3.4.1 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(g+h x)^4} \, dx\) [301]

Optimal. Leaf size=284 \[ -\frac {B (b c-a d) n}{6 (b g-a h) (d g-c h) (g+h x)^2}-\frac {B (b c-a d) (2 b d g-b c h-a d h) n}{3 (b g-a h)^2 (d g-c h)^2 (g+h x)}+\frac {b^3 B n \log (a+b x)}{3 h (b g-a h)^3}-\frac {B d^3 n \log (c+d x)}{3 h (d g-c h)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}+\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log (g+h x)}{3 (b g-a h)^3 (d g-c h)^3} \]

[Out]

-1/6*B*(-a*d+b*c)*n/(-a*h+b*g)/(-c*h+d*g)/(h*x+g)^2-1/3*B*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n/(-a*h+b*g)^2/(-c
*h+d*g)^2/(h*x+g)+1/3*b^3*B*n*ln(b*x+a)/h/(-a*h+b*g)^3-1/3*B*d^3*n*ln(d*x+c)/h/(-c*h+d*g)^3+1/3*(-A-B*ln(e*(b*
x+a)^n/((d*x+c)^n)))/h/(h*x+g)^3+1/3*B*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d
^2*g^2))*n*ln(h*x+g)/(-a*h+b*g)^3/(-c*h+d*g)^3

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Rubi [A]
time = 0.29, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 84} \begin {gather*} \frac {B n (b c-a d) \log (g+h x) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{3 (b g-a h)^3 (d g-c h)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 h (g+h x)^3}+\frac {b^3 B n \log (a+b x)}{3 h (b g-a h)^3}-\frac {B n (b c-a d) (-a d h-b c h+2 b d g)}{3 (g+h x) (b g-a h)^2 (d g-c h)^2}-\frac {B n (b c-a d)}{6 (g+h x)^2 (b g-a h) (d g-c h)}-\frac {B d^3 n \log (c+d x)}{3 h (d g-c h)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(B*(b*c - a*d)*n)/((b*g - a*h)*(d*g - c*h)*(g + h*x)^2) - (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n)/(3*
(b*g - a*h)^2*(d*g - c*h)^2*(g + h*x)) + (b^3*B*n*Log[a + b*x])/(3*h*(b*g - a*h)^3) - (B*d^3*n*Log[c + d*x])/(
3*h*(d*g - c*h)^3) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*h*(g + h*x)^3) + (B*(b*c - a*d)*(a^2*d^2*h^2
- a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[g + h*x])/(3*(b*g - a*h)^3*(d*g - c*h)^
3)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps

\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx &=\int \left (\frac {A}{(g+h x)^4}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4}\right ) \, dx\\ &=-\frac {A}{3 h (g+h x)^3}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4} \, dx\\ &=-\frac {A}{3 h (g+h x)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x) (g+h x)^3} \, dx}{3 h}\\ &=-\frac {A}{3 h (g+h x)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}+\frac {(B (b c-a d) n) \int \left (\frac {b^4}{(b c-a d) (b g-a h)^3 (a+b x)}+\frac {d^4}{(b c-a d) (-d g+c h)^3 (c+d x)}+\frac {h^2}{(b g-a h) (d g-c h) (g+h x)^3}-\frac {h^2 (-2 b d g+b c h+a d h)}{(b g-a h)^2 (d g-c h)^2 (g+h x)^2}+\frac {h^2 \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right )}{(b g-a h)^3 (d g-c h)^3 (g+h x)}\right ) \, dx}{3 h}\\ &=-\frac {A}{3 h (g+h x)^3}-\frac {B (b c-a d) n}{6 (b g-a h) (d g-c h) (g+h x)^2}-\frac {B (b c-a d) (2 b d g-b c h-a d h) n}{3 (b g-a h)^2 (d g-c h)^2 (g+h x)}+\frac {b^3 B n \log (a+b x)}{3 h (b g-a h)^3}-\frac {B d^3 n \log (c+d x)}{3 h (d g-c h)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}+\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log (g+h x)}{3 (b g-a h)^3 (d g-c h)^3}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 330, normalized size = 1.16 \begin {gather*} \frac {1}{6} \left (\frac {B (-b c+a d) n}{(b g-a h) (d g-c h) (g+h x)^2}+\frac {2 B (b c-a d) (-2 b d g+b c h+a d h) n}{(b g-a h)^2 (d g-c h)^2 (g+h x)}-\frac {2 b^3 B n \log (a+b x)}{h (-b g+a h)^3}-\frac {2 B n \log (a+b x)}{h (g+h x)^3}+\frac {2 B d^3 n \log (c+d x)}{h (-d g+c h)^3}+\frac {2 B n \log (c+d x)}{h (g+h x)^3}-\frac {2 \left (A-B n \log (a+b x)+B n \log (c+d x)+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{h (g+h x)^3}+\frac {2 B (b c-a d) \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log (g+h x)}{(b g-a h)^3 (d g-c h)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.74, size = 9645, normalized size = 33.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(9645\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^4,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (273) = 546\).
time = 0.38, size = 914, normalized size = 3.22 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, b^{3} n e \log \left (b x + a\right )}{b^{3} g^{3} h - 3 \, a b^{2} g^{2} h^{2} + 3 \, a^{2} b g h^{3} - a^{3} h^{4}} - \frac {2 \, d^{3} n e \log \left (d x + c\right )}{d^{3} g^{3} h - 3 \, c d^{2} g^{2} h^{2} + 3 \, c^{2} d g h^{3} - c^{3} h^{4}} + \frac {2 \, {\left (3 \, a b^{2} d^{3} g^{2} n - 3 \, a^{2} b d^{3} g h n + a^{3} d^{3} h^{2} n - {\left (3 \, c d^{2} g^{2} n - 3 \, c^{2} d g h n + c^{3} h^{2} n\right )} b^{3}\right )} e \log \left (h x + g\right )}{{\left (d^{3} g^{3} h^{3} - 3 \, c d^{2} g^{2} h^{4} + 3 \, c^{2} d g h^{5} - c^{3} h^{6}\right )} a^{3} - 3 \, {\left (d^{3} g^{4} h^{2} - 3 \, c d^{2} g^{3} h^{3} + 3 \, c^{2} d g^{2} h^{4} - c^{3} g h^{5}\right )} a^{2} b + 3 \, {\left (d^{3} g^{5} h - 3 \, c d^{2} g^{4} h^{2} + 3 \, c^{2} d g^{3} h^{3} - c^{3} g^{2} h^{4}\right )} a b^{2} - {\left (d^{3} g^{6} - 3 \, c d^{2} g^{5} h + 3 \, c^{2} d g^{4} h^{2} - c^{3} g^{3} h^{3}\right )} b^{3}} + \frac {2 \, {\left (2 \, a b d^{2} g h n - a^{2} d^{2} h^{2} n - {\left (2 \, c d g h n - c^{2} h^{2} n\right )} b^{2}\right )} x e - {\left ({\left (3 \, d^{2} g h n - c d h^{2} n\right )} a^{2} - {\left (5 \, d^{2} g^{2} n - c^{2} h^{2} n\right )} a b + {\left (5 \, c d g^{2} n - 3 \, c^{2} g h n\right )} b^{2}\right )} e}{{\left (d^{2} g^{4} h^{2} - 2 \, c d g^{3} h^{3} + c^{2} g^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{5} h - 2 \, c d g^{4} h^{2} + c^{2} g^{3} h^{3}\right )} a b + {\left (d^{2} g^{6} - 2 \, c d g^{5} h + c^{2} g^{4} h^{2}\right )} b^{2} + {\left ({\left (d^{2} g^{2} h^{4} - 2 \, c d g h^{5} + c^{2} h^{6}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h^{3} - 2 \, c d g^{2} h^{4} + c^{2} g h^{5}\right )} a b + {\left (d^{2} g^{4} h^{2} - 2 \, c d g^{3} h^{3} + c^{2} g^{2} h^{4}\right )} b^{2}\right )} x^{2} + 2 \, {\left ({\left (d^{2} g^{3} h^{3} - 2 \, c d g^{2} h^{4} + c^{2} g h^{5}\right )} a^{2} - 2 \, {\left (d^{2} g^{4} h^{2} - 2 \, c d g^{3} h^{3} + c^{2} g^{2} h^{4}\right )} a b + {\left (d^{2} g^{5} h - 2 \, c d g^{4} h^{2} + c^{2} g^{3} h^{3}\right )} b^{2}\right )} x}\right )} B e^{\left (-1\right )} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} - \frac {A}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1512 vs. \(2 (273) = 546\).
time = 3.25, size = 1512, normalized size = 5.32 \begin {gather*} \frac {B b^{4} n \log \left ({\left | b x + a \right |}\right )}{3 \, {\left (b^{4} g^{3} h - 3 \, a b^{3} g^{2} h^{2} + 3 \, a^{2} b^{2} g h^{3} - a^{3} b h^{4}\right )}} - \frac {B d^{4} n \log \left ({\left | d x + c \right |}\right )}{3 \, {\left (d^{4} g^{3} h - 3 \, c d^{3} g^{2} h^{2} + 3 \, c^{2} d^{2} g h^{3} - c^{3} d h^{4}\right )}} - \frac {B n \log \left (b x + a\right )}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} + \frac {B n \log \left (d x + c\right )}{3 \, {\left (h^{4} x^{3} + 3 \, g h^{3} x^{2} + 3 \, g^{2} h^{2} x + g^{3} h\right )}} + \frac {{\left (3 \, B b^{3} c d^{2} g^{2} n - 3 \, B a b^{2} d^{3} g^{2} n - 3 \, B b^{3} c^{2} d g h n + 3 \, B a^{2} b d^{3} g h n + B b^{3} c^{3} h^{2} n - B a^{3} d^{3} h^{2} n\right )} \log \left (h x + g\right )}{3 \, {\left (b^{3} d^{3} g^{6} - 3 \, b^{3} c d^{2} g^{5} h - 3 \, a b^{2} d^{3} g^{5} h + 3 \, b^{3} c^{2} d g^{4} h^{2} + 9 \, a b^{2} c d^{2} g^{4} h^{2} + 3 \, a^{2} b d^{3} g^{4} h^{2} - b^{3} c^{3} g^{3} h^{3} - 9 \, a b^{2} c^{2} d g^{3} h^{3} - 9 \, a^{2} b c d^{2} g^{3} h^{3} - a^{3} d^{3} g^{3} h^{3} + 3 \, a b^{2} c^{3} g^{2} h^{4} + 9 \, a^{2} b c^{2} d g^{2} h^{4} + 3 \, a^{3} c d^{2} g^{2} h^{4} - 3 \, a^{2} b c^{3} g h^{5} - 3 \, a^{3} c^{2} d g h^{5} + a^{3} c^{3} h^{6}\right )}} - \frac {4 \, B b^{2} c d g h^{3} n x^{2} - 4 \, B a b d^{2} g h^{3} n x^{2} - 2 \, B b^{2} c^{2} h^{4} n x^{2} + 2 \, B a^{2} d^{2} h^{4} n x^{2} + 9 \, B b^{2} c d g^{2} h^{2} n x - 9 \, B a b d^{2} g^{2} h^{2} n x - 5 \, B b^{2} c^{2} g h^{3} n x + 5 \, B a^{2} d^{2} g h^{3} n x + B a b c^{2} h^{4} n x - B a^{2} c d h^{4} n x + 5 \, B b^{2} c d g^{3} h n - 5 \, B a b d^{2} g^{3} h n - 3 \, B b^{2} c^{2} g^{2} h^{2} n + 3 \, B a^{2} d^{2} g^{2} h^{2} n + B a b c^{2} g h^{3} n - B a^{2} c d g h^{3} n + 2 \, A b^{2} d^{2} g^{4} + 2 \, B b^{2} d^{2} g^{4} - 4 \, A b^{2} c d g^{3} h - 4 \, B b^{2} c d g^{3} h - 4 \, A a b d^{2} g^{3} h - 4 \, B a b d^{2} g^{3} h + 2 \, A b^{2} c^{2} g^{2} h^{2} + 2 \, B b^{2} c^{2} g^{2} h^{2} + 8 \, A a b c d g^{2} h^{2} + 8 \, B a b c d g^{2} h^{2} + 2 \, A a^{2} d^{2} g^{2} h^{2} + 2 \, B a^{2} d^{2} g^{2} h^{2} - 4 \, A a b c^{2} g h^{3} - 4 \, B a b c^{2} g h^{3} - 4 \, A a^{2} c d g h^{3} - 4 \, B a^{2} c d g h^{3} + 2 \, A a^{2} c^{2} h^{4} + 2 \, B a^{2} c^{2} h^{4}}{6 \, {\left (b^{2} d^{2} g^{4} h^{4} x^{3} - 2 \, b^{2} c d g^{3} h^{5} x^{3} - 2 \, a b d^{2} g^{3} h^{5} x^{3} + b^{2} c^{2} g^{2} h^{6} x^{3} + 4 \, a b c d g^{2} h^{6} x^{3} + a^{2} d^{2} g^{2} h^{6} x^{3} - 2 \, a b c^{2} g h^{7} x^{3} - 2 \, a^{2} c d g h^{7} x^{3} + a^{2} c^{2} h^{8} x^{3} + 3 \, b^{2} d^{2} g^{5} h^{3} x^{2} - 6 \, b^{2} c d g^{4} h^{4} x^{2} - 6 \, a b d^{2} g^{4} h^{4} x^{2} + 3 \, b^{2} c^{2} g^{3} h^{5} x^{2} + 12 \, a b c d g^{3} h^{5} x^{2} + 3 \, a^{2} d^{2} g^{3} h^{5} x^{2} - 6 \, a b c^{2} g^{2} h^{6} x^{2} - 6 \, a^{2} c d g^{2} h^{6} x^{2} + 3 \, a^{2} c^{2} g h^{7} x^{2} + 3 \, b^{2} d^{2} g^{6} h^{2} x - 6 \, b^{2} c d g^{5} h^{3} x - 6 \, a b d^{2} g^{5} h^{3} x + 3 \, b^{2} c^{2} g^{4} h^{4} x + 12 \, a b c d g^{4} h^{4} x + 3 \, a^{2} d^{2} g^{4} h^{4} x - 6 \, a b c^{2} g^{3} h^{5} x - 6 \, a^{2} c d g^{3} h^{5} x + 3 \, a^{2} c^{2} g^{2} h^{6} x + b^{2} d^{2} g^{7} h - 2 \, b^{2} c d g^{6} h^{2} - 2 \, a b d^{2} g^{6} h^{2} + b^{2} c^{2} g^{5} h^{3} + 4 \, a b c d g^{5} h^{3} + a^{2} d^{2} g^{5} h^{3} - 2 \, a b c^{2} g^{4} h^{4} - 2 \, a^{2} c d g^{4} h^{4} + a^{2} c^{2} g^{3} h^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*b^4*n*log(abs(b*x + a))/(b^4*g^3*h - 3*a*b^3*g^2*h^2 + 3*a^2*b^2*g*h^3 - a^3*b*h^4) - 1/3*B*d^4*n*log(ab
s(d*x + c))/(d^4*g^3*h - 3*c*d^3*g^2*h^2 + 3*c^2*d^2*g*h^3 - c^3*d*h^4) - 1/3*B*n*log(b*x + a)/(h^4*x^3 + 3*g*
h^3*x^2 + 3*g^2*h^2*x + g^3*h) + 1/3*B*n*log(d*x + c)/(h^4*x^3 + 3*g*h^3*x^2 + 3*g^2*h^2*x + g^3*h) + 1/3*(3*B
*b^3*c*d^2*g^2*n - 3*B*a*b^2*d^3*g^2*n - 3*B*b^3*c^2*d*g*h*n + 3*B*a^2*b*d^3*g*h*n + B*b^3*c^3*h^2*n - B*a^3*d
^3*h^2*n)*log(h*x + g)/(b^3*d^3*g^6 - 3*b^3*c*d^2*g^5*h - 3*a*b^2*d^3*g^5*h + 3*b^3*c^2*d*g^4*h^2 + 9*a*b^2*c*
d^2*g^4*h^2 + 3*a^2*b*d^3*g^4*h^2 - b^3*c^3*g^3*h^3 - 9*a*b^2*c^2*d*g^3*h^3 - 9*a^2*b*c*d^2*g^3*h^3 - a^3*d^3*
g^3*h^3 + 3*a*b^2*c^3*g^2*h^4 + 9*a^2*b*c^2*d*g^2*h^4 + 3*a^3*c*d^2*g^2*h^4 - 3*a^2*b*c^3*g*h^5 - 3*a^3*c^2*d*
g*h^5 + a^3*c^3*h^6) - 1/6*(4*B*b^2*c*d*g*h^3*n*x^2 - 4*B*a*b*d^2*g*h^3*n*x^2 - 2*B*b^2*c^2*h^4*n*x^2 + 2*B*a^
2*d^2*h^4*n*x^2 + 9*B*b^2*c*d*g^2*h^2*n*x - 9*B*a*b*d^2*g^2*h^2*n*x - 5*B*b^2*c^2*g*h^3*n*x + 5*B*a^2*d^2*g*h^
3*n*x + B*a*b*c^2*h^4*n*x - B*a^2*c*d*h^4*n*x + 5*B*b^2*c*d*g^3*h*n - 5*B*a*b*d^2*g^3*h*n - 3*B*b^2*c^2*g^2*h^
2*n + 3*B*a^2*d^2*g^2*h^2*n + B*a*b*c^2*g*h^3*n - B*a^2*c*d*g*h^3*n + 2*A*b^2*d^2*g^4 + 2*B*b^2*d^2*g^4 - 4*A*
b^2*c*d*g^3*h - 4*B*b^2*c*d*g^3*h - 4*A*a*b*d^2*g^3*h - 4*B*a*b*d^2*g^3*h + 2*A*b^2*c^2*g^2*h^2 + 2*B*b^2*c^2*
g^2*h^2 + 8*A*a*b*c*d*g^2*h^2 + 8*B*a*b*c*d*g^2*h^2 + 2*A*a^2*d^2*g^2*h^2 + 2*B*a^2*d^2*g^2*h^2 - 4*A*a*b*c^2*
g*h^3 - 4*B*a*b*c^2*g*h^3 - 4*A*a^2*c*d*g*h^3 - 4*B*a^2*c*d*g*h^3 + 2*A*a^2*c^2*h^4 + 2*B*a^2*c^2*h^4)/(b^2*d^
2*g^4*h^4*x^3 - 2*b^2*c*d*g^3*h^5*x^3 - 2*a*b*d^2*g^3*h^5*x^3 + b^2*c^2*g^2*h^6*x^3 + 4*a*b*c*d*g^2*h^6*x^3 +
a^2*d^2*g^2*h^6*x^3 - 2*a*b*c^2*g*h^7*x^3 - 2*a^2*c*d*g*h^7*x^3 + a^2*c^2*h^8*x^3 + 3*b^2*d^2*g^5*h^3*x^2 - 6*
b^2*c*d*g^4*h^4*x^2 - 6*a*b*d^2*g^4*h^4*x^2 + 3*b^2*c^2*g^3*h^5*x^2 + 12*a*b*c*d*g^3*h^5*x^2 + 3*a^2*d^2*g^3*h
^5*x^2 - 6*a*b*c^2*g^2*h^6*x^2 - 6*a^2*c*d*g^2*h^6*x^2 + 3*a^2*c^2*g*h^7*x^2 + 3*b^2*d^2*g^6*h^2*x - 6*b^2*c*d
*g^5*h^3*x - 6*a*b*d^2*g^5*h^3*x + 3*b^2*c^2*g^4*h^4*x + 12*a*b*c*d*g^4*h^4*x + 3*a^2*d^2*g^4*h^4*x - 6*a*b*c^
2*g^3*h^5*x - 6*a^2*c*d*g^3*h^5*x + 3*a^2*c^2*g^2*h^6*x + b^2*d^2*g^7*h - 2*b^2*c*d*g^6*h^2 - 2*a*b*d^2*g^6*h^
2 + b^2*c^2*g^5*h^3 + 4*a*b*c*d*g^5*h^3 + a^2*d^2*g^5*h^3 - 2*a*b*c^2*g^4*h^4 - 2*a^2*c*d*g^4*h^4 + a^2*c^2*g^
3*h^5)

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Mupad [B]
time = 9.26, size = 1183, normalized size = 4.17 \begin {gather*} \frac {B\,d^3\,n\,\ln \left (c+d\,x\right )}{3\,c^3\,h^4-9\,c^2\,d\,g\,h^3+9\,c\,d^2\,g^2\,h^2-3\,d^3\,g^3\,h}-\frac {\ln \left (g+h\,x\right )\,\left (h^2\,\left (B\,a^3\,d^3\,n-B\,b^3\,c^3\,n\right )-h\,\left (3\,B\,a^2\,b\,d^3\,g\,n-3\,B\,b^3\,c^2\,d\,g\,n\right )+3\,B\,a\,b^2\,d^3\,g^2\,n-3\,B\,b^3\,c\,d^2\,g^2\,n\right )}{3\,a^3\,c^3\,h^6-9\,a^3\,c^2\,d\,g\,h^5+9\,a^3\,c\,d^2\,g^2\,h^4-3\,a^3\,d^3\,g^3\,h^3-9\,a^2\,b\,c^3\,g\,h^5+27\,a^2\,b\,c^2\,d\,g^2\,h^4-27\,a^2\,b\,c\,d^2\,g^3\,h^3+9\,a^2\,b\,d^3\,g^4\,h^2+9\,a\,b^2\,c^3\,g^2\,h^4-27\,a\,b^2\,c^2\,d\,g^3\,h^3+27\,a\,b^2\,c\,d^2\,g^4\,h^2-9\,a\,b^2\,d^3\,g^5\,h-3\,b^3\,c^3\,g^3\,h^3+9\,b^3\,c^2\,d\,g^4\,h^2-9\,b^3\,c\,d^2\,g^5\,h+3\,b^3\,d^3\,g^6}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{3\,h\,\left (g^3+3\,g^2\,h\,x+3\,g\,h^2\,x^2+h^3\,x^3\right )}-\frac {B\,b^3\,n\,\ln \left (a+b\,x\right )}{3\,a^3\,h^4-9\,a^2\,b\,g\,h^3+9\,a\,b^2\,g^2\,h^2-3\,b^3\,g^3\,h}-\frac {\frac {2\,A\,a^2\,c^2\,h^4+2\,A\,b^2\,d^2\,g^4+2\,A\,a^2\,d^2\,g^2\,h^2+2\,A\,b^2\,c^2\,g^2\,h^2+3\,B\,a^2\,d^2\,g^2\,h^2\,n-3\,B\,b^2\,c^2\,g^2\,h^2\,n-4\,A\,a\,b\,c^2\,g\,h^3-4\,A\,a\,b\,d^2\,g^3\,h-4\,A\,a^2\,c\,d\,g\,h^3-4\,A\,b^2\,c\,d\,g^3\,h+8\,A\,a\,b\,c\,d\,g^2\,h^2+B\,a\,b\,c^2\,g\,h^3\,n-5\,B\,a\,b\,d^2\,g^3\,h\,n-B\,a^2\,c\,d\,g\,h^3\,n+5\,B\,b^2\,c\,d\,g^3\,h\,n}{2\,\left (a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4\right )}+\frac {x\,\left (-B\,n\,a^2\,c\,d\,h^4+5\,B\,n\,a^2\,d^2\,g\,h^3+B\,n\,a\,b\,c^2\,h^4-9\,B\,n\,a\,b\,d^2\,g^2\,h^2-5\,B\,n\,b^2\,c^2\,g\,h^3+9\,B\,n\,b^2\,c\,d\,g^2\,h^2\right )}{2\,\left (a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4\right )}+\frac {x^2\,\left (B\,n\,a^2\,d^2\,h^4-2\,B\,g\,n\,a\,b\,d^2\,h^3-B\,n\,b^2\,c^2\,h^4+2\,B\,g\,n\,b^2\,c\,d\,h^3\right )}{a^2\,c^2\,h^4-2\,a^2\,c\,d\,g\,h^3+a^2\,d^2\,g^2\,h^2-2\,a\,b\,c^2\,g\,h^3+4\,a\,b\,c\,d\,g^2\,h^2-2\,a\,b\,d^2\,g^3\,h+b^2\,c^2\,g^2\,h^2-2\,b^2\,c\,d\,g^3\,h+b^2\,d^2\,g^4}}{3\,g^3\,h+9\,g^2\,h^2\,x+9\,g\,h^3\,x^2+3\,h^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(g + h*x)^4,x)

[Out]

(B*d^3*n*log(c + d*x))/(3*c^3*h^4 - 3*d^3*g^3*h + 9*c*d^2*g^2*h^2 - 9*c^2*d*g*h^3) - (log(g + h*x)*(h^2*(B*a^3
*d^3*n - B*b^3*c^3*n) - h*(3*B*a^2*b*d^3*g*n - 3*B*b^3*c^2*d*g*n) + 3*B*a*b^2*d^3*g^2*n - 3*B*b^3*c*d^2*g^2*n)
)/(3*a^3*c^3*h^6 + 3*b^3*d^3*g^6 - 3*a^3*d^3*g^3*h^3 - 3*b^3*c^3*g^3*h^3 - 9*a^2*b*c^3*g*h^5 - 9*a*b^2*d^3*g^5
*h - 9*a^3*c^2*d*g*h^5 - 9*b^3*c*d^2*g^5*h + 9*a*b^2*c^3*g^2*h^4 + 9*a^2*b*d^3*g^4*h^2 + 9*a^3*c*d^2*g^2*h^4 +
9*b^3*c^2*d*g^4*h^2 + 27*a*b^2*c*d^2*g^4*h^2 - 27*a*b^2*c^2*d*g^3*h^3 - 27*a^2*b*c*d^2*g^3*h^3 + 27*a^2*b*c^2
*d*g^2*h^4) - (B*log((e*(a + b*x)^n)/(c + d*x)^n))/(3*h*(g^3 + h^3*x^3 + 3*g^2*h*x + 3*g*h^2*x^2)) - (B*b^3*n*
log(a + b*x))/(3*a^3*h^4 - 3*b^3*g^3*h + 9*a*b^2*g^2*h^2 - 9*a^2*b*g*h^3) - ((2*A*a^2*c^2*h^4 + 2*A*b^2*d^2*g^
4 + 2*A*a^2*d^2*g^2*h^2 + 2*A*b^2*c^2*g^2*h^2 + 3*B*a^2*d^2*g^2*h^2*n - 3*B*b^2*c^2*g^2*h^2*n - 4*A*a*b*c^2*g*
h^3 - 4*A*a*b*d^2*g^3*h - 4*A*a^2*c*d*g*h^3 - 4*A*b^2*c*d*g^3*h + 8*A*a*b*c*d*g^2*h^2 + B*a*b*c^2*g*h^3*n - 5*
B*a*b*d^2*g^3*h*n - B*a^2*c*d*g*h^3*n + 5*B*b^2*c*d*g^3*h*n)/(2*(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 +
b^2*c^2*g^2*h^2 - 2*a*b*c^2*g*h^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*g^3*h + 4*a*b*c*d*g^2*h^2))
+ (x*(B*a*b*c^2*h^4*n - B*a^2*c*d*h^4*n + 5*B*a^2*d^2*g*h^3*n - 5*B*b^2*c^2*g*h^3*n - 9*B*a*b*d^2*g^2*h^2*n +
9*B*b^2*c*d*g^2*h^2*n))/(2*(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g^2*h^2 - 2*a*b*c^2*g*h^3 -
2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*g^3*h + 4*a*b*c*d*g^2*h^2)) + (x^2*(B*a^2*d^2*h^4*n - B*b^2*c^2
*h^4*n - 2*B*a*b*d^2*g*h^3*n + 2*B*b^2*c*d*g*h^3*n))/(a^2*c^2*h^4 + b^2*d^2*g^4 + a^2*d^2*g^2*h^2 + b^2*c^2*g^
2*h^2 - 2*a*b*c^2*g*h^3 - 2*a*b*d^2*g^3*h - 2*a^2*c*d*g*h^3 - 2*b^2*c*d*g^3*h + 4*a*b*c*d*g^2*h^2))/(3*g^3*h +
3*h^4*x^3 + 9*g^2*h^2*x + 9*g*h^3*x^2)

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