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

Optimal. Leaf size=283 \[ -\frac{g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac{b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2}+\frac{B g i^3 n (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g i^3 n (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac{B g i^3 n (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac{B g i^3 n x (b c-a d)^4}{20 b^3 d}+\frac{B g i^3 n (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac{B g i^3 n (c+d x)^4 (b c-a d)}{20 d^2} \]

[Out]

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

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Rubi [A]  time = 0.381663, antiderivative size = 243, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 43} \[ -\frac{g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac{b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2}+\frac{B g i^3 n (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g i^3 n (b c-a d)^5 \log (a+b x)}{20 b^4 d^2}+\frac{B g i^3 n x (b c-a d)^4}{20 b^3 d}+\frac{B g i^3 n (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac{B g i^3 n (c+d x)^4 (b c-a d)}{20 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (129 c+129 d x)^3 (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac{(-b c+a d) g (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{b g (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{129 d}\right ) \, dx\\ &=\frac{(b g) \int (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{129 d}+\frac{((-b c+a d) g) \int (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}\\ &=-\frac{2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac{2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac{(b B g n) \int \frac{35723051649 (b c-a d) (c+d x)^4}{a+b x} \, dx}{83205 d^2}+\frac{(B (b c-a d) g n) \int \frac{276922881 (b c-a d) (c+d x)^3}{a+b x} \, dx}{516 d^2}\\ &=-\frac{2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac{2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac{(2146689 b B (b c-a d) g n) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^2}+\frac{\left (2146689 B (b c-a d)^2 g n\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d^2}\\ &=-\frac{2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac{2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac{(2146689 b B (b c-a d) g n) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^2}+\frac{\left (2146689 B (b c-a d)^2 g n\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{4 d^2}\\ &=\frac{2146689 B (b c-a d)^4 g n x}{20 b^3 d}+\frac{2146689 B (b c-a d)^3 g n (c+d x)^2}{40 b^2 d^2}+\frac{715563 B (b c-a d)^2 g n (c+d x)^3}{20 b d^2}-\frac{2146689 B (b c-a d) g n (c+d x)^4}{20 d^2}+\frac{2146689 B (b c-a d)^5 g n \log (a+b x)}{20 b^4 d^2}-\frac{2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac{2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}\\ \end{align*}

Mathematica [A]  time = 0.214849, size = 269, normalized size = 0.95 \[ \frac{g i^3 \left (24 b (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-30 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{5 B n (b c-a d)^2 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^4}-\frac{2 B n (b c-a d) \left (6 b^2 (c+d x)^2 (b c-a d)^2+4 b^3 (c+d x)^3 (b c-a d)+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{b^4}\right )}{120 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*((5*B*(b*c - a*d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b
*c - a*d)^3*Log[a + b*x]))/b^4 - (2*B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2
+ 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]))/b^4 - 30*(b*c - a*d)*(c
+ d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(1
20*d^2)

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Maple [F]  time = 0.536, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.50127, size = 1509, normalized size = 5.33 \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)*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/5*B*b*d^3*g*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b*d^3*g*i^3*x^5 + 3/4*B*b*c*d^2*g*i^3*x^4
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a*d^3*g*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A
*b*c*d^2*g*i^3*x^4 + 1/4*A*a*d^3*g*i^3*x^4 + B*b*c^2*d*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a*
c*d^2*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*b*c^2*d*g*i^3*x^3 + A*a*c*d^2*g*i^3*x^3 + 1/2*B*b*c
^3*g*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a*c^2*d*g*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x +
c))^n) + 1/2*A*b*c^3*g*i^3*x^2 + 3/2*A*a*c^2*d*g*i^3*x^2 + 1/60*B*b*d^3*g*i^3*n*(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)) - 1/8*B*b*c*d^2*g*i^3*n*(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)) - 1/24*B*a*d^3*g*i^3*n*(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)) + 1/2*B*b*c^2*d*g*i^3*n*(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)) + 1/2*B*a*
c*d^2*g*i^3*n*(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)) - 1/2*B*b*c^3*g*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) -
3/2*B*a*c^2*d*g*i^3*n*(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*c^3*g*i^3*n*(a
*log(b*x + a)/b - c*log(d*x + c)/d) + B*a*c^3*g*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*c^3*g*i^3*x

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Fricas [B]  time = 0.796294, size = 1501, normalized size = 5.3 \begin{align*} \frac{24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \,{\left (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} - B a^{5} d^{5}\right )} g i^{3} n \log \left (b x + a\right ) + 6 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} n \log \left (d x + c\right ) - 6 \,{\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g i^{3} n - 5 \,{\left (3 \, A b^{5} c d^{4} + A a b^{4} d^{5}\right )} g i^{3}\right )} x^{4} - 2 \,{\left ({\left (11 \, B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} - B a^{2} b^{3} d^{5}\right )} g i^{3} n - 60 \,{\left (A b^{5} c^{2} d^{3} + A a b^{4} c d^{4}\right )} g i^{3}\right )} x^{3} - 3 \,{\left ({\left (9 \, B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g i^{3} n - 20 \,{\left (A b^{5} c^{3} d^{2} + 3 \, A a b^{4} c^{2} d^{3}\right )} g i^{3}\right )} x^{2} + 6 \,{\left (20 \, A a b^{4} c^{3} d^{2} g i^{3} -{\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 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} n\right )} x + 6 \,{\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \,{\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \,{\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \,{\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (e\right ) + 6 \,{\left (4 \, B b^{5} d^{5} g i^{3} n x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} n x + 5 \,{\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} n x^{4} + 20 \,{\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} n x^{3} + 10 \,{\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} n x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{120 \, b^{4} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [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)*(d*i*x+c*i)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

Timed out

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

[Out]

Timed out