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

Optimal. Leaf size=357 \[ -\frac {B g^2 x^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )\right )}{5 b^3 d^3}+\frac {2 B g x (b c-a d) \left (a^3 d^3 g^3-a^2 b d^2 g^2 (5 d f-c g)+a b^2 d g \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )-\left (b^3 \left (-c^3 g^3+5 c^2 d f g^2-10 c d^2 f^2 g+10 d^3 f^3\right )\right )\right )}{5 b^4 d^4}+\frac {(f+g x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 g}-\frac {2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac {2 B g^3 x^3 (b c-a d) (-a d g-b c g+5 b d f)}{15 b^2 d^2}-\frac {B g^4 x^4 (b c-a d)}{10 b d}+\frac {2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

[Out]

2/5*B*(-a*d+b*c)*g*(a^3*d^3*g^3-a^2*b*d^2*g^2*(-c*g+5*d*f)+a*b^2*d*g*(c^2*g^2-5*c*d*f*g+10*d^2*f^2)-b^3*(-c^3*
g^3+5*c^2*d*f*g^2-10*c*d^2*f^2*g+10*d^3*f^3))*x/b^4/d^4-1/5*B*(-a*d+b*c)*g^2*(a^2*d^2*g^2-a*b*d*g*(-c*g+5*d*f)
+b^2*(c^2*g^2-5*c*d*f*g+10*d^2*f^2))*x^2/b^3/d^3-2/15*B*(-a*d+b*c)*g^3*(-a*d*g-b*c*g+5*b*d*f)*x^3/b^2/d^2-1/10
*B*(-a*d+b*c)*g^4*x^4/b/d-2/5*B*(-a*g+b*f)^5*ln(b*x+a)/b^5/g+1/5*(g*x+f)^5*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/g+2
/5*B*(-c*g+d*f)^5*ln(d*x+c)/d^5/g

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Rubi [A]  time = 0.50, antiderivative size = 341, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2525, 12, 72} \[ -\frac {B g^2 x^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )\right )}{5 b^3 d^3}+\frac {2 B g x \left (-10 a^2 b^2 d^4 f^2 g+5 a^3 b d^4 f g^2-a^4 d^4 g^3+10 a b^3 d^4 f^3+b^4 (-c) \left (5 c^2 d f g^2-c^3 g^3-10 c d^2 f^2 g+10 d^3 f^3\right )\right )}{5 b^4 d^4}+\frac {(f+g x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 g}-\frac {2 B g^3 x^3 (b c-a d) (-a d g-b c g+5 b d f)}{15 b^2 d^2}-\frac {2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac {B g^4 x^4 (b c-a d)}{10 b d}+\frac {2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]),x]

[Out]

(2*B*g*(10*a*b^3*d^4*f^3 - 10*a^2*b^2*d^4*f^2*g + 5*a^3*b*d^4*f*g^2 - a^4*d^4*g^3 - b^4*c*(10*d^3*f^3 - 10*c*d
^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2 - a*b*d*g*(5*d*f - c*g)
+ b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*x^2)/(5*b^3*d^3) - (2*B*(b*c - a*d)*g^3*(5*b*d*f - b*c*g - a*d*g)*x^
3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*x^4)/(10*b*d) - (2*B*(b*f - a*g)^5*Log[a + b*x])/(5*b^5*g) + ((f + g*x)^5
*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*g) + (2*B*(d*f - c*g)^5*Log[c + d*x])/(5*d^5*g)

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]

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]

Rubi steps

\begin {align*} \int (f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx &=\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac {B \int \frac {2 (b c-a d) (f+g x)^5}{(a+b x) (c+d x)} \, dx}{5 g}\\ &=\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac {(2 B (b c-a d)) \int \frac {(f+g x)^5}{(a+b x) (c+d x)} \, dx}{5 g}\\ &=\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac {(2 B (b c-a d)) \int \left (\frac {g^2 \left (-a^3 d^3 g^3+a^2 b d^2 g^2 (5 d f-c g)-a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )+b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right )}{b^4 d^4}+\frac {g^3 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac {g^4 (5 b d f-b c g-a d g) x^2}{b^2 d^2}+\frac {g^5 x^3}{b d}+\frac {(b f-a g)^5}{b^4 (b c-a d) (a+b x)}+\frac {(d f-c g)^5}{d^4 (-b c+a d) (c+d x)}\right ) \, dx}{5 g}\\ &=\frac {2 B (b c-a d) g \left (a^3 d^3 g^3-a^2 b d^2 g^2 (5 d f-c g)+a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )-b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right ) x}{5 b^4 d^4}-\frac {B (b c-a d) g^2 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) x^2}{5 b^3 d^3}-\frac {2 B (b c-a d) g^3 (5 b d f-b c g-a d g) x^3}{15 b^2 d^2}-\frac {B (b c-a d) g^4 x^4}{10 b d}-\frac {2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}+\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}+\frac {2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 282, normalized size = 0.79 \[ \frac {\frac {B g^2 x (a d-b c) \left (-12 a^3 d^3 g^3+6 a^2 b d^2 g^2 (-2 c g+10 d f+d g x)-2 a b^2 d g \left (6 c^2 g^2-3 c d g (10 f+g x)+d^2 \left (60 f^2+15 f g x+2 g^2 x^2\right )\right )+b^3 \left (-12 c^3 g^3+6 c^2 d g^2 (10 f+g x)-2 c d^2 g \left (60 f^2+15 f g x+2 g^2 x^2\right )+d^3 \left (120 f^3+60 f^2 g x+20 f g^2 x^2+3 g^3 x^3\right )\right )\right )}{6 b^4 d^4}+(f+g x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-\frac {2 B (b f-a g)^5 \log (a+b x)}{b^5}+\frac {2 B (d f-c g)^5 \log (c+d x)}{d^5}}{5 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]),x]

[Out]

((B*(-(b*c) + a*d)*g^2*x*(-12*a^3*d^3*g^3 + 6*a^2*b*d^2*g^2*(10*d*f - 2*c*g + d*g*x) - 2*a*b^2*d*g*(6*c^2*g^2
- 3*c*d*g*(10*f + g*x) + d^2*(60*f^2 + 15*f*g*x + 2*g^2*x^2)) + b^3*(-12*c^3*g^3 + 6*c^2*d*g^2*(10*f + g*x) -
2*c*d^2*g*(60*f^2 + 15*f*g*x + 2*g^2*x^2) + d^3*(120*f^3 + 60*f^2*g*x + 20*f*g^2*x^2 + 3*g^3*x^3))))/(6*b^4*d^
4) - (2*B*(b*f - a*g)^5*Log[a + b*x])/b^5 + (f + g*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (2*B*(d*f -
c*g)^5*Log[c + d*x])/d^5)/(5*g)

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fricas [A]  time = 2.35, size = 660, normalized size = 1.85 \[ \frac {6 \, A b^{5} d^{5} g^{4} x^{5} + 3 \, {\left (10 \, A b^{5} d^{5} f g^{3} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4}\right )} x^{4} + 4 \, {\left (15 \, A b^{5} d^{5} f^{2} g^{2} - 5 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f g^{3} + {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g^{4}\right )} x^{3} + 6 \, {\left (10 \, A b^{5} d^{5} f^{3} g - 10 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f^{2} g^{2} + 5 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} f g^{3} - {\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} g^{4}\right )} x^{2} + 6 \, {\left (5 \, A b^{5} d^{5} f^{4} - 20 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f^{3} g + 20 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} f^{2} g^{2} - 10 \, {\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} f g^{3} + 2 \, {\left (B b^{5} c^{4} d - B a^{4} b d^{5}\right )} g^{4}\right )} x + 12 \, {\left (5 \, B a b^{4} d^{5} f^{4} - 10 \, B a^{2} b^{3} d^{5} f^{3} g + 10 \, B a^{3} b^{2} d^{5} f^{2} g^{2} - 5 \, B a^{4} b d^{5} f g^{3} + B a^{5} d^{5} g^{4}\right )} \log \left (b x + a\right ) - 12 \, {\left (5 \, B b^{5} c d^{4} f^{4} - 10 \, B b^{5} c^{2} d^{3} f^{3} g + 10 \, B b^{5} c^{3} d^{2} f^{2} g^{2} - 5 \, B b^{5} c^{4} d f g^{3} + B b^{5} c^{5} g^{4}\right )} \log \left (d x + c\right ) + 6 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B b^{5} d^{5} f g^{3} x^{4} + 10 \, B b^{5} d^{5} f^{2} g^{2} x^{3} + 10 \, B b^{5} d^{5} f^{3} g x^{2} + 5 \, B b^{5} d^{5} f^{4} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{30 \, b^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(6*A*b^5*d^5*g^4*x^5 + 3*(10*A*b^5*d^5*f*g^3 - (B*b^5*c*d^4 - B*a*b^4*d^5)*g^4)*x^4 + 4*(15*A*b^5*d^5*f^2
*g^2 - 5*(B*b^5*c*d^4 - B*a*b^4*d^5)*f*g^3 + (B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*g^4)*x^3 + 6*(10*A*b^5*d^5*f^3*g
- 10*(B*b^5*c*d^4 - B*a*b^4*d^5)*f^2*g^2 + 5*(B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*f*g^3 - (B*b^5*c^3*d^2 - B*a^3*b^
2*d^5)*g^4)*x^2 + 6*(5*A*b^5*d^5*f^4 - 20*(B*b^5*c*d^4 - B*a*b^4*d^5)*f^3*g + 20*(B*b^5*c^2*d^3 - B*a^2*b^3*d^
5)*f^2*g^2 - 10*(B*b^5*c^3*d^2 - B*a^3*b^2*d^5)*f*g^3 + 2*(B*b^5*c^4*d - B*a^4*b*d^5)*g^4)*x + 12*(5*B*a*b^4*d
^5*f^4 - 10*B*a^2*b^3*d^5*f^3*g + 10*B*a^3*b^2*d^5*f^2*g^2 - 5*B*a^4*b*d^5*f*g^3 + B*a^5*d^5*g^4)*log(b*x + a)
- 12*(5*B*b^5*c*d^4*f^4 - 10*B*b^5*c^2*d^3*f^3*g + 10*B*b^5*c^3*d^2*f^2*g^2 - 5*B*b^5*c^4*d*f*g^3 + B*b^5*c^5
*g^4)*log(d*x + c) + 6*(B*b^5*d^5*g^4*x^5 + 5*B*b^5*d^5*f*g^3*x^4 + 10*B*b^5*d^5*f^2*g^2*x^3 + 10*B*b^5*d^5*f^
3*g*x^2 + 5*B*b^5*d^5*f^4*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/(b^5*d^5)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.15, size = 2438, normalized size = 6.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x*f^4+1/5*A*x^5*g^4-8*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c*f^3*g-12/d^2*B/b*ln(1/(d*x+c)*a*
d-1/(d*x+c)*b*c+b)*a*c^2*f^2*g^2+16/d^3*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^4*f*g^3+2/d^3*B/b/(a
*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c^4*g^4-8/d^2*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^3*
b*f^3*g+12/d^3*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^4*b*f^2*g^2+8/d^3*B*g^3*a/b*ln(1/(d*x+c)*a*d-1/
(d*x+c)*b*c+b)*c^3*f+8/d*B*g/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c*f^3-24/d^2*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1
/(d*x+c)*b*c+b)*a*c^3*f^2*g^2+16/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^2*f^3*g-8/d^4*B/(a*d-b*c)
*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^5*b*f*g^3+5/6/d^5*B*g^4*c^5+B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)
*x*f^4+1/5*B*g^4*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*x^5+1/d*A*c*f^4+1/5/d^5*A*c^5*g^4+A*x^4*f*g^3+2*A
*x^2*f^3*g+2*A*x^3*f^2*g^2-8/d^2*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c^3*f*g^3+12/d*B/b/(a*d-b
*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c^2*f^2*g^2-2/d^2*B/b*a*c^2*f^2*g^2+2/3/d^3*B*g^3*a/b*c^3*f+1/d^2*B*
g^3*a^2/b^2*c^2*f+4/d*B*g/b*a*f^3*c+2/d*B*g^3*a^3/b^3*f*c-4/d*B*g^2*a^2/b^2*f^2*c+4*B*g/b^2*ln(1/(d*x+c))*a^2*
f^3+2*B*g^3*a^4/b^4*ln(1/(d*x+c))*f-2*B*g^3*a^4/b^4*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*f+2/3*B*g^3*a/b*f*x^3-1/
10/d*B*g^4*c*x^4-2/5*B*g^4*a^4/b^4*x+1/10*B*g^4*a/b*x^4+1/5*B*g^4*a^3/b^3*x^2-2/15*B*g^4*a^2/b^2*x^3-4/d^4*B/(
a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^5*g^4+2/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^2*b*f
^4+2*d*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*f^4-2/d^4*B*g^4*a/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+
b)*c^4+2/d^5*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^6*b*g^4+2/5/d^5*B*ln(1/(d*x+c))*c^5*g^4+8/5/d^5*B
*g^4*c^5*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)+2*B*g*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*f^3*x^2+B*g^3*ln(
(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*f*x^4+2*B*g^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*f^2*x^3-2/5
*B*g^4*a^5/b^5*ln(1/(d*x+c))+2/5*B*g^4*a^5/b^5*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)-2*B/b*ln(1/(d*x+c))*a*f^4+2/1
5/d^2*B*g^4*c^2*x^3-1/5/d^3*B*g^4*c^3*x^2+2/5/d^4*B*g^4*c^4*x+1/d*B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e
)*c*f^4+2/d*B*ln(1/(d*x+c))*c*f^4+1/5/d^5*B*g^4*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^5-11/3/d^4*B*g^3
*c^4*f+6/d^3*B*c^3*f^2*g^2-4/d^2*B*c^2*f^3*g-1/d^4*A*c^4*f*g^3+2/d^3*A*c^3*f^2*g^2-2/d^2*A*c^2*f^3*g-1/10/d^4*
B*g^4*a/b*c^4-1/5/d^2*B*g^4*a^3/b^3*c^2-2/15/d^3*B*g^4*a^2/b^2*c^3-2/5/d*B*g^4*a^4/b^4*c-4/d*B*g*c*f^3*x-4*B*g
^2*a^3/b^3*ln(1/(d*x+c))*f^2-4*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c*f^4+4*B*g^2*a^3/b^3*ln(1/(d*x
+c)*a*d-1/(d*x+c)*b*c+b)*f^2-4*B*g/b^2*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*f^3+4/d^3*B*ln(1/(d*x+c))*c^3*f^2
*g^2-4/d^2*B*ln(1/(d*x+c))*c^2*f^3*g-6/d^4*B*g^3*c^4*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*f+8/d^3*B*ln(1/(d*x+c)*
a*d-1/(d*x+c)*b*c+b)*c^3*f^2*g^2-4/d^2*B*g*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^2*f^3-1/d^4*B*g^3*ln((1/(d*x+c)
*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^4*f+2/d^3*B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^3*f^2*g^2-2/d^2*B*l
n((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^2*f^3*g+4*B*g/b*a*f^3*x+2*B*g^3*a^3/b^3*f*x-4*B*g^2*a^2/b^2*f^2*x
+2*B*g^2*a/b*f^2*x^2-2/d^4*B*ln(1/(d*x+c))*c^4*f*g^3-B*g^3*a^2/b^2*f*x^2+4/d^2*B*c^2*f^2*g^2*x+1/d^2*B*g^3*c^2
*f*x^2-2/d^3*B*g^3*c^3*f*x-2/3/d*B*g^3*c*f*x^3-2/d*B*g^2*c*f^2*x^2

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maxima [B]  time = 1.17, size = 855, normalized size = 2.39 \[ \frac {1}{5} \, A g^{4} x^{5} + A f g^{3} x^{4} + 2 \, A f^{2} g^{2} x^{3} + 2 \, A f^{3} g x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{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 ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B f^{4} + 2 \, {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{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 ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B f^{3} g + 2 \, {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{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 ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B f^{2} g^{2} + \frac {1}{3} \, {\left (3 \, x^{4} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{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 ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B f g^{3} + \frac {1}{30} \, {\left (6 \, x^{5} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{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 ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B g^{4} + A f^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*A*g^4*x^5 + A*f*g^3*x^4 + 2*A*f^2*g^2*x^3 + 2*A*f^3*g*x^2 + (x*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2
*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a*log(b*x + a)/b - 2*c*log(d*x + c)/
d)*B*f^4 + 2*(x^2*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x
^2 + 2*c*d*x + c^2)) - 2*a^2*log(b*x + a)/b^2 + 2*c^2*log(d*x + c)/d^2 - 2*(b*c - a*d)*x/(b*d))*B*f^3*g + 2*(x
^3*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x +
c^2)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/
(b^2*d^2))*B*f^2*g^2 + 1/3*(3*x^4*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2
) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2
*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*f*g^3 + 1/30*(6*x^5*log(b^2*
e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 12*
a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4
)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*g^4 + A*f^4*x

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mupad [B]  time = 5.33, size = 1403, normalized size = 3.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((e*(a + b*x)^2)/(c + d*x)^2)*((B*g^4*x^5)/5 + B*f^4*x + 2*B*f^2*g^2*x^3 + 2*B*f^3*g*x^2 + B*f*g^3*x^4) + x
^2*((20*A*a*c*f*g^3 + 20*A*b*d*f^3*g + 30*A*a*d*f^2*g^2 + 30*A*b*c*f^2*g^2 + 20*B*a*d*f^2*g^2 - 20*B*b*c*f^2*g
^2)/(10*b*d) + ((5*a*d + 5*b*c)*((((5*A*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(5
*b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d) - (5*A*a*c*g^4 + 20*A*a*d*f*g^3 + 20*A*b*c*f
*g^3 + 10*B*a*d*f*g^3 - 10*B*b*c*f*g^3 + 30*A*b*d*f^2*g^2)/(5*b*d) + (A*a*c*g^4)/(b*d)))/(10*b*d) - (a*c*((5*A
*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(5*b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d)
))/(2*b*d)) + x^4*((5*A*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(20*b*d) - (A*g^4*
(5*a*d + 5*b*c))/(20*b*d)) + x*((5*A*b*d*f^4 + 20*A*a*d*f^3*g + 20*A*b*c*f^3*g + 20*B*a*d*f^3*g - 20*B*b*c*f^3
*g + 30*A*a*c*f^2*g^2)/(5*b*d) - ((5*a*d + 5*b*c)*((20*A*a*c*f*g^3 + 20*A*b*d*f^3*g + 30*A*a*d*f^2*g^2 + 30*A*
b*c*f^2*g^2 + 20*B*a*d*f^2*g^2 - 20*B*b*c*f^2*g^2)/(5*b*d) + ((5*a*d + 5*b*c)*((((5*A*a*d*g^4 + 5*A*b*c*g^4 +
2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(5*b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d
) - (5*A*a*c*g^4 + 20*A*a*d*f*g^3 + 20*A*b*c*f*g^3 + 10*B*a*d*f*g^3 - 10*B*b*c*f*g^3 + 30*A*b*d*f^2*g^2)/(5*b*
d) + (A*a*c*g^4)/(b*d)))/(5*b*d) - (a*c*((5*A*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g
^3)/(5*b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d)))/(b*d)))/(5*b*d) + (a*c*((((5*A*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d
*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(5*b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(5*b*d) - (5*
A*a*c*g^4 + 20*A*a*d*f*g^3 + 20*A*b*c*f*g^3 + 10*B*a*d*f*g^3 - 10*B*b*c*f*g^3 + 30*A*b*d*f^2*g^2)/(5*b*d) + (A
*a*c*g^4)/(b*d)))/(b*d)) - x^3*((((5*A*a*d*g^4 + 5*A*b*c*g^4 + 2*B*a*d*g^4 - 2*B*b*c*g^4 + 20*A*b*d*f*g^3)/(5*
b*d) - (A*g^4*(5*a*d + 5*b*c))/(5*b*d))*(5*a*d + 5*b*c))/(15*b*d) - (5*A*a*c*g^4 + 20*A*a*d*f*g^3 + 20*A*b*c*f
*g^3 + 10*B*a*d*f*g^3 - 10*B*b*c*f*g^3 + 30*A*b*d*f^2*g^2)/(15*b*d) + (A*a*c*g^4)/(3*b*d)) + (A*g^4*x^5)/5 + (
log(a + b*x)*((2*B*a^5*g^4)/5 + 2*B*a*b^4*f^4 - 4*B*a^2*b^3*f^3*g + 4*B*a^3*b^2*f^2*g^2 - 2*B*a^4*b*f*g^3))/b^
5 - (log(c + d*x)*(2*B*c^5*g^4 + 10*B*c*d^4*f^4 - 20*B*c^2*d^3*f^3*g + 20*B*c^3*d^2*f^2*g^2 - 10*B*c^4*d*f*g^3
))/(5*d^5)

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sympy [B]  time = 26.59, size = 1477, normalized size = 4.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**4*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)

[Out]

A*g**4*x**5/5 + 2*B*a*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2 - 10*a*b**3*f**3*g + 5*b**4*f**4)*
log(x + (2*B*a**5*c*d**4*g**4 - 10*B*a**4*b*c*d**4*f*g**3 + 20*B*a**3*b**2*c*d**4*f**2*g**2 - 20*B*a**2*b**3*c
*d**4*f**3*g + 2*B*a**2*d**5*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2 - 10*a*b**3*f**3*g + 5*b**4
*f**4)/b + 2*B*a*b**4*c**5*g**4 - 10*B*a*b**4*c**4*d*f*g**3 + 20*B*a*b**4*c**3*d**2*f**2*g**2 - 20*B*a*b**4*c*
*2*d**3*f**3*g + 20*B*a*b**4*c*d**4*f**4 - 2*B*a*c*d**4*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2
- 10*a*b**3*f**3*g + 5*b**4*f**4))/(2*B*a**5*d**5*g**4 - 10*B*a**4*b*d**5*f*g**3 + 20*B*a**3*b**2*d**5*f**2*g*
*2 - 20*B*a**2*b**3*d**5*f**3*g + 10*B*a*b**4*d**5*f**4 + 2*B*b**5*c**5*g**4 - 10*B*b**5*c**4*d*f*g**3 + 20*B*
b**5*c**3*d**2*f**2*g**2 - 20*B*b**5*c**2*d**3*f**3*g + 10*B*b**5*c*d**4*f**4))/(5*b**5) - 2*B*c*(c**4*g**4 -
5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**4)*log(x + (2*B*a**5*c*d**4*g**4 - 10*
B*a**4*b*c*d**4*f*g**3 + 20*B*a**3*b**2*c*d**4*f**2*g**2 - 20*B*a**2*b**3*c*d**4*f**3*g + 2*B*a*b**4*c**5*g**4
- 10*B*a*b**4*c**4*d*f*g**3 + 20*B*a*b**4*c**3*d**2*f**2*g**2 - 20*B*a*b**4*c**2*d**3*f**3*g + 20*B*a*b**4*c*
d**4*f**4 - 2*B*a*b**4*c*(c**4*g**4 - 5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**
4) + 2*B*b**5*c**2*(c**4*g**4 - 5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**4)/d)/
(2*B*a**5*d**5*g**4 - 10*B*a**4*b*d**5*f*g**3 + 20*B*a**3*b**2*d**5*f**2*g**2 - 20*B*a**2*b**3*d**5*f**3*g + 1
0*B*a*b**4*d**5*f**4 + 2*B*b**5*c**5*g**4 - 10*B*b**5*c**4*d*f*g**3 + 20*B*b**5*c**3*d**2*f**2*g**2 - 20*B*b**
5*c**2*d**3*f**3*g + 10*B*b**5*c*d**4*f**4))/(5*d**5) + x**4*(A*f*g**3 + B*a*g**4/(10*b) - B*c*g**4/(10*d)) +
x**3*(2*A*f**2*g**2 - 2*B*a**2*g**4/(15*b**2) + 2*B*a*f*g**3/(3*b) + 2*B*c**2*g**4/(15*d**2) - 2*B*c*f*g**3/(3
*d)) + x**2*(2*A*f**3*g + B*a**3*g**4/(5*b**3) - B*a**2*f*g**3/b**2 + 2*B*a*f**2*g**2/b - B*c**3*g**4/(5*d**3)
+ B*c**2*f*g**3/d**2 - 2*B*c*f**2*g**2/d) + x*(A*f**4 - 2*B*a**4*g**4/(5*b**4) + 2*B*a**3*f*g**3/b**3 - 4*B*a
**2*f**2*g**2/b**2 + 4*B*a*f**3*g/b + 2*B*c**4*g**4/(5*d**4) - 2*B*c**3*f*g**3/d**3 + 4*B*c**2*f**2*g**2/d**2
- 4*B*c*f**3*g/d) + (B*f**4*x + 2*B*f**3*g*x**2 + 2*B*f**2*g**2*x**3 + B*f*g**3*x**4 + B*g**4*x**5/5)*log(e*(a
+ b*x)**2/(c + d*x)**2)

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