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

Optimal. Leaf size=125 \[ \frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}+\frac {B g (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

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

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Rubi [A]  time = 0.35, antiderivative size = 213, normalized size of antiderivative = 1.70, number of steps used = 14, number of rules used = 11, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2528, 2486, 31, 2524, 12, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B g (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac {g (b c-a d) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}-\frac {B g (b c-a d) \log ^2(c+d x)}{2 d^2 i}+\frac {B g (b c-a d) \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}-\frac {B g (b c-a d) \log (c+d x)}{d^2 i}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d i}+\frac {A b g x}{d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 2524

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

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]

Rubi steps

\begin {align*} \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{33 c+33 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{33 d}+\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{33 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{33 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(b B g) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{33 d}+\frac {(B (b c-a d) g) \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}{33 d^2}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {1}{c+d x} \, dx}{33 d}+\frac {(B (b c-a d) g) \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}{33 d^2 e}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(B (b c-a d) g) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{33 d^2 e}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(b B (b c-a d) g) \int \frac {\log (c+d x)}{a+b x} \, dx}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {\log (c+d x)}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {(B (b c-a d) g) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {B (b c-a d) g \log ^2(c+d x)}{66 d^2}-\frac {(B (b c-a d) g) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{33 d^2}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {B (b c-a d) g \log ^2(c+d x)}{66 d^2}+\frac {B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{33 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 162, normalized size = 1.30 \[ \frac {g \left (-2 (b c-a d) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+B (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b c-a d) \log (c+d x)+2 A b d x\right )}{2 d^2 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b g x + A a g + {\left (B b g x + B a g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.13, size = 895, normalized size = 7.16 \[ \frac {B \,a^{2} e g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right ) i}-\frac {2 B a b c e g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right ) d i}+\frac {B \,b^{2} c^{2} e g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right ) d^{2} i}+\frac {B a b e g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d i}-\frac {B \,b^{2} c e g \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d^{2} i}+\frac {A a b e g}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d i}-\frac {A \,b^{2} c e g}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d^{2} i}-\frac {B a g \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{d i}+\frac {B b c g \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{d^{2} i}-\frac {A a g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{d i}+\frac {A b c g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{d^{2} i}-\frac {B a g \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{d i}-\frac {B a g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{d i}+\frac {B b c g \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{d^{2} i}+\frac {B b c g \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{d^{2} i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.68, size = 221, normalized size = 1.77 \[ A b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A a g \log \left (d i x + c i\right )}{d i} - \frac {{\left (b c g - a d g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac {{\left (a d g \log \relax (e) - {\left (g \log \relax (e) + g\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} - \frac {2 \, B b d g x \log \left (d x + c\right ) - 2 \, B b d g x \log \relax (e) - {\left (b c g - a d g\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (B b d g x + B a d g\right )} \log \left (b x + a\right )}{2 \, d^{2} i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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