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

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

[Out]

-(B*(b*c - a*d)*n)/(6*(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)

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Rubi [A]  time = 0.536117, antiderivative size = 296, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6742, 2492, 72} \[ \frac{B n (b c-a d) \log (g+h x) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{3 (b g-a h)^3 (d g-c h)^3}+\frac{b^3 B n \log (a+b x)}{3 h (b g-a h)^3}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h (g+h x)^3}-\frac{B n (b c-a d) (-a d h-b c h+2 b d g)}{3 (g+h x) (b g-a h)^2 (d g-c h)^2}-\frac{B n (b c-a d)}{6 (g+h x)^2 (b g-a h) (d g-c h)}-\frac{A}{3 h (g+h x)^3}-\frac{B d^3 n \log (c+d x)}{3 h (d g-c h)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(3*h*(g + h*x)^3) - (B*(b*c - a*d)*n)/(6*(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) - (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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*
(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && NeQ[m, -1]

Rule 72

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]

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*}

Mathematica [A]  time = 1.34277, size = 273, normalized size = 0.96 \[ -\frac{B n (b c-a d) \left (-\frac{2 h \log (g+h x) \left (a^2 d^2 h^2+a b d h (c h-3 d g)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{(b g-a h)^3 (d g-c h)^3}-\frac{2 b^3 \log (a+b x)}{(b c-a d) (b g-a h)^3}+\frac{2 d^3 \log (c+d x)}{(b c-a d) (d g-c h)^3}-\frac{2 h (a d h+b c h-2 b d g)}{(g+h x) (b g-a h)^2 (d g-c h)^2}+\frac{h}{(g+h x)^2 (b g-a h) (d g-c h)}\right )+\frac{2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}+\frac{2 A}{(g+h x)^3}}{6 h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.925, size = 9645, normalized size = 34. \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

result too large to display

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Maxima [B]  time = 1.63639, size = 1242, normalized size = 4.37 \begin{align*} \text{result too large to display} \end{align*}

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*e*n*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*e*n*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*e*g^2*n - 3*a^2*b*d^3*e*g*h*n + a^3*d^
3*e*h^2*n - (3*c*d^2*e*g^2*n - 3*c^2*d*e*g*h*n + c^3*e*h^2*n)*b^3)*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) - ((3*d^2*e*g*h*n - c*d*e*h^2*n)*a^2 - (5*d^2*e*g^2*n - c^2*e*h^2*n)*a*b + (5*c*d*e*g^2*
n - 3*c^2*e*g*h*n)*b^2 - 2*(2*a*b*d^2*e*g*h*n - a^2*d^2*e*h^2*n - (2*c*d*e*g*h*n - c^2*e*h^2*n)*b^2)*x)/((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 - 1/3*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)]  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((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)]  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((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]  time = 4.52414, size = 2041, normalized size = 7.19 \begin{align*} \text{result too large to display} \end{align*}

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)