3.3.70 \(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(f+g x)^4} \, dx\) [270]

Optimal. Leaf size=277 \[ -\frac {B (b c-a d)}{3 (b f-a g) (d f-c g) (f+g x)^2}-\frac {2 B (b c-a d) (2 b d f-b c g-a d g)}{3 (b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {2 b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 g (f+g x)^3}-\frac {2 B d^3 \log (c+d x)}{3 g (d f-c g)^3}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^3 (d f-c g)^3} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2548, 84} \begin {gather*} \frac {2 B (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 (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{3 g (f+g x)^3}+\frac {2 b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {2 B (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 (b c-a d)}{3 (f+g x)^2 (b f-a g) (d f-c g)}-\frac {2 B d^3 \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)^2)/(c + d*x)^2])/(f + g*x)^4,x]

[Out]

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

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Mathematica [A]
time = 0.47, size = 263, normalized size = 0.95 \begin {gather*} \frac {-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^3}+2 B (b c-a d) \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)^2)/(c + d*x)^2])/(f + g*x)^4,x]

[Out]

(-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^3) + 2*B*(b*c - a*d)*(-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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(2562\) vs. \(2(268)=536\).
time = 0.88, size = 2563, normalized size = 9.25

method	result	size
risch	\(\text {Expression too large to display}\)	\(2293\)
derivativedivides	\(\text {Expression too large to display}\)	\(2563\)
default	\(\text {Expression too large to display}\)	\(2563\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (266) = 532\).
time = 0.36, size = 903, normalized size = 3.26 \begin {gather*} \frac {1}{3} \, {\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} - \frac {\log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g}\right )} B - \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)^2/(d*x+c)^2))/(g*x+f)^4,x, algorithm="maxima")

[Out]

1/3*(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) - log(b^
2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(g^
4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g))*B - 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)^2/(d*x+c)^2))/(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)**2/(d*x+c)**2))/(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. 1391 vs. \(2 (266) = 532\).
time = 2.35, size = 1391, normalized size = 5.02 \begin {gather*} \frac {2 \, B b^{4} \log \left ({\left | b x + a \right |}\right )}{3 \, {\left (b^{4} f^{3} g - 3 \, a b^{3} f^{2} g^{2} + 3 \, a^{2} b^{2} f g^{3} - a^{3} b g^{4}\right )}} - \frac {2 \, B d^{4} \log \left ({\left | d x + c \right |}\right )}{3 \, {\left (d^{4} f^{3} g - 3 \, c d^{3} f^{2} g^{2} + 3 \, c^{2} d^{2} f g^{3} - c^{3} d g^{4}\right )}} + \frac {2 \, {\left (3 \, B b^{3} c d^{2} f^{2} - 3 \, B a b^{2} d^{3} f^{2} - 3 \, B b^{3} c^{2} d f g + 3 \, B a^{2} b d^{3} f g + B b^{3} c^{3} g^{2} - B a^{3} d^{3} g^{2}\right )} \log \left (g x + f\right )}{3 \, {\left (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}\right )}} - \frac {B \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} - \frac {4 \, B b^{2} c d f g^{3} x^{2} - 4 \, B a b d^{2} f g^{3} x^{2} - 2 \, B b^{2} c^{2} g^{4} x^{2} + 2 \, B a^{2} d^{2} g^{4} x^{2} + 9 \, B b^{2} c d f^{2} g^{2} x - 9 \, B a b d^{2} f^{2} g^{2} x - 5 \, B b^{2} c^{2} f g^{3} x + 5 \, B a^{2} d^{2} f g^{3} x + B a b c^{2} g^{4} x - B a^{2} c d g^{4} x + A b^{2} d^{2} f^{4} + B b^{2} d^{2} f^{4} - 2 \, A b^{2} c d f^{3} g + 3 \, B b^{2} c d f^{3} g - 2 \, A a b d^{2} f^{3} g - 7 \, B a b d^{2} f^{3} g + A b^{2} c^{2} f^{2} g^{2} - 2 \, B b^{2} c^{2} f^{2} g^{2} + 4 \, A a b c d f^{2} g^{2} + 4 \, B a b c d f^{2} g^{2} + A a^{2} d^{2} f^{2} g^{2} + 4 \, B a^{2} d^{2} f^{2} g^{2} - 2 \, A a b c^{2} f g^{3} - B a b c^{2} f g^{3} - 2 \, A a^{2} c d f g^{3} - 3 \, B a^{2} c d f g^{3} + A a^{2} c^{2} g^{4} + B a^{2} c^{2} g^{4}}{3 \, {\left (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}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 11.58, size = 1147, normalized size = 4.14 \begin {gather*} \frac {\ln \left (f+g\,x\right )\,\left (g\,\left (6\,B\,a^2\,b\,d^3\,f-6\,B\,b^3\,c^2\,d\,f\right )-g^2\,\left (2\,B\,a^3\,d^3-2\,B\,b^3\,c^3\right )-6\,B\,a\,b^2\,d^3\,f^2+6\,B\,b^3\,c\,d^2\,f^2\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 {\frac {A\,a^2\,c^2\,g^4+A\,b^2\,d^2\,f^4+A\,a^2\,d^2\,f^2\,g^2+A\,b^2\,c^2\,f^2\,g^2+3\,B\,a^2\,d^2\,f^2\,g^2-3\,B\,b^2\,c^2\,f^2\,g^2-2\,A\,a\,b\,c^2\,f\,g^3-2\,A\,a\,b\,d^2\,f^3\,g+B\,a\,b\,c^2\,f\,g^3-2\,A\,a^2\,c\,d\,f\,g^3-5\,B\,a\,b\,d^2\,f^3\,g-2\,A\,b^2\,c\,d\,f^3\,g-B\,a^2\,c\,d\,f\,g^3+5\,B\,b^2\,c\,d\,f^3\,g+4\,A\,a\,b\,c\,d\,f^2\,g^2}{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}+\frac {2\,x^2\,\left (B\,a^2\,d^2\,g^4-2\,B\,f\,a\,b\,d^2\,g^3-B\,b^2\,c^2\,g^4+2\,B\,f\,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}+\frac {x\,\left (-B\,a^2\,c\,d\,g^4+5\,B\,a^2\,d^2\,f\,g^3+B\,a\,b\,c^2\,g^4-9\,B\,a\,b\,d^2\,f^2\,g^2-5\,B\,b^2\,c^2\,f\,g^3+9\,B\,b^2\,c\,d\,f^2\,g^2\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}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{3\,g\,\left (f^3+3\,f^2\,g\,x+3\,f\,g^2\,x^2+g^3\,x^3\right )}-\frac {2\,B\,b^3\,\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 {2\,B\,d^3\,\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(f + g*x)^4,x)

[Out]

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

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