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

Optimal. Leaf size=212 \[ \frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A-B\right )}{20 b^2}+\frac{g^3 i (a+b x)^4 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b}+\frac{B g^3 i (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g^3 i (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}-\frac{B g^3 i (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac{B g^3 i x (b c-a d)^4}{20 b d^3} \]

[Out]

-(B*(b*c - a*d)^4*g^3*i*x)/(20*b*d^3) + (B*(b*c - a*d)^3*g^3*i*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d)^2*g^
3*i*(a + b*x)^3)/(60*b^2*d) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b) + ((b*c
- a*d)*g^3*i*(a + b*x)^4*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^2) + (B*(b*c - a*d)^5*g^3*i*Log[c +
d*x])/(20*b^2*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.345444, antiderivative size = 232, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 43} \[ \frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^2}+\frac{d g^3 i (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^2}+\frac{B g^3 i (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g^3 i (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}-\frac{B g^3 i (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac{B g^3 i (a+b x)^4 (b c-a d)}{20 b^2}-\frac{B g^3 i x (b c-a d)^4}{20 b d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)^4*g^3*i*x)/(20*b*d^3) + (B*(b*c - a*d)^3*g^3*i*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d)^2*g^
3*i*(a + b*x)^3)/(60*b^2*d) - (B*(b*c - a*d)*g^3*i*(a + b*x)^4)/(20*b^2) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2) + (d*g^3*i*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^2)
+ (B*(b*c - a*d)^5*g^3*i*Log[c + d*x])/(20*b^2*d^4)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 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]

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 (c+d x) (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{(b c-a d) (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}+\frac{d (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g}\right ) \, dx\\ &=\frac{(b c-a d) \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b}+\frac{d \int (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}\\ &=\frac{(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac{d g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac{(B d) \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b^2 g^2}-\frac{(B (b c-a d)) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b^2 g}\\ &=\frac{(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac{d g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac{\left (B d (b c-a d) g^3\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b^2}-\frac{\left (B (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b^2}\\ &=\frac{(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac{d g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^2}-\frac{\left (B d (b c-a d) g^3\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^2}-\frac{\left (B (b c-a d)^2 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^2}\\ &=-\frac{B (b c-a d)^4 g^3 x}{20 b d^3}+\frac{B (b c-a d)^3 g^3 (a+b x)^2}{40 b^2 d^2}-\frac{B (b c-a d)^2 g^3 (a+b x)^3}{60 b^2 d}-\frac{B (b c-a d) g^3 (a+b x)^4}{20 b^2}+\frac{(b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2}+\frac{d g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^2}+\frac{B (b c-a d)^5 g^3 \log (c+d x)}{20 b^2 d^4}\\ \end{align*}

Mathematica [A]  time = 0.22024, size = 261, normalized size = 1.23 \[ \frac{g^3 i \left (24 d (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+30 (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{5 B (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{d^4}+\frac{2 B (b c-a d) \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )}{d^4}\right )}{120 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i*(30*(b*c - a*d)*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*d*(a + b*x)^5*(A + B*Log[(e*(a +
b*x))/(c + d*x)]) - (5*B*(b*c - a*d)^2*(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]))/d^4 + (2*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]))/d^4))/(120*b^
2)

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Maple [B]  time = 0.221, size = 7284, normalized size = 34.4 \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*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

result too large to display

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Maxima [B]  time = 1.81904, size = 1380, normalized size = 6.51 \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)^3*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/5*A*b^3*d*g^3*i*x^5 + 1/4*A*b^3*c*g^3*i*x^4 + 3/4*A*a*b^2*d*g^3*i*x^4 + A*a*b^2*c*g^3*i*x^3 + A*a^2*b*d*g^3*
i*x^3 + 3/2*A*a^2*b*c*g^3*i*x^2 + 1/2*A*a^3*d*g^3*i*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*c*g^3*i + 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*c*g^3*i + 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*c*g^3*i + 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*l
og(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*c*g^3*i + 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*a^3*d*g^3*i + 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^2*b*d*g
^3*i + 1/8*(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^2*d*g^
3*i + 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^3*d*g^3*i + A*a^3*c*g^3*i*x

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Fricas [B]  time = 1.35829, size = 1057, normalized size = 4.99 \begin{align*} \frac{24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \,{\left ({\left (5 \, A - B\right )} b^{5} c d^{4} +{\left (15 \, A + B\right )} a b^{4} d^{5}\right )} g^{3} i x^{4} - 2 \,{\left (B b^{5} c^{2} d^{3} - 10 \,{\left (6 \, A - B\right )} a b^{4} c d^{4} -{\left (60 \, A + 11 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 3 \,{\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 5 \,{\left (12 \, A - B\right )} a^{2} b^{3} c d^{4} +{\left (20 \, A + 9 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{3} i x^{2} - 6 \,{\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} - 5 \,{\left (4 \, A + B\right )} a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i x + 6 \,{\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i \log \left (b x + a\right ) + 6 \,{\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}\right )} g^{3} i \log \left (d x + c\right ) + 6 \,{\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \,{\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \,{\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \,{\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{120 \, b^{2} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*A*b^5*d^5*g^3*i*x^5 + 6*((5*A - B)*b^5*c*d^4 + (15*A + B)*a*b^4*d^5)*g^3*i*x^4 - 2*(B*b^5*c^2*d^3 -
10*(6*A - B)*a*b^4*c*d^4 - (60*A + 11*B)*a^2*b^3*d^5)*g^3*i*x^3 + 3*(B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 + 5*(12
*A - B)*a^2*b^3*c*d^4 + (20*A + 9*B)*a^3*b^2*d^5)*g^3*i*x^2 - 6*(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 - 5*(4*A + B)*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^3*i*x + 6*(5*B*a^4*b*c*d^4 - B*a^5*d^5)*g^3*i*log(b*x +
a) + 6*(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)*g^3*i*log(d*x + c) + 6*(4*
B*b^5*d^5*g^3*i*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*x + 5*(B*b^5*c*d^4 + 3*B*a*b^4*d^5)*g^3*i*x^4 + 20*(B*a*b^4*c*d
^4 + B*a^2*b^3*d^5)*g^3*i*x^3 + 10*(3*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*x^2)*log((b*e*x + a*e)/(d*x + c))
)/(b^2*d^4)

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Sympy [B]  time = 9.87056, size = 1187, normalized size = 5.6 \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)**3*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**3*d*g**3*i*x**5/5 - B*a**4*g**3*i*(a*d - 5*b*c)*log(x + (B*a**5*c*d**4*g**3*i + B*a**5*d**4*g**3*i*(a*d -
5*b*c)/b - 15*B*a**4*b*c**2*d**3*g**3*i - B*a**4*c*d**3*g**3*i*(a*d - 5*b*c) + 10*B*a**3*b**2*c**3*d**2*g**3*
i - 5*B*a**2*b**3*c**4*d*g**3*i + B*a*b**4*c**5*g**3*i)/(B*a**5*d**5*g**3*i - 5*B*a**4*b*c*d**4*g**3*i - 10*B*
a**3*b**2*c**2*d**3*g**3*i + 10*B*a**2*b**3*c**3*d**2*g**3*i - 5*B*a*b**4*c**4*d*g**3*i + B*b**5*c**5*g**3*i))
/(20*b**2) - B*c**2*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3)*log(x + (B*a**5*c*d
**4*g**3*i - 15*B*a**4*b*c**2*d**3*g**3*i + 10*B*a**3*b**2*c**3*d**2*g**3*i - 5*B*a**2*b**3*c**4*d*g**3*i + B*
a*b**4*c**5*g**3*i + B*a*b*c**2*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3) - B*b**
2*c**3*g**3*i*(10*a**3*d**3 - 10*a**2*b*c*d**2 + 5*a*b**2*c**2*d - b**3*c**3)/d)/(B*a**5*d**5*g**3*i - 5*B*a**
4*b*c*d**4*g**3*i - 10*B*a**3*b**2*c**2*d**3*g**3*i + 10*B*a**2*b**3*c**3*d**2*g**3*i - 5*B*a*b**4*c**4*d*g**3
*i + B*b**5*c**5*g**3*i))/(20*d**4) + x**4*(3*A*a*b**2*d*g**3*i/4 + A*b**3*c*g**3*i/4 + B*a*b**2*d*g**3*i/20 -
B*b**3*c*g**3*i/20) + (B*a**3*c*g**3*i*x + B*a**3*d*g**3*i*x**2/2 + 3*B*a**2*b*c*g**3*i*x**2/2 + B*a**2*b*d*g
**3*i*x**3 + B*a*b**2*c*g**3*i*x**3 + 3*B*a*b**2*d*g**3*i*x**4/4 + B*b**3*c*g**3*i*x**4/4 + B*b**3*d*g**3*i*x*
*5/5)*log(e*(a + b*x)/(c + d*x)) + x**3*(60*A*a**2*b*d**2*g**3*i + 60*A*a*b**2*c*d*g**3*i + 11*B*a**2*b*d**2*g
**3*i - 10*B*a*b**2*c*d*g**3*i - B*b**3*c**2*g**3*i)/(60*d) + x**2*(20*A*a**3*d**3*g**3*i + 60*A*a**2*b*c*d**2
*g**3*i + 9*B*a**3*d**3*g**3*i - 5*B*a**2*b*c*d**2*g**3*i - 5*B*a*b**2*c**2*d*g**3*i + B*b**3*c**3*g**3*i)/(40
*d**2) + x*(20*A*a**3*b*c*d**3*g**3*i + B*a**4*d**4*g**3*i + 5*B*a**3*b*c*d**3*g**3*i - 10*B*a**2*b**2*c**2*d*
*2*g**3*i + 5*B*a*b**3*c**3*d*g**3*i - B*b**4*c**4*g**3*i)/(20*b*d**3)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out