3.89 \(\int (a g+b g x)^3 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=149 \[ \frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b}-\frac{B g^3 x (b c-a d)^3}{4 d^3}+\frac{B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac{B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac{B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]

[Out]

-(B*(b*c - a*d)^3*g^3*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*(a + b*x)^
3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + (B*(b*c - a*d)^4*g^3*Log[c + d*x]
)/(4*b*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0962254, antiderivative size = 149, 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} \[ \frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b}-\frac{B g^3 x (b c-a d)^3}{4 d^3}+\frac{B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac{B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac{B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

-(B*(b*c - a*d)^3*g^3*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*(a + b*x)^
3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + (B*(b*c - a*d)^4*g^3*Log[c + d*x]
)/(4*b*d^4)

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)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b}-\frac{B \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b g}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac{B (b c-a d)^3 g^3 x}{4 d^3}+\frac{B (b c-a d)^2 g^3 (a+b x)^2}{8 b d^2}-\frac{B (b c-a d) g^3 (a+b x)^3}{12 b d}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b}+\frac{B (b c-a d)^4 g^3 \log (c+d x)}{4 b d^4}\\ \end{align*}

Mathematica [A]  time = 0.0977591, size = 120, normalized size = 0.81 \[ \frac{g^3 \left ((a+b x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{6 d^4}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.18, size = 5556, normalized size = 37.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 1.22249, size = 593, normalized size = 3.98 \begin{align*} \frac{1}{4} \, A b^{3} g^{3} x^{4} + A a b^{2} g^{3} x^{3} + \frac{3}{2} \, A a^{2} b g^{3} 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^{3} g^{3} + \frac{3}{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 a^{2} b g^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\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 a b^{2} g^{3} + \frac{1}{24} \,{\left (6 \, x^{4} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac{6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac{2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} g^{3} + A a^{3} g^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/4*A*b^3*g^3*x^4 + A*a*b^2*g^3*x^3 + 3/2*A*a^2*b*g^3*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^3*g^3 + 3/2*(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*a^2*b*g^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) +
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*a*b^2*g^3 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c
)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B
*b^3*g^3 + A*a^3*g^3*x

________________________________________________________________________________________

Fricas [B]  time = 1.13677, size = 664, normalized size = 4.46 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{3} x^{4} + 6 \, B a^{4} d^{4} g^{3} \log \left (b x + a\right ) - 2 \,{\left (B b^{4} c d^{3} -{\left (12 \, A + B\right )} a b^{3} d^{4}\right )} g^{3} x^{3} + 3 \,{\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \,{\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} g^{3} x^{2} - 6 \,{\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} -{\left (4 \, A + 3 \, B\right )} a^{3} b d^{4}\right )} g^{3} x + 6 \,{\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} \log \left (d x + c\right ) + 6 \,{\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{24 \, b d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/24*(6*A*b^4*d^4*g^3*x^4 + 6*B*a^4*d^4*g^3*log(b*x + a) - 2*(B*b^4*c*d^3 - (12*A + B)*a*b^3*d^4)*g^3*x^3 + 3*
(B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 + 3*(4*A + B)*a^2*b^2*d^4)*g^3*x^2 - 6*(B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 6*B
*a^2*b^2*c*d^3 - (4*A + 3*B)*a^3*b*d^4)*g^3*x + 6*(B*b^4*c^4 - 4*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2 - 4*B*a^3
*b*c*d^3)*g^3*log(d*x + c) + 6*(B*b^4*d^4*g^3*x^4 + 4*B*a*b^3*d^4*g^3*x^3 + 6*B*a^2*b^2*d^4*g^3*x^2 + 4*B*a^3*
b*d^4*g^3*x)*log((b*e*x + a*e)/(d*x + c)))/(b*d^4)

________________________________________________________________________________________

Sympy [B]  time = 7.13052, size = 719, normalized size = 4.83 \begin{align*} \frac{A b^{3} g^{3} x^{4}}{4} + \frac{B a^{4} g^{3} \log{\left (x + \frac{\frac{B a^{5} d^{4} g^{3}}{b} + 4 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 b} - \frac{B c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{5 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3} - B a c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) + \frac{B b c^{2} g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 d^{4}} + \left (B a^{3} g^{3} x + \frac{3 B a^{2} b g^{3} x^{2}}{2} + B a b^{2} g^{3} x^{3} + \frac{B b^{3} g^{3} x^{4}}{4}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )} + \frac{x^{3} \left (12 A a b^{2} d g^{3} + B a b^{2} d g^{3} - B b^{3} c g^{3}\right )}{12 d} + \frac{x^{2} \left (12 A a^{2} b d^{2} g^{3} + 3 B a^{2} b d^{2} g^{3} - 4 B a b^{2} c d g^{3} + B b^{3} c^{2} g^{3}\right )}{8 d^{2}} + \frac{x \left (4 A a^{3} d^{3} g^{3} + 3 B a^{3} d^{3} g^{3} - 6 B a^{2} b c d^{2} g^{3} + 4 B a b^{2} c^{2} d g^{3} - B b^{3} c^{3} g^{3}\right )}{4 d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**3*g**3*x**4/4 + B*a**4*g**3*log(x + (B*a**5*d**4*g**3/b + 4*B*a**4*c*d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**
3 + 4*B*a**2*b**2*c**3*d*g**3 - B*a*b**3*c**4*g**3)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2
*c**2*d**2*g**3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(4*b) - B*c*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*
a*b*c*d + b**2*c**2)*log(x + (5*B*a**4*c*d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**3 + 4*B*a**2*b**2*c**3*d*g**3 - B
*a*b**3*c**4*g**3 - B*a*c*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2) + B*b*c**2*g**3*(2*a*d - b*
c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2)/d)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2*c**2*d*
*2*g**3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(4*d**4) + (B*a**3*g**3*x + 3*B*a**2*b*g**3*x**2/2 + B*a
*b**2*g**3*x**3 + B*b**3*g**3*x**4/4)*log(e*(a + b*x)/(c + d*x)) + x**3*(12*A*a*b**2*d*g**3 + B*a*b**2*d*g**3
- B*b**3*c*g**3)/(12*d) + x**2*(12*A*a**2*b*d**2*g**3 + 3*B*a**2*b*d**2*g**3 - 4*B*a*b**2*c*d*g**3 + B*b**3*c*
*2*g**3)/(8*d**2) + x*(4*A*a**3*d**3*g**3 + 3*B*a**3*d**3*g**3 - 6*B*a**2*b*c*d**2*g**3 + 4*B*a*b**2*c**2*d*g*
*3 - B*b**3*c**3*g**3)/(4*d**3)

________________________________________________________________________________________

Giac [B]  time = 21.8869, size = 458, normalized size = 3.07 \begin{align*} \frac{B a^{4} g^{3} \log \left (b x + a\right )}{4 \, b} + \frac{1}{4} \,{\left (A b^{3} g^{3} + B b^{3} g^{3}\right )} x^{4} - \frac{{\left (B b^{3} c g^{3} - 12 \, A a b^{2} d g^{3} - 13 \, B a b^{2} d g^{3}\right )} x^{3}}{12 \, d} + \frac{1}{4} \,{\left (B b^{3} g^{3} x^{4} + 4 \, B a b^{2} g^{3} x^{3} + 6 \, B a^{2} b g^{3} x^{2} + 4 \, B a^{3} g^{3} x\right )} \log \left (\frac{b x + a}{d x + c}\right ) + \frac{{\left (B b^{3} c^{2} g^{3} - 4 \, B a b^{2} c d g^{3} + 12 \, A a^{2} b d^{2} g^{3} + 15 \, B a^{2} b d^{2} g^{3}\right )} x^{2}}{8 \, d^{2}} - \frac{{\left (B b^{3} c^{3} g^{3} - 4 \, B a b^{2} c^{2} d g^{3} + 6 \, B a^{2} b c d^{2} g^{3} - 4 \, A a^{3} d^{3} g^{3} - 7 \, B a^{3} d^{3} g^{3}\right )} x}{4 \, d^{3}} + \frac{{\left (B b^{3} c^{4} g^{3} - 4 \, B a b^{2} c^{3} d g^{3} + 6 \, B a^{2} b c^{2} d^{2} g^{3} - 4 \, B a^{3} c d^{3} g^{3}\right )} \log \left (d x + c\right )}{4 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

1/4*B*a^4*g^3*log(b*x + a)/b + 1/4*(A*b^3*g^3 + B*b^3*g^3)*x^4 - 1/12*(B*b^3*c*g^3 - 12*A*a*b^2*d*g^3 - 13*B*a
*b^2*d*g^3)*x^3/d + 1/4*(B*b^3*g^3*x^4 + 4*B*a*b^2*g^3*x^3 + 6*B*a^2*b*g^3*x^2 + 4*B*a^3*g^3*x)*log((b*x + a)/
(d*x + c)) + 1/8*(B*b^3*c^2*g^3 - 4*B*a*b^2*c*d*g^3 + 12*A*a^2*b*d^2*g^3 + 15*B*a^2*b*d^2*g^3)*x^2/d^2 - 1/4*(
B*b^3*c^3*g^3 - 4*B*a*b^2*c^2*d*g^3 + 6*B*a^2*b*c*d^2*g^3 - 4*A*a^3*d^3*g^3 - 7*B*a^3*d^3*g^3)*x/d^3 + 1/4*(B*
b^3*c^4*g^3 - 4*B*a*b^2*c^3*d*g^3 + 6*B*a^2*b*c^2*d^2*g^3 - 4*B*a^3*c*d^3*g^3)*log(d*x + c)/d^4