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

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

[Out]

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

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Rubi [A]  time = 0.0528924, antiderivative size = 81, normalized size of antiderivative = 1., 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} \[ \frac{g (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b}+\frac{B g (b c-a d)^2 \log (c+d x)}{2 b d^2}-\frac{B g x (b c-a d)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.0342394, size = 69, normalized size = 0.85 \[ \frac{g \left ((a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{B (a d-b c) ((a d-b c) \log (c+d x)+b d x)}{d^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.158, size = 1544, normalized size = 19.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10547, size = 194, normalized size = 2.4 \begin{align*} \frac{1}{2} \, A b g x^{2} +{\left (x \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\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{b e x}{d x + c} + \frac{a e}{d x + c}\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 \end{align*}

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))),x, algorithm="maxima")

[Out]

1/2*A*b*g*x^2 + (x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a*g + 1/2*(x^
2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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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Fricas [A]  time = 1.05546, size = 278, normalized size = 3.43 \begin{align*} \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{b e x + a e}{d x + c}\right )}{2 \, b d^{2}} \end{align*}

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))),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((b*e*x + a*e)/(d*x + c)))/(b*d^2)

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Sympy [B]  time = 4.44988, size = 257, normalized size = 3.17 \begin{align*} \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}} + \left (B a g x + \frac{B b g x^{2}}{2}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )} + \frac{x \left (2 A a d g + B a d g - B b c g\right )}{2 d} \end{align*}

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

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Giac [A]  time = 1.80428, size = 150, normalized size = 1.85 \begin{align*} \frac{B a^{2} g \log \left (b x + a\right )}{2 \, b} + \frac{1}{2} \,{\left (A b g + B b g\right )} x^{2} + \frac{1}{2} \,{\left (B b g x^{2} + 2 \, B a g x\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b c g - 2 \, A a d g - 3 \, B a d g\right )} x}{2 \, d} + \frac{{\left (B b c^{2} g - 2 \, B a c d g\right )} \log \left (d x + c\right )}{2 \, d^{2}} \end{align*}

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))),x, algorithm="giac")

[Out]

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