3.175 \(\int (a g+b g x)^2 (A+B \log (\frac{e (c+d x)}{a+b x})) \, dx\)

Optimal. Leaf size=118 \[ \frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{3 b}-\frac{B g^2 x (b c-a d)^2}{3 d^2}+\frac{B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac{B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

[Out]

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

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Rubi [A]  time = 0.0806846, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 43} \[ \frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{3 b}-\frac{B g^2 x (b c-a d)^2}{3 d^2}+\frac{B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac{B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \, dx &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{3 b}-\frac{B \int \frac{(-b c+a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b g}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{3 b}+\frac{\left (B (b c-a d) g^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b}\\ &=\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{3 b}+\frac{\left (B (b c-a d) g^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac{B (b c-a d)^2 g^2 x}{3 d^2}+\frac{B (b c-a d) g^2 (a+b x)^2}{6 b d}+\frac{B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3}+\frac{g^2 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0506713, size = 99, normalized size = 0.84 \[ \frac{g^2 \left (\frac{B (b c-a d) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.159, size = 1537, normalized size = 13. \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)^2*(A+B*ln(e*(d*x+c)/(b*x+a))),x)

[Out]

-1/3/b*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^3+1/3/b*e^3*B*g^2*ln(d
*e/b-e*(a*d-b*c)/b/(b*x+a))*d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^6/(b*x+a)^3+1/3*b^5*e^3*B*g^2*ln(d*e/b-e*(a
*d-b*c)/b/(b*x+a))/d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*c^6/(b*x+a)^3-b*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+
a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*c^2*a*d-20/3*b^2*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*
d+b*e/(b*x+a)*c)^3*a^3/(b*x+a)^3*c^3-B*g^2/d*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a^2*c+1/3*b^2*e^3*A*g^2/(
-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*c^3-1/3*b^2*B*g^2/d^3*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*c^3-e*B*g^2/(-e/
(b*x+a)*a*d+b*e/(b*x+a)*c)*a^2*c-2*b^4*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/d^2/(-e/(b*x+a)*a*d+b*e/(b*x+
a)*c)^3*a*c^5/(b*x+a)^3+5*b*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^4/(
b*x+a)^3*c^2+1/2*e^2*B*g^2*d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a^2*c+5*b^3*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*
x+a))/d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^2*c^4/(b*x+a)^3+e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^2/(-e/(
b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^2*c+b*e*B*g^2/d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)*c^2*a-b*e^3*A*g^2/(-e/(b*x+a)*a*d
+b*e/(b*x+a)*c)^3*a*c^2*d-1/3*b^2*e*B*g^2/d^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)*c^3+1/3/b*e*B*g^2/(-e/(b*x+a)*a*d
+b*e/(b*x+a)*c)*a^3*d-1/3/b*e^3*A*g^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^3*d^3-1/6/b*e^2*B*g^2*d^2/(-e/(b*x+a)
*a*d+b*e/(b*x+a)*c)^2*a^3-1/2*b*e^2*B*g^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a*c^2+1/6*b^2*e^2*B*g^2/d/(-e/(b*x+
a)*a*d+b*e/(b*x+a)*c)^2*c^3+b*B*g^2/d^2*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a*c^2+1/3*b^2*e^3*B*g^2*ln(d*e
/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*c^3+e^3*A*g^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^2*
c*d^2+1/3/b*B*g^2*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a^3-2*e^3*B*g^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^2/
(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^5/(b*x+a)^3*c

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Maxima [B]  time = 1.20332, size = 375, normalized size = 3.18 \begin{align*} \frac{1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} 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^{2} g^{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 a b g^{2} + \frac{1}{6} \,{\left (2 \, x^{3} \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\right ) - \frac{2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac{2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.06778, size = 467, normalized size = 3.96 \begin{align*} \frac{2 \, A b^{3} d^{3} g^{2} x^{3} - 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) +{\left (B b^{3} c d^{2} +{\left (6 \, A - B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} - 2 \,{\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} -{\left (3 \, A - 2 \, B\right )} a^{2} b d^{3}\right )} g^{2} x + 2 \,{\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + 2 \,{\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac{d e x + c e}{b x + a}\right )}{6 \, b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.00286, size = 503, normalized size = 4.26 \begin{align*} \frac{A b^{2} g^{2} x^{3}}{3} - \frac{B a^{3} g^{2} \log{\left (x + \frac{\frac{B a^{4} d^{3} g^{2}}{b} + 3 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 b} + \frac{B c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{4 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2} - B a c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac{B b c^{2} g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac{B b^{2} g^{2} x^{3}}{3}\right ) \log{\left (\frac{e \left (c + d x\right )}{a + b x} \right )} + \frac{x^{2} \left (6 A a b d g^{2} - B a b d g^{2} + B b^{2} c g^{2}\right )}{6 d} + \frac{x \left (3 A a^{2} d^{2} g^{2} - 2 B a^{2} d^{2} g^{2} + 3 B a b c d g^{2} - B b^{2} c^{2} g^{2}\right )}{3 d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 4.9494, size = 305, normalized size = 2.58 \begin{align*} -\frac{B a^{3} g^{2} \log \left (b x + a\right )}{3 \, b} + \frac{1}{3} \,{\left (A b^{2} g^{2} + B b^{2} g^{2}\right )} x^{3} + \frac{{\left (B b^{2} c g^{2} + 6 \, A a b d g^{2} + 5 \, B a b d g^{2}\right )} x^{2}}{6 \, d} + \frac{1}{3} \,{\left (B b^{2} g^{2} x^{3} + 3 \, B a b g^{2} x^{2} + 3 \, B a^{2} g^{2} x\right )} \log \left (\frac{d x + c}{b x + a}\right ) - \frac{{\left (B b^{2} c^{2} g^{2} - 3 \, B a b c d g^{2} - 3 \, A a^{2} d^{2} g^{2} - B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} + \frac{{\left (B b^{2} c^{3} g^{2} - 3 \, B a b c^{2} d g^{2} + 3 \, B a^{2} c d^{2} g^{2}\right )} \log \left (d x + c\right )}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="giac")

[Out]

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