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

Optimal. Leaf size=81 \[ \frac {g (a+b x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b}-\frac {B g (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac {B g x (b c-a d)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2525, 12, 43} \[ \frac {g (a+b x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b}-\frac {B g (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac {B g x (b c-a d)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 43

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 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]

Rubi steps

\begin {align*} \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b}-\frac {B \int \frac {(b c-a d) g^2 (-a-b x)}{c+d x} \, dx}{2 b g}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b}-\frac {(B (b c-a d) g) \int \frac {-a-b x}{c+d x} \, dx}{2 b}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b}-\frac {(B (b c-a d) g) \int \left (-\frac {b}{d}+\frac {b c-a d}{d (c+d x)}\right ) \, dx}{2 b}\\ &=\frac {B (b c-a d) g x}{2 d}-\frac {B (b c-a d)^2 g \log (c+d x)}{2 b d^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 69, normalized size = 0.85 \[ \frac {g \left ((a+b x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+\frac {B (b c-a d) ((a d-b c) \log (c+d x)+b d x)}{d^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.96, size = 127, normalized size = 1.57 \[ \frac {A b^{2} d^{2} g x^{2} - B a^{2} d^{2} g \log \left (b x + a\right ) + {\left (B b^{2} c d + {\left (2 \, A - B\right )} a b d^{2}\right )} g x - {\left (B b^{2} c^{2} - 2 \, B a b c d\right )} g \log \left (d x + c\right ) + {\left (B b^{2} d^{2} g x^{2} + 2 \, B a b d^{2} g x\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{2 \, b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.67, size = 1395, normalized size = 17.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*b^3*c^3*d^2*g*e^3*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) - 3*B*a*b^2*c^2*d^3*g*e^3*log(-d*e + (d*x*e + c
*e)*b/(b*x + a)) + 3*B*a^2*b*c*d^4*g*e^3*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) - B*a^3*d^5*g*e^3*log(-d*e + (d
*x*e + c*e)*b/(b*x + a)) - 2*(d*x*e + c*e)*B*b^4*c^3*d*g*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) +
6*(d*x*e + c*e)*B*a*b^3*c^2*d^2*g*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) - 6*(d*x*e + c*e)*B*a^2
*b^2*c*d^3*g*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) + 2*(d*x*e + c*e)*B*a^3*b*d^4*g*e^2*log(-d*e
+ (d*x*e + c*e)*b/(b*x + a))/(b*x + a) + (d*x*e + c*e)^2*B*b^5*c^3*g*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(
b*x + a)^2 - 3*(d*x*e + c*e)^2*B*a*b^4*c^2*d*g*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 + 3*(d*x*e
+ c*e)^2*B*a^2*b^3*c*d^2*g*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 - (d*x*e + c*e)^2*B*a^3*b^2*d^3
*g*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 + 2*(d*x*e + c*e)*B*b^4*c^3*d*g*e^2*log((d*x*e + c*e)/(
b*x + a))/(b*x + a) - 6*(d*x*e + c*e)*B*a*b^3*c^2*d^2*g*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a) + 6*(d*x*e
+ c*e)*B*a^2*b^2*c*d^3*g*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 2*(d*x*e + c*e)*B*a^3*b*d^4*g*e^2*log((d
*x*e + c*e)/(b*x + a))/(b*x + a) - (d*x*e + c*e)^2*B*b^5*c^3*g*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 + 3*
(d*x*e + c*e)^2*B*a*b^4*c^2*d*g*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 - 3*(d*x*e + c*e)^2*B*a^2*b^3*c*d^2
*g*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 + (d*x*e + c*e)^2*B*a^3*b^2*d^3*g*e*log((d*x*e + c*e)/(b*x + a))
/(b*x + a)^2 + A*b^3*c^3*d^2*g*e^3 - B*b^3*c^3*d^2*g*e^3 - 3*A*a*b^2*c^2*d^3*g*e^3 + 3*B*a*b^2*c^2*d^3*g*e^3 +
3*A*a^2*b*c*d^4*g*e^3 - 3*B*a^2*b*c*d^4*g*e^3 - A*a^3*d^5*g*e^3 + B*a^3*d^5*g*e^3 + (d*x*e + c*e)*B*b^4*c^3*d
*g*e^2/(b*x + a) - 3*(d*x*e + c*e)*B*a*b^3*c^2*d^2*g*e^2/(b*x + a) + 3*(d*x*e + c*e)*B*a^2*b^2*c*d^3*g*e^2/(b*
x + a) - (d*x*e + c*e)*B*a^3*b*d^4*g*e^2/(b*x + a))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*
(b*c - a*d)))/(b*d^4*e^2 - 2*(d*x*e + c*e)*b^2*d^3*e/(b*x + a) + (d*x*e + c*e)^2*b^3*d^2/(b*x + a)^2)

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maple [B]  time = 0.14, size = 951, normalized size = 11.74 \[ -\frac {B \,a^{4} d^{2} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} \left (b x +a \right )^{2} b}+\frac {2 B \,a^{3} c d \,e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} \left (b x +a \right )^{2}}-\frac {3 B \,a^{2} b \,c^{2} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} \left (b x +a \right )^{2}}+\frac {2 B a \,b^{2} c^{3} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} \left (b x +a \right )^{2} d}-\frac {B \,b^{3} c^{4} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} \left (b x +a \right )^{2} d^{2}}+\frac {B \,a^{2} d^{2} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} b}-\frac {B a c d \,e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2}}+\frac {B b \,c^{2} e^{2} g \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2}}+\frac {A \,a^{2} d^{2} e^{2} g}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2} b}-\frac {A a c d \,e^{2} g}{\left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2}}+\frac {A b \,c^{2} e^{2} g}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right )^{2}}+\frac {B \,a^{2} d e g}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right ) b}-\frac {B a c e g}{-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}}+\frac {B b \,c^{2} e g}{2 \left (-\frac {a d e}{b x +a}+\frac {b c e}{b x +a}\right ) d}+\frac {B \,a^{2} g \ln \left (-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b \right )}{2 b}-\frac {B a c g \ln \left (-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b \right )}{d}+\frac {B b \,c^{2} g \ln \left (-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b \right )}{2 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.12, size = 143, normalized size = 1.77 \[ \frac {1}{2} \, A b g x^{2} + {\left (x \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {a \log \left (b x + a\right )}{b} + \frac {c \log \left (d x + c\right )}{d}\right )} B a g + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} B b g + A a g x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.31, size = 126, normalized size = 1.56 \[ x\,\left (\frac {g\,\left (4\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,g\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )+\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (\frac {B\,b\,g\,x^2}{2}+B\,a\,g\,x\right )-\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^2\,g-2\,B\,a\,c\,d\,g\right )}{2\,d^2}+\frac {A\,b\,g\,x^2}{2}-\frac {B\,a^2\,g\,\ln \left (a+b\,x\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((g*(4*A*a*d + 2*A*b*c - B*a*d + B*b*c))/(2*d) - (A*g*(2*a*d + 2*b*c))/(2*d)) + log((e*(c + d*x))/(a + b*x))
*((B*b*g*x^2)/2 + B*a*g*x) - (log(c + d*x)*(B*b*c^2*g - 2*B*a*c*d*g))/(2*d^2) + (A*b*g*x^2)/2 - (B*a^2*g*log(a
+ b*x))/(2*b)

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sympy [B]  time = 1.92, size = 253, normalized size = 3.12 \[ \frac {A b g x^{2}}{2} - \frac {B a^{2} g \log {\left (x + \frac {\frac {B a^{3} d^{2} g}{b} + 2 B a^{2} c d g - B a b c^{2} g}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{2 b} + \frac {B c g \left (2 a d - b c\right ) \log {\left (x + \frac {3 B a^{2} c d g - B a b c^{2} g - B a c g \left (2 a d - b c\right ) + \frac {B b c^{2} g \left (2 a d - b c\right )}{d}}{B a^{2} d^{2} g + 2 B a b c d g - B b^{2} c^{2} g} \right )}}{2 d^{2}} + x \left (A a g - \frac {B a g}{2} + \frac {B b c g}{2 d}\right ) + \left (B a g x + \frac {B b g x^{2}}{2}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*g*x**2/2 - B*a**2*g*log(x + (B*a**3*d**2*g/b + 2*B*a**2*c*d*g - B*a*b*c**2*g)/(B*a**2*d**2*g + 2*B*a*b*c*d
*g - B*b**2*c**2*g))/(2*b) + B*c*g*(2*a*d - b*c)*log(x + (3*B*a**2*c*d*g - B*a*b*c**2*g - B*a*c*g*(2*a*d - b*c
) + B*b*c**2*g*(2*a*d - b*c)/d)/(B*a**2*d**2*g + 2*B*a*b*c*d*g - B*b**2*c**2*g))/(2*d**2) + x*(A*a*g - B*a*g/2
+ B*b*c*g/(2*d)) + (B*a*g*x + B*b*g*x**2/2)*log(e*(c + d*x)/(a + b*x))

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