3.2.53 \(\int \frac {(a g+b g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(c i+d i x)^3} \, dx\) [153]

Optimal. Leaf size=89 \[ -\frac {B g n (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d) i^3 (c+d x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {2561, 2341} \begin {gather*} \frac {g (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g n (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(153 c+153 d x)^3} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d (c+d x)^3}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d (c+d x)^2}\right ) \, dx\\ &=\frac {(b g) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3581577 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3581577 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7163154 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d^2 (c+d x)}+\frac {(b B g n) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3581577 d^2}-\frac {(B (b c-a d) g n) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7163154 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7163154 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d^2 (c+d x)}+\frac {(b B (b c-a d) g n) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3581577 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7163154 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7163154 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d^2 (c+d x)}+\frac {(b B (b c-a d) g n) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3581577 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7163154 d^2}\\ &=-\frac {B (b c-a d) g n}{14326308 d^2 (c+d x)^2}+\frac {b B g n}{7163154 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x)}{7163154 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7163154 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3581577 d^2 (c+d x)}-\frac {b^2 B g n \log (c+d x)}{7163154 d^2 (b c-a d)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(89)=178\).
time = 0.11, size = 215, normalized size = 2.42 \begin {gather*} \frac {g \left (\frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^2 (c+d x)^2}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 (c+d x)}+\frac {b B n \left (\frac {1}{c+d x}+\frac {b \log (a+b x)}{b c-a d}-\frac {b \log (c+d x)}{b c-a d}\right )}{d^2}-\frac {B n \left (\frac {b c-a d}{(c+d x)^2}+\frac {2 b}{c+d x}+\frac {2 b^2 \log (a+b x)}{b c-a d}-\frac {2 b^2 \log (c+d x)}{b c-a d}\right )}{4 d^2}\right )}{i^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (d i x +c i \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (80) = 160\).
time = 0.31, size = 530, normalized size = 5.96 \begin {gather*} -{\left (\frac {b^{2} \log \left (b x + a\right )}{2 i \, b^{2} c^{2} d - 4 i \, a b c d^{2} + 2 i \, a^{2} d^{3}} - \frac {b^{2} \log \left (d x + c\right )}{2 i \, b^{2} c^{2} d - 4 i \, a b c d^{2} + 2 i \, a^{2} d^{3}} - \frac {2 \, b d x + 3 \, b c - a d}{-4 i \, b c^{3} d + 4 i \, a c^{2} d^{2} - 4 \, {\left (i \, b c d^{3} - i \, a d^{4}\right )} x^{2} - 8 \, {\left (i \, b c^{2} d^{2} - i \, a c d^{3}\right )} x}\right )} B a g n - B b g n {\left (\frac {{\left (b^{2} c - 2 \, a b d\right )} \log \left (b x + a\right )}{2 i \, b^{2} c^{2} d^{2} - 4 i \, a b c d^{3} + 2 i \, a^{2} d^{4}} - \frac {{\left (b^{2} c - 2 \, a b d\right )} \log \left (d x + c\right )}{2 i \, b^{2} c^{2} d^{2} - 4 i \, a b c d^{3} + 2 i \, a^{2} d^{4}} - \frac {b c^{2} - 3 \, a c d + 2 \, {\left (b c d - 2 \, a d^{2}\right )} x}{-4 i \, b c^{3} d^{2} + 4 i \, a c^{2} d^{3} - 4 \, {\left (i \, b c d^{4} - i \, a d^{5}\right )} x^{2} - 8 \, {\left (i \, b c^{2} d^{3} - i \, a c d^{4}\right )} x}\right )} + \frac {{\left (2 \, d x + c\right )} B b g \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{2 i \, d^{4} x^{2} + 4 i \, c d^{3} x + 2 i \, c^{2} d^{2}} + \frac {{\left (2 \, d x + c\right )} A b g}{2 i \, d^{4} x^{2} + 4 i \, c d^{3} x + 2 i \, c^{2} d^{2}} + \frac {B a g \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{2 i \, d^{3} x^{2} + 4 i \, c d^{2} x + 2 i \, c^{2} d} + \frac {A a g}{2 i \, d^{3} x^{2} + 4 i \, c d^{2} x + 2 i \, c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (80) = 160\).
time = 0.42, size = 220, normalized size = 2.47 \begin {gather*} -\frac {{\left (-i \, B b^{2} c^{2} + i \, B a^{2} d^{2}\right )} g n - 2 \, {\left ({\left (-i \, A - i \, B\right )} b^{2} c^{2} + {\left (i \, A + i \, B\right )} a^{2} d^{2}\right )} g - 2 \, {\left ({\left (i \, B b^{2} c d - i \, B a b d^{2}\right )} g n + 2 \, {\left ({\left (-i \, A - i \, B\right )} b^{2} c d + {\left (i \, A + i \, B\right )} a b d^{2}\right )} g\right )} x - 2 \, {\left (i \, B b^{2} d^{2} g n x^{2} + 2 i \, B a b d^{2} g n x + i \, B a^{2} d^{2} g n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((-I*B*b^2*c^2 + I*B*a^2*d^2)*g*n - 2*((-I*A - I*B)*b^2*c^2 + (I*A + I*B)*a^2*d^2)*g - 2*((I*B*b^2*c*d -
I*B*a*b*d^2)*g*n + 2*((-I*A - I*B)*b^2*c*d + (I*A + I*B)*a*b*d^2)*g)*x - 2*(I*B*b^2*d^2*g*n*x^2 + 2*I*B*a*b*d^
2*g*n*x + I*B*a^2*d^2*g*n)*log((b*x + a)/(d*x + c)))/(b*c^3*d^2 - a*c^2*d^3 + (b*c*d^4 - a*d^5)*x^2 + 2*(b*c^2
*d^3 - a*c*d^4)*x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 3.63, size = 93, normalized size = 1.04 \begin {gather*} -\frac {1}{4} \, {\left (-\frac {2 i \, {\left (b x + a\right )}^{2} B g n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {{\left (i \, B g n - 2 i \, A g - 2 i \, B g\right )} {\left (b x + a\right )}^{2}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-2*I*(b*x + a)^2*B*g*n*log((b*x + a)/(d*x + c))/(d*x + c)^2 + (I*B*g*n - 2*I*A*g - 2*I*B*g)*(b*x + a)^2/
(d*x + c)^2)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

________________________________________________________________________________________

Mupad [B]
time = 5.49, size = 205, normalized size = 2.30 \begin {gather*} -\frac {x\,\left (2\,A\,b\,d\,g-B\,b\,d\,g\,n\right )+A\,a\,d\,g+A\,b\,c\,g-\frac {B\,a\,d\,g\,n}{2}-\frac {B\,b\,c\,g\,n}{2}}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,a\,g}{2\,d}+\frac {B\,b\,c\,g}{2\,d^2}+\frac {B\,b\,g\,x}{d}\right )}{c^2\,i^3+2\,c\,d\,i^3\,x+d^2\,i^3\,x^2}+\frac {B\,b^2\,g\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________