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

Optimal. Leaf size=283 \[ \frac{B n (b c-a d) \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 g (f+g x)^3}+\frac{b^3 B n \log (a+b x)}{3 g (b f-a g)^3}-\frac{B n (b c-a d) (-a d g-b c g+2 b d f)}{3 (f+g x) (b f-a g)^2 (d f-c g)^2}-\frac{B n (b c-a d)}{6 (f+g x)^2 (b f-a g) (d f-c g)}-\frac{B d^3 n \log (c+d x)}{3 g (d f-c g)^3} \]

[Out]

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

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Rubi [A]  time = 0.457508, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 72} \[ \frac{B n (b c-a d) \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 g (f+g x)^3}+\frac{b^3 B n \log (a+b x)}{3 g (b f-a g)^3}-\frac{B n (b c-a d) (-a d g-b c g+2 b d f)}{3 (f+g x) (b f-a g)^2 (d f-c g)^2}-\frac{B n (b c-a d)}{6 (f+g x)^2 (b f-a g) (d f-c g)}-\frac{B d^3 n \log (c+d x)}{3 g (d f-c g)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.02113, size = 264, normalized size = 0.93 \[ \frac{B n (b c-a d) \left (\frac{g \log (f+g x) \left (a^2 d^2 g^2+a b d g (c g-3 d f)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(b f-a g)^3 (d f-c g)^3}+\frac{b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}+\frac{d^3 \log (c+d x)}{(b c-a d) (c g-d f)^3}+\frac{g (a d g+b c g-2 b d f)}{(f+g x) (b f-a g)^2 (d f-c g)^2}-\frac{g}{2 (f+g x)^2 (b f-a g) (d f-c g)}\right )-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{(f+g x)^3}}{3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.507, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{4}} \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((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x)

[Out]

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

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Maxima [B]  time = 1.54857, size = 1150, normalized size = 4.06 \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)/(d*x+c))^n))/(g*x+f)^4,x, algorithm="maxima")

[Out]

1/6*(2*b^3*log(b*x + a)/(b^3*f^3*g - 3*a*b^2*f^2*g^2 + 3*a^2*b*f*g^3 - a^3*g^4) - 2*d^3*log(d*x + c)/(d^3*f^3*
g - 3*c*d^2*f^2*g^2 + 3*c^2*d*f*g^3 - c^3*g^4) + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2 - 3*(b^3*c^2*d - a^2*b*d^3)*
f*g + (b^3*c^3 - a^3*d^3)*g^2)*log(g*x + f)/(b^3*d^3*f^6 + a^3*c^3*g^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^5*g + 3*(
b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^4*g^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^3*g^3 +
3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^2*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*f*g^5) - (5*(b^2*c*d - a*b*d^2)
*f^2 - 3*(b^2*c^2 - a^2*d^2)*f*g + (a*b*c^2 - a^2*c*d)*g^2 + 2*(2*(b^2*c*d - a*b*d^2)*f*g - (b^2*c^2 - a^2*d^2
)*g^2)*x)/(b^2*d^2*f^6 + a^2*c^2*f^2*g^4 - 2*(b^2*c*d + a*b*d^2)*f^5*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^4*g
^2 - 2*(a*b*c^2 + a^2*c*d)*f^3*g^3 + (b^2*d^2*f^4*g^2 + a^2*c^2*g^6 - 2*(b^2*c*d + a*b*d^2)*f^3*g^3 + (b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*f^2*g^4 - 2*(a*b*c^2 + a^2*c*d)*f*g^5)*x^2 + 2*(b^2*d^2*f^5*g + a^2*c^2*f*g^5 - 2*(b^2
*c*d + a*b*d^2)*f^4*g^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3*g^3 - 2*(a*b*c^2 + a^2*c*d)*f^2*g^4)*x))*B*n - 1
/3*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - 1/3*A/(g^4*x^3 + 3
*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g)

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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)/(d*x+c))^n))/(g*x+f)^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)/(d*x+c))**n))/(g*x+f)**4,x)

[Out]

Timed out

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Giac [B]  time = 3.99896, size = 2925, normalized size = 10.34 \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)/(d*x+c))^n))/(g*x+f)^4,x, algorithm="giac")

[Out]

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