3.1.65 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(f+g x)^4} \, dx\) [65]

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

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2547, 84} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(B*(b*c - a*d)*n)/((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 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 2547

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x
_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(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] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]

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*}

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Mathematica [A]
time = 0.52, size = 264, normalized size = 0.93 \begin {gather*} \frac {-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^3}+B (b c-a d) n \left (-\frac {g}{2 (b f-a g) (d f-c g) (f+g x)^2}+\frac {g (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\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) (-d f+c g)^3}+\frac {g \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 ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}\right )}{3 g} \end {gather*}

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*(-1/2*g/((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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (g x +f \right )^{4}}\, dx\]

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] Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (272) = 544\).
time = 0.36, size = 853, normalized size = 3.01 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, b^{3} \log \left (b x + a\right )}{b^{3} f^{3} g - 3 \, a b^{2} f^{2} g^{2} + 3 \, a^{2} b f g^{3} - a^{3} g^{4}} - \frac {2 \, d^{3} \log \left (d x + c\right )}{d^{3} f^{3} g - 3 \, c d^{2} f^{2} g^{2} + 3 \, c^{2} d f g^{3} - c^{3} g^{4}} + \frac {2 \, {\left (3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f^{2} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} f g + {\left (b^{3} c^{3} - a^{3} d^{3}\right )} g^{2}\right )} \log \left (g x + f\right )}{b^{3} d^{3} f^{6} + a^{3} c^{3} g^{6} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f^{5} g + 3 \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{4} g^{2} - {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f^{3} g^{3} + 3 \, {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f^{2} g^{4} - 3 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} f g^{5}} - \frac {5 \, {\left (b^{2} c d - a b d^{2}\right )} f^{2} - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f g + {\left (a b c^{2} - a^{2} c d\right )} g^{2} + 2 \, {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f g - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g^{2}\right )} x}{b^{2} d^{2} f^{6} + a^{2} c^{2} f^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{5} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{4} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3} g^{3} + {\left (b^{2} d^{2} f^{4} g^{2} + a^{2} c^{2} g^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} f^{5} g + a^{2} c^{2} f g^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{4} g^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3} g^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} g^{4}\right )} x}\right )} B n - \frac {B \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} - \frac {A}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} \end {gather*}

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((b*x/(d*x + c) + a/(d*x + c))^n*e)/(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)] 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)/(d*x+c))^n))/(g*x+f)^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)/(d*x+c))**n))/(g*x+f)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 9570 vs. \(2 (272) = 544\).
time = 3.23, size = 9570, normalized size = 33.82 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/6*(2*(3*B*b^4*c^2*d^2*f^2*n - 6*B*a*b^3*c*d^3*f^2*n + 3*B*a^2*b^2*d^4*f^2*n - 3*B*b^4*c^3*d*f*g*n + 3*B*a*b^
3*c^2*d^2*f*g*n + 3*B*a^2*b^2*c*d^3*f*g*n - 3*B*a^3*b*d^4*f*g*n + B*b^4*c^4*g^2*n - B*a*b^3*c^3*d*g^2*n - B*a^
3*b*c*d^3*g^2*n + B*a^4*d^4*g^2*n)*log(-b*f + (b*x + a)*d*f/(d*x + c) + a*g - (b*x + a)*c*g/(d*x + 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) + 2*(3*B*
b^4*c^2*d^2*f^2*n - 6*B*a*b^3*c*d^3*f^2*n - 6*(b*x + a)*B*b^3*c^2*d^3*f^2*n/(d*x + c) + 3*B*a^2*b^2*d^4*f^2*n
+ 12*(b*x + a)*B*a*b^2*c*d^4*f^2*n/(d*x + c) + 3*(b*x + a)^2*B*b^2*c^2*d^4*f^2*n/(d*x + c)^2 - 6*(b*x + a)*B*a
^2*b*d^5*f^2*n/(d*x + c) - 6*(b*x + a)^2*B*a*b*c*d^5*f^2*n/(d*x + c)^2 + 3*(b*x + a)^2*B*a^2*d^6*f^2*n/(d*x +
c)^2 - 3*B*b^4*c^3*d*f*g*n + 3*B*a*b^3*c^2*d^2*f*g*n + 9*(b*x + a)*B*b^3*c^3*d^2*f*g*n/(d*x + c) + 3*B*a^2*b^2
*c*d^3*f*g*n - 15*(b*x + a)*B*a*b^2*c^2*d^3*f*g*n/(d*x + c) - 6*(b*x + a)^2*B*b^2*c^3*d^3*f*g*n/(d*x + c)^2 -
3*B*a^3*b*d^4*f*g*n + 3*(b*x + a)*B*a^2*b*c*d^4*f*g*n/(d*x + c) + 12*(b*x + a)^2*B*a*b*c^2*d^4*f*g*n/(d*x + c)
^2 + 3*(b*x + a)*B*a^3*d^5*f*g*n/(d*x + c) - 6*(b*x + a)^2*B*a^2*c*d^5*f*g*n/(d*x + c)^2 + B*b^4*c^4*g^2*n - B
*a*b^3*c^3*d*g^2*n - 3*(b*x + a)*B*b^3*c^4*d*g^2*n/(d*x + c) + 3*(b*x + a)*B*a*b^2*c^3*d^2*g^2*n/(d*x + c) + 3
*(b*x + a)^2*B*b^2*c^4*d^2*g^2*n/(d*x + c)^2 - B*a^3*b*c*d^3*g^2*n + 3*(b*x + a)*B*a^2*b*c^2*d^3*g^2*n/(d*x +
c) - 6*(b*x + a)^2*B*a*b*c^3*d^3*g^2*n/(d*x + c)^2 + B*a^4*d^4*g^2*n - 3*(b*x + a)*B*a^3*c*d^4*g^2*n/(d*x + c)
+ 3*(b*x + a)^2*B*a^2*c^2*d^4*g^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(b^3*d^3*f^6 - 3*(b*x + a)*b^2*d^4*
f^6/(d*x + c) + 3*(b*x + a)^2*b*d^5*f^6/(d*x + c)^2 - (b*x + a)^3*d^6*f^6/(d*x + c)^3 - 3*b^3*c*d^2*f^5*g - 3*
a*b^2*d^3*f^5*g + 12*(b*x + a)*b^2*c*d^3*f^5*g/(d*x + c) + 6*(b*x + a)*a*b*d^4*f^5*g/(d*x + c) - 15*(b*x + a)^
2*b*c*d^4*f^5*g/(d*x + c)^2 - 3*(b*x + a)^2*a*d^5*f^5*g/(d*x + c)^2 + 6*(b*x + a)^3*c*d^5*f^5*g/(d*x + c)^3 +
3*b^3*c^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 - 18*(b*x + a)*b^2*c^2*d^2*f^4*g^2/(d*x + c) + 3*a^2*b*d^3*f^4*g^2
- 24*(b*x + a)*a*b*c*d^3*f^4*g^2/(d*x + c) + 30*(b*x + a)^2*b*c^2*d^3*f^4*g^2/(d*x + c)^2 - 3*(b*x + a)*a^2*d
^4*f^4*g^2/(d*x + c) + 15*(b*x + a)^2*a*c*d^4*f^4*g^2/(d*x + c)^2 - 15*(b*x + a)^3*c^2*d^4*f^4*g^2/(d*x + c)^3
- b^3*c^3*f^3*g^3 - 9*a*b^2*c^2*d*f^3*g^3 + 12*(b*x + a)*b^2*c^3*d*f^3*g^3/(d*x + c) - 9*a^2*b*c*d^2*f^3*g^3
+ 36*(b*x + a)*a*b*c^2*d^2*f^3*g^3/(d*x + c) - 30*(b*x + a)^2*b*c^3*d^2*f^3*g^3/(d*x + c)^2 - a^3*d^3*f^3*g^3
+ 12*(b*x + a)*a^2*c*d^3*f^3*g^3/(d*x + c) - 30*(b*x + a)^2*a*c^2*d^3*f^3*g^3/(d*x + c)^2 + 20*(b*x + a)^3*c^3
*d^3*f^3*g^3/(d*x + c)^3 + 3*a*b^2*c^3*f^2*g^4 - 3*(b*x + a)*b^2*c^4*f^2*g^4/(d*x + c) + 9*a^2*b*c^2*d*f^2*g^4
- 24*(b*x + a)*a*b*c^3*d*f^2*g^4/(d*x + c) + 15*(b*x + a)^2*b*c^4*d*f^2*g^4/(d*x + c)^2 + 3*a^3*c*d^2*f^2*g^4
- 18*(b*x + a)*a^2*c^2*d^2*f^2*g^4/(d*x + c) + 30*(b*x + a)^2*a*c^3*d^2*f^2*g^4/(d*x + c)^2 - 15*(b*x + a)^3*
c^4*d^2*f^2*g^4/(d*x + c)^3 - 3*a^2*b*c^3*f*g^5 + 6*(b*x + a)*a*b*c^4*f*g^5/(d*x + c) - 3*(b*x + a)^2*b*c^5*f*
g^5/(d*x + c)^2 - 3*a^3*c^2*d*f*g^5 + 12*(b*x + a)*a^2*c^3*d*f*g^5/(d*x + c) - 15*(b*x + a)^2*a*c^4*d*f*g^5/(d
*x + c)^2 + 6*(b*x + a)^3*c^5*d*f*g^5/(d*x + c)^3 + a^3*c^3*g^6 - 3*(b*x + a)*a^2*c^4*g^6/(d*x + c) + 3*(b*x +
a)^2*a*c^5*g^6/(d*x + c)^2 - (b*x + a)^3*c^6*g^6/(d*x + c)^3) - 2*(3*B*b^4*c^2*d^2*f^2*n - 6*B*a*b^3*c*d^3*f^
2*n + 3*B*a^2*b^2*d^4*f^2*n - 3*B*b^4*c^3*d*f*g*n + 3*B*a*b^3*c^2*d^2*f*g*n + 3*B*a^2*b^2*c*d^3*f*g*n - 3*B*a^
3*b*d^4*f*g*n + B*b^4*c^4*g^2*n - B*a*b^3*c^3*d*g^2*n - B*a^3*b*c*d^3*g^2*n + B*a^4*d^4*g^2*n)*log((b*x + a)/(
d*x + 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) + (6*B*b^6*c^3*d*f^3*g*n - 18*B*a*b^5*c^2*d^2*f^3*g*n - 12*(b*x + a)*B*b^5*c^3*d^2*f^3*g*n/(d*x + c) +
18*B*a^2*b^4*c*d^3*f^3*g*n + 36*(b*x + a)*B*a*b^4*c^2*d^3*f^3*g*n/(d*x + c) + 6*(b*x + a)^2*B*b^4*c^3*d^3*f^3
*g*n/(d*x + c)^2 - 6*B*a^3*b^3*d^4*f^3*g*n - 36*(b*x + a)*B*a^2*b^3*c*d^4*f^3*g*n/(d*x + c) - 18*(b*x + a)^2*B
*a*b^3*c^2*d^4*f^3*g*n/(d*x + c)^2 + 12*(b*x + a)*B*a^3*b^2*d^5*f^3*g*n/(d*x + c) + 18*(b*x + a)^2*B*a^2*b^2*c
*d^5*f^3*g*n/(d*x + c)^2 - 6*(b*x + a)^2*B*a^3*b*d^6*f^3*g*n/(d*x + c)^2 - 3*B*b^6*c^4*f^2*g^2*n - 6*B*a*b^5*c
^3*d*f^2*g^2*n + 17*(b*x + a)*B*b^5*c^4*d*f^2*g^2*n/(d*x + c) + 36*B*a^2*b^4*c^2*d^2*f^2*g^2*n - 32*(b*x + a)*
B*a*b^4*c^3*d^2*f^2*g^2*n/(d*x + c) - 14*(b*x + a)^2*B*b^4*c^4*d^2*f^2*g^2*n/(d*x + c)^2 - 42*B*a^3*b^3*c*d^3*
f^2*g^2*n - 6*(b*x + a)*B*a^2*b^3*c^2*d^3*f^2*g...

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Mupad [B]
time = 9.23, size = 1182, normalized size = 4.18 \begin {gather*} \frac {B\,d^3\,n\,\ln \left (c+d\,x\right )}{3\,c^3\,g^4-9\,c^2\,d\,f\,g^3+9\,c\,d^2\,f^2\,g^2-3\,d^3\,f^3\,g}-\frac {\ln \left (f+g\,x\right )\,\left (g^2\,\left (B\,a^3\,d^3\,n-B\,b^3\,c^3\,n\right )-g\,\left (3\,B\,a^2\,b\,d^3\,f\,n-3\,B\,b^3\,c^2\,d\,f\,n\right )+3\,B\,a\,b^2\,d^3\,f^2\,n-3\,B\,b^3\,c\,d^2\,f^2\,n\right )}{3\,a^3\,c^3\,g^6-9\,a^3\,c^2\,d\,f\,g^5+9\,a^3\,c\,d^2\,f^2\,g^4-3\,a^3\,d^3\,f^3\,g^3-9\,a^2\,b\,c^3\,f\,g^5+27\,a^2\,b\,c^2\,d\,f^2\,g^4-27\,a^2\,b\,c\,d^2\,f^3\,g^3+9\,a^2\,b\,d^3\,f^4\,g^2+9\,a\,b^2\,c^3\,f^2\,g^4-27\,a\,b^2\,c^2\,d\,f^3\,g^3+27\,a\,b^2\,c\,d^2\,f^4\,g^2-9\,a\,b^2\,d^3\,f^5\,g-3\,b^3\,c^3\,f^3\,g^3+9\,b^3\,c^2\,d\,f^4\,g^2-9\,b^3\,c\,d^2\,f^5\,g+3\,b^3\,d^3\,f^6}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{3\,g\,\left (f^3+3\,f^2\,g\,x+3\,f\,g^2\,x^2+g^3\,x^3\right )}-\frac {B\,b^3\,n\,\ln \left (a+b\,x\right )}{3\,a^3\,g^4-9\,a^2\,b\,f\,g^3+9\,a\,b^2\,f^2\,g^2-3\,b^3\,f^3\,g}-\frac {\frac {2\,A\,a^2\,c^2\,g^4+2\,A\,b^2\,d^2\,f^4+2\,A\,a^2\,d^2\,f^2\,g^2+2\,A\,b^2\,c^2\,f^2\,g^2+3\,B\,a^2\,d^2\,f^2\,g^2\,n-3\,B\,b^2\,c^2\,f^2\,g^2\,n-4\,A\,a\,b\,c^2\,f\,g^3-4\,A\,a\,b\,d^2\,f^3\,g-4\,A\,a^2\,c\,d\,f\,g^3-4\,A\,b^2\,c\,d\,f^3\,g+8\,A\,a\,b\,c\,d\,f^2\,g^2+B\,a\,b\,c^2\,f\,g^3\,n-5\,B\,a\,b\,d^2\,f^3\,g\,n-B\,a^2\,c\,d\,f\,g^3\,n+5\,B\,b^2\,c\,d\,f^3\,g\,n}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}+\frac {x\,\left (-B\,n\,a^2\,c\,d\,g^4+5\,B\,n\,a^2\,d^2\,f\,g^3+B\,n\,a\,b\,c^2\,g^4-9\,B\,n\,a\,b\,d^2\,f^2\,g^2-5\,B\,n\,b^2\,c^2\,f\,g^3+9\,B\,n\,b^2\,c\,d\,f^2\,g^2\right )}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}+\frac {x^2\,\left (B\,n\,a^2\,d^2\,g^4-2\,B\,f\,n\,a\,b\,d^2\,g^3-B\,n\,b^2\,c^2\,g^4+2\,B\,f\,n\,b^2\,c\,d\,g^3\right )}{a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4}}{3\,f^3\,g+9\,f^2\,g^2\,x+9\,f\,g^3\,x^2+3\,g^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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