3.88 \(\int (a g+b g x)^4 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=180 \[ \frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b}+\frac{B g^4 x (b c-a d)^4}{5 d^4}-\frac{B g^4 (a+b x)^2 (b c-a d)^3}{10 b d^3}+\frac{B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 (a+b x)^4 (b c-a d)}{20 b d} \]

[Out]

(B*(b*c - a*d)^4*g^4*x)/(5*d^4) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*g^4*(a + b*x
)^3)/(15*b*d^2) - (B*(b*c - a*d)*g^4*(a + b*x)^4)/(20*b*d) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*
x)]))/(5*b) - (B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5)

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Rubi [A]  time = 0.123745, antiderivative size = 180, 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} \[ \frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b}+\frac{B g^4 x (b c-a d)^4}{5 d^4}-\frac{B g^4 (a+b x)^2 (b c-a d)^3}{10 b d^3}+\frac{B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 (a+b x)^4 (b c-a d)}{20 b d} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(B*(b*c - a*d)^4*g^4*x)/(5*d^4) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*g^4*(a + b*x
)^3)/(15*b*d^2) - (B*(b*c - a*d)*g^4*(a + b*x)^4)/(20*b*d) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*
x)]))/(5*b) - (B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5)

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)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b}-\frac{B \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b g}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b}-\frac{\left (B (b c-a d) g^4\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b}-\frac{\left (B (b c-a d) g^4\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac{B (b c-a d)^4 g^4 x}{5 d^4}-\frac{B (b c-a d)^3 g^4 (a+b x)^2}{10 b d^3}+\frac{B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac{B (b c-a d) g^4 (a+b x)^4}{20 b d}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b}-\frac{B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}\\ \end{align*}

Mathematica [A]  time = 0.1062, size = 142, normalized size = 0.79 \[ \frac{g^4 \left ((a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{B (b c-a d) \left (6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (a d-b c)-12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (c+d x)+3 d^4 (a+b x)^4\right )}{12 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - (B*(b*c - a*d)*(-12*b*d*(b*c - a*d)^3*x + 6*d^2*(b*c
- a*d)^2*(a + b*x)^2 + 4*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 3*d^4*(a + b*x)^4 + 12*(b*c - a*d)^4*Log[c + d*x]))/
(12*d^5)))/(5*b)

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Maple [B]  time = 0.225, size = 8417, normalized size = 46.8 \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)^4*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

result too large to display

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Maxima [B]  time = 1.2001, size = 841, normalized size = 4.67 \begin{align*} \frac{1}{5} \, A b^{4} g^{4} x^{5} + A a b^{3} g^{4} x^{4} + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, A a^{3} b g^{4} 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^{4} g^{4} + 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^{3} b g^{4} +{\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^{2} b^{2} g^{4} + \frac{1}{6} \,{\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 a b^{3} g^{4} + \frac{1}{60} \,{\left (12 \, x^{5} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac{12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac{3 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{4} g^{4} + A a^{4} g^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/5*A*b^4*g^4*x^5 + A*a*b^3*g^4*x^4 + 2*A*a^2*b^2*g^4*x^3 + 2*A*a^3*b*g^4*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^4*g^4 + 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^3*b*g^4 + (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^2*b^2*g^4 + 1/6*(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*a*b^3*g^4 + 1/60*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12*c
^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*
b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*b^4*g^4 + A*a^4*g^4*x

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Fricas [B]  time = 1.36423, size = 910, normalized size = 5.06 \begin{align*} \frac{12 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \,{\left (B b^{5} c d^{4} -{\left (20 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \,{\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + 2 \,{\left (15 \, A + 2 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \,{\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 2 \,{\left (10 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 12 \,{\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} +{\left (5 \, A + 4 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 12 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 12 \,{\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{60 \, b d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/60*(12*A*b^5*d^5*g^4*x^5 + 12*B*a^5*d^5*g^4*log(b*x + a) - 3*(B*b^5*c*d^4 - (20*A + B)*a*b^4*d^5)*g^4*x^4 +
4*(B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + 2*(15*A + 2*B)*a^2*b^3*d^5)*g^4*x^3 - 6*(B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^
3 + 10*B*a^2*b^3*c*d^4 - 2*(10*A + 3*B)*a^3*b^2*d^5)*g^4*x^2 + 12*(B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2*
b^3*c^2*d^3 - 10*B*a^3*b^2*c*d^4 + (5*A + 4*B)*a^4*b*d^5)*g^4*x - 12*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b
^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3 + 5*B*a^4*b*c*d^4)*g^4*log(d*x + c) + 12*(B*b^5*d^5*g^4*x^5 + 5*B*a*b^4*d^5*
g^4*x^4 + 10*B*a^2*b^3*d^5*g^4*x^3 + 10*B*a^3*b^2*d^5*g^4*x^2 + 5*B*a^4*b*d^5*g^4*x)*log((b*e*x + a*e)/(d*x +
c)))/(b*d^5)

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Sympy [B]  time = 10.0173, size = 993, normalized size = 5.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**4*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**4*g**4*x**5/5 + B*a**5*g**4*log(x + (B*a**6*d**5*g**4/b + 5*B*a**5*c*d**4*g**4 - 10*B*a**4*b*c**2*d**3*g*
*4 + 10*B*a**3*b**2*c**3*d**2*g**4 - 5*B*a**2*b**3*c**4*d*g**4 + B*a*b**4*c**5*g**4)/(B*a**5*d**5*g**4 + 5*B*a
**4*b*c*d**4*g**4 - 10*B*a**3*b**2*c**2*d**3*g**4 + 10*B*a**2*b**3*c**3*d**2*g**4 - 5*B*a*b**4*c**4*d*g**4 + B
*b**5*c**5*g**4))/(5*b) - B*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d
+ b**4*c**4)*log(x + (6*B*a**5*c*d**4*g**4 - 10*B*a**4*b*c**2*d**3*g**4 + 10*B*a**3*b**2*c**3*d**2*g**4 - 5*B*
a**2*b**3*c**4*d*g**4 + B*a*b**4*c**5*g**4 - B*a*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d*
*2 - 5*a*b**3*c**3*d + b**4*c**4) + B*b*c**2*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5
*a*b**3*c**3*d + b**4*c**4)/d)/(B*a**5*d**5*g**4 + 5*B*a**4*b*c*d**4*g**4 - 10*B*a**3*b**2*c**2*d**3*g**4 + 10
*B*a**2*b**3*c**3*d**2*g**4 - 5*B*a*b**4*c**4*d*g**4 + B*b**5*c**5*g**4))/(5*d**5) + (B*a**4*g**4*x + 2*B*a**3
*b*g**4*x**2 + 2*B*a**2*b**2*g**4*x**3 + B*a*b**3*g**4*x**4 + B*b**4*g**4*x**5/5)*log(e*(a + b*x)/(c + d*x)) +
x**4*(20*A*a*b**3*d*g**4 + B*a*b**3*d*g**4 - B*b**4*c*g**4)/(20*d) + x**3*(30*A*a**2*b**2*d**2*g**4 + 4*B*a**
2*b**2*d**2*g**4 - 5*B*a*b**3*c*d*g**4 + B*b**4*c**2*g**4)/(15*d**2) + x**2*(20*A*a**3*b*d**3*g**4 + 6*B*a**3*
b*d**3*g**4 - 10*B*a**2*b**2*c*d**2*g**4 + 5*B*a*b**3*c**2*d*g**4 - B*b**4*c**3*g**4)/(10*d**3) + x*(5*A*a**4*
d**4*g**4 + 4*B*a**4*d**4*g**4 - 10*B*a**3*b*c*d**3*g**4 + 10*B*a**2*b**2*c**2*d**2*g**4 - 5*B*a*b**3*c**3*d*g
**4 + B*b**4*c**4*g**4)/(5*d**4)

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Giac [B]  time = 99.387, size = 639, normalized size = 3.55 \begin{align*} \frac{B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} + \frac{1}{5} \,{\left (A b^{4} g^{4} + B b^{4} g^{4}\right )} x^{5} - \frac{{\left (B b^{4} c g^{4} - 20 \, A a b^{3} d g^{4} - 21 \, B a b^{3} d g^{4}\right )} x^{4}}{20 \, d} + \frac{{\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} + 30 \, A a^{2} b^{2} d^{2} g^{4} + 34 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac{1}{5} \,{\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} - 20 \, A a^{3} b d^{3} g^{4} - 26 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{10 \, d^{3}} + \frac{{\left (B b^{4} c^{4} g^{4} - 5 \, B a b^{3} c^{3} d g^{4} + 10 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 10 \, B a^{3} b c d^{3} g^{4} + 5 \, A a^{4} d^{4} g^{4} + 9 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} - \frac{{\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

1/5*B*a^5*g^4*log(b*x + a)/b + 1/5*(A*b^4*g^4 + B*b^4*g^4)*x^5 - 1/20*(B*b^4*c*g^4 - 20*A*a*b^3*d*g^4 - 21*B*a
*b^3*d*g^4)*x^4/d + 1/15*(B*b^4*c^2*g^4 - 5*B*a*b^3*c*d*g^4 + 30*A*a^2*b^2*d^2*g^4 + 34*B*a^2*b^2*d^2*g^4)*x^3
/d^2 + 1/5*(B*b^4*g^4*x^5 + 5*B*a*b^3*g^4*x^4 + 10*B*a^2*b^2*g^4*x^3 + 10*B*a^3*b*g^4*x^2 + 5*B*a^4*g^4*x)*log
((b*x + a)/(d*x + c)) - 1/10*(B*b^4*c^3*g^4 - 5*B*a*b^3*c^2*d*g^4 + 10*B*a^2*b^2*c*d^2*g^4 - 20*A*a^3*b*d^3*g^
4 - 26*B*a^3*b*d^3*g^4)*x^2/d^3 + 1/5*(B*b^4*c^4*g^4 - 5*B*a*b^3*c^3*d*g^4 + 10*B*a^2*b^2*c^2*d^2*g^4 - 10*B*a
^3*b*c*d^3*g^4 + 5*A*a^4*d^4*g^4 + 9*B*a^4*d^4*g^4)*x/d^4 - 1/5*(B*b^4*c^5*g^4 - 5*B*a*b^3*c^4*d*g^4 + 10*B*a^
2*b^2*c^3*d^2*g^4 - 10*B*a^3*b*c^2*d^3*g^4 + 5*B*a^4*c*d^4*g^4)*log(-d*x - c)/d^5