}-\frac {A b g^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {b B g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^3 (c+d x)}-\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i^3 (c+d x)^2}-\frac {b^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^3}-\frac {b^2 B g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3} \]

[Out]

1/4*B*g^2*(b*x+a)^2/d/i^3/(d*x+c)^2-A*b*g^2*(b*x+a)/d^2/i^3/(d*x+c)+b*B*g^2*(b*x+a)/d^2/i^3/(d*x+c)-b*B*g^2*(b
*x+a)*ln(e*(b*x+a)/(d*x+c))/d^2/i^3/(d*x+c)-1/2*g^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i^3/(d*x+c)^2-b^2*
g^2*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3/i^3-b^2*B*g^2*polylog(2,d*(b*x+a)/b/(d*x+c))/d^3/
i^3

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Rubi [A]
time = 0.16, antiderivative size = 251, 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, 45, 2393, 2332, 2341, 2354, 2438} \begin {gather*} -\frac {b^2 B g^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}-\frac {b^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^3}-\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^3 (c+d x)^2}-\frac {A b g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^3 (c+d x)}+\frac {b B g^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {B g^2 (a+b x)^2}{4 d i^3 (c+d x)^2} \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)^3,x]

[Out]

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

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 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 )}{(48 c+48 d x)^3} \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{110592 d^2 (c+d x)^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^2 (c+d x)^2}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{110592 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{110592 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)^2} \, dx}{55296 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)^3} \, dx}{110592 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b^2 B 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}{110592 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{55296 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)^3} \, dx}{221184 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b B (b c-a d)^2 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{55296 d^3}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{221184 d^3}-\frac {\left (b^2 B 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}{110592 d^3 e}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b B (b c-a d)^2 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}{55296 d^3}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{221184 d^3}-\frac {\left (b^2 B 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}{110592 d^3 e}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b^3 B g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{110592 d^2}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{110592 d^2}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 B g^2 \log ^2(c+d x)}{221184 d^3}+\frac {\left (b^2 B 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 )}{110592 d^3}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 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 )}{221184 d^3 (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 )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 B g^2 \log ^2(c+d x)}{221184 d^3}-\frac {b^2 B g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{110592 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 245, normalized size = 0.98 \begin {gather*} \frac {g^2 \left (\frac {B (b c-a d)^2}{(c+d x)^2}-\frac {6 b B (b c-a d)}{c+d x}-6 b^2 B \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}+\frac {8 b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+6 b^2 B \log (c+d x)+4 b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 b^2 B \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 )}{4 d^3 i^3} \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)^3,x]

[Out]

(g^2*((B*(b*c - a*d)^2)/(c + d*x)^2 - (6*b*B*(b*c - a*d))/(c + d*x) - 6*b^2*B*Log[a + b*x] - (2*(b*c - a*d)^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 + (8*b*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c
+ d*x) + 6*b^2*B*Log[c + d*x] + 4*b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 2*b^2*B*((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)])))/(4*d^3*i^3
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(605\) vs. \(2(247)=494\).
time = 1.43, size = 606, normalized size = 2.41

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {g^{2} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}+\frac {g^{2} d A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right ) e^{3} i^{3}}+\frac {g^{2} A \,b^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d \left (a d -c b \right ) e \,i^{3}}+\frac {g^{2} d B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a d -c b \right ) e^{3} i^{3}}-\frac {g^{2} d B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (a d -c b \right ) e^{3} i^{3}}+\frac {g^{2} B b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}-\frac {g^{2} B b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}+\frac {g^{2} B \,b^{2} \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d \left (a d -c b \right ) e \,i^{3}}+\frac {g^{2} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d \left (a d -c b \right ) e \,i^{3}}\right )}{d^{2}}\)	\(606\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {g^{2} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}+\frac {g^{2} d A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right ) e^{3} i^{3}}+\frac {g^{2} A \,b^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d \left (a d -c b \right ) e \,i^{3}}+\frac {g^{2} d B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (a d -c b \right ) e^{3} i^{3}}-\frac {g^{2} d B \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (a d -c b \right ) e^{3} i^{3}}+\frac {g^{2} B b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}-\frac {g^{2} B b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right ) e^{2} i^{3}}+\frac {g^{2} B \,b^{2} \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d \left (a d -c b \right ) e \,i^{3}}+\frac {g^{2} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{d \left (a d -c b \right ) e \,i^{3}}\right )}{d^{2}}\)	\(606\)
risch	\(\text {Expression too large to display}\)	\(1758\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to 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)^3,x, algorithm="maxima")

[Out]

-(b^2*log(b*x + a)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - b^2*log(d*x + c)/(2*I*b^2*c^2*d - 4*I*a*b*c
*d^2 + 2*I*a^2*d^3) - (2*b*d*x + 3*b*c - a*d)/(-4*I*b*c^3*d + 4*I*a*c^2*d^2 - 4*(I*b*c*d^3 - I*a*d^4)*x^2 - 8*
(I*b*c^2*d^2 - I*a*c*d^3)*x) - log(b*x*e/(d*x + c) + a*e/(d*x + c))/(2*I*d^3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d))*B
*a^2*g^2 - 2*B*a*b*g^2*((b^2*c - 2*a*b*d)*log(b*x + a)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (b^2*
c - 2*a*b*d)*log(d*x + c)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (2*d*x + c)*log(b*x*e/(d*x + c) +
a*e/(d*x + c))/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) - (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/(-4*I*b*c
^3*d^2 + 4*I*a*c^2*d^3 - 4*(I*b*c*d^4 - I*a*d^5)*x^2 - 8*(I*b*c^2*d^3 - I*a*c*d^4)*x)) - A*b^2*g^2*((4*c*d*x +
3*c^2)/(2*I*d^5*x^2 + 4*I*c*d^4*x + 2*I*c^2*d^3) - I*log(d*x + c)/d^3) - 1/2*B*b^2*g^2*(((I*d^2*x^2 + 2*I*c*d
*x + I*c^2)*log(d*x + c)^2 + (4*I*c*d*x + 3*I*c^2)*log(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3) - 2*integrate
(1/2*(2*I*d^2*x^2*log(b*x + a) + 2*I*d^2*x^2 + 4*I*c*d*x + 3*I*c^2)/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3
*d^2), x)) + 2*(2*d*x + c)*A*a*b*g^2/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + A*a^2*g^2/(2*I*d^3*x^2 + 4*I*
c*d^2*x + 2*I*c^2*d)

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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)^3,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^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), 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)**3,x)

[Out]

Timed out

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Giac [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)^3,x, algorithm="giac")

[Out]

integrate((b*g*x + a*g)^2*(B*log((b*x + a)*e/(d*x + c)) + A)/(I*d*x + I*c)^3, x)

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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 )}^3} \,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)^3,x)

[Out]

int(((a*g + b*g*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x)^3, x)

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