3.3.30 \(\int (f+g x)^4 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [230]

Optimal. Leaf size=355 \[ \frac {B (b c-a d) g \left (a^3 d^3 g^3-a^2 b d^2 g^2 (5 d f-c g)+a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )-b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right ) x}{5 b^4 d^4}-\frac {B (b c-a d) g^2 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) x^2}{10 b^3 d^3}-\frac {B (b c-a d) g^3 (5 b d f-b c g-a d g) x^3}{15 b^2 d^2}-\frac {B (b c-a d) g^4 x^4}{20 b d}-\frac {B (b f-a g)^5 \log (a+b x)}{5 b^5 g}+\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 g}+\frac {B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2548, 84} \begin {gather*} -\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 )}{10 b^3 d^3}+\frac {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)}{c+d x}\right )+A\right )}{5 g}-\frac {B (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac {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)}{20 b d}+\frac {B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)*g*(a^3*d^3*g^3 - a^2*b*d^2*g^2*(5*d*f - c*g) + a*b^2*d*g*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2) - b
^3*(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)/(10*b^3*d^3) - (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)/(20*b*d) - (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))/(c + d*x)]))/(5*g) + (B*(d*f - c*g)^5*Log[c + d*x])/(5*d^5*g)

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 (f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac {(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 g}-\frac {B \int \frac {(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)}{c+d x}\right )\right )}{5 g}-\frac {(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)}{c+d x}\right )\right )}{5 g}-\frac {(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 {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}{10 b^3 d^3}-\frac {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}{20 b d}-\frac {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)}{c+d x}\right )\right )}{5 g}+\frac {B (d f-c g)^5 \log (c+d x)}{5 d^5 g}\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 279, normalized size = 0.79 \begin {gather*} \frac {\frac {B (-b c+a d) g^2 x \left (-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 \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 )}{12 b^4 d^4}-\frac {B (b f-a g)^5 \log (a+b x)}{b^5}+(f+g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {B (d f-c g)^5 \log (c+d x)}{d^5}}{5 g} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]),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))))/(12*b^4*d
^4) - (B*(b*f - a*g)^5*Log[a + b*x])/b^5 + (f + g*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (B*(d*f - c*g)^5
*Log[c + d*x])/d^5)/(5*g)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(14847\) vs. \(2(341)=682\).
time = 0.46, size = 14848, normalized size = 41.83

method	result	size
risch	\(-\frac {g^{4} B \,a^{4} x}{5 b^{4}}+\frac {g^{4} B \,c^{4} x}{5 d^{4}}-\frac {g^{3} B \ln \left (-b x -a \right ) a^{4} f}{b^{4}}+\frac {2 g^{2} B \ln \left (-b x -a \right ) a^{3} f^{2}}{b^{3}}-\frac {2 g B \ln \left (-b x -a \right ) a^{2} f^{3}}{b^{2}}+\frac {g^{3} B \ln \left (d x +c \right ) c^{4} f}{d^{4}}+\frac {\left (g x +f \right )^{5} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{5 g}+\frac {g^{4} A \,x^{5}}{5}-\frac {g^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}+\frac {g^{4} B \ln \left (-b x -a \right ) a^{5}}{5 b^{5}}+\frac {B \ln \left (-b x -a \right ) a \,f^{4}}{b}-\frac {B \ln \left (d x +c \right ) c \,f^{4}}{d}+\frac {B \ln \left (d x +c \right ) f^{5}}{5 g}-\frac {B \ln \left (-b x -a \right ) f^{5}}{5 g}+\frac {g^{3} B a f \,x^{3}}{3 b}-\frac {g^{3} B c f \,x^{3}}{3 d}-\frac {g^{3} B \,a^{2} f \,x^{2}}{2 b^{2}}+\frac {g^{2} B a \,f^{2} x^{2}}{b}+\frac {g^{3} B \,c^{2} f \,x^{2}}{2 d^{2}}-\frac {g^{2} B c \,f^{2} x^{2}}{d}+\frac {g^{3} B \,a^{3} f x}{b^{3}}-\frac {2 g^{2} B \,a^{2} f^{2} x}{b^{2}}+\frac {2 g B a \,f^{3} x}{b}-\frac {g^{3} B \,c^{3} f x}{d^{3}}+\frac {2 g^{2} B \,c^{2} f^{2} x}{d^{2}}-\frac {2 g B c \,f^{3} x}{d}-\frac {2 g^{2} B \ln \left (d x +c \right ) c^{3} f^{2}}{d^{3}}+\frac {2 g B \ln \left (d x +c \right ) c^{2} f^{3}}{d^{2}}+2 g A \,f^{3} x^{2}+\frac {g^{4} B \,a^{3} x^{2}}{10 b^{3}}-\frac {g^{4} B \,c^{3} x^{2}}{10 d^{3}}+A \,f^{4} x +g^{3} A f \,x^{4}+\frac {g^{4} B a \,x^{4}}{20 b}-\frac {g^{4} B c \,x^{4}}{20 d}+2 g^{2} A \,f^{2} x^{3}-\frac {g^{4} B \,a^{2} x^{3}}{15 b^{2}}+\frac {g^{4} B \,c^{2} x^{3}}{15 d^{2}}\)	\(594\)
derivativedivides	\(\text {Expression too large to display}\)	\(14848\)
default	\(\text {Expression too large to display}\)	\(14848\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [A]
time = 0.30, size = 603, normalized size = 1.70 \begin {gather*} \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 x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B f^{4} + 2 \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B f^{3} g + {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\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}{6} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\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}{60} \, {\left (12 \, x^{5} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(A+B*log(e*(b*x+a)/(d*x+c))),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*x*e/(d*x + c) + a*e/(d*x + c)) + a*lo
g(b*x + a)/b - c*log(d*x + c)/d)*B*f^4 + 2*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 +
c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*f^3*g + (2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 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 + 1/6*(6*x^4*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - 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 + 1/60
*(12*x^5*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 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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Fricas [A]
time = 0.78, size = 635, normalized size = 1.79 \begin {gather*} \frac {12 \, A b^{5} d^{5} g^{4} x^{5} + 3 \, {\left (20 \, 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 (30 \, 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 (20 \, 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} + 12 \, {\left (5 \, A b^{5} d^{5} f^{4} - 10 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f^{3} g + 10 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} f^{2} g^{2} - 5 \, {\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} f g^{3} + {\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 ) + 12 \, {\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 {{\left (b x + a\right )} e}{d x + c}\right )}{60 \, b^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*A*b^5*d^5*g^4*x^5 + 3*(20*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*(30*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*(20*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 + 12*(5*A*b^5*d^5*f^4 - 10*(B*b^5*c*d^4 - B*a*b^4*d^5)*f^3*g + 10*(B*b^5*c^2*d^3 - B*a^2*b^3*
d^5)*f^2*g^2 - 5*(B*b^5*c^3*d^2 - B*a^3*b^2*d^5)*f*g^3 + (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) + 12*(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*x + a)*e/(d*x + c)))/(b^5*d^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1436 vs. \(2 (337) = 674\).
time = 89.01, size = 1436, normalized size = 4.05 \begin {gather*} \frac {A g^{4} x^{5}}{5} + \frac {B a \left (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}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{4} - 5 B a^{4} b c d^{4} f g^{3} + 10 B a^{3} b^{2} c d^{4} f^{2} g^{2} - 10 B a^{2} b^{3} c d^{4} f^{3} g + \frac {B a^{2} d^{5} \left (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}\right )}{b} + B a b^{4} c^{5} g^{4} - 5 B a b^{4} c^{4} d f g^{3} + 10 B a b^{4} c^{3} d^{2} f^{2} g^{2} - 10 B a b^{4} c^{2} d^{3} f^{3} g + 10 B a b^{4} c d^{4} f^{4} - B a c d^{4} \left (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}\right )}{B a^{5} d^{5} g^{4} - 5 B a^{4} b d^{5} f g^{3} + 10 B a^{3} b^{2} d^{5} f^{2} g^{2} - 10 B a^{2} b^{3} d^{5} f^{3} g + 5 B a b^{4} d^{5} f^{4} + B b^{5} c^{5} g^{4} - 5 B b^{5} c^{4} d f g^{3} + 10 B b^{5} c^{3} d^{2} f^{2} g^{2} - 10 B b^{5} c^{2} d^{3} f^{3} g + 5 B b^{5} c d^{4} f^{4}} \right )}}{5 b^{5}} - \frac {B c \left (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}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{4} - 5 B a^{4} b c d^{4} f g^{3} + 10 B a^{3} b^{2} c d^{4} f^{2} g^{2} - 10 B a^{2} b^{3} c d^{4} f^{3} g + B a b^{4} c^{5} g^{4} - 5 B a b^{4} c^{4} d f g^{3} + 10 B a b^{4} c^{3} d^{2} f^{2} g^{2} - 10 B a b^{4} c^{2} d^{3} f^{3} g + 10 B a b^{4} c d^{4} f^{4} - B a b^{4} c \left (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}\right ) + \frac {B b^{5} c^{2} \left (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}\right )}{d}}{B a^{5} d^{5} g^{4} - 5 B a^{4} b d^{5} f g^{3} + 10 B a^{3} b^{2} d^{5} f^{2} g^{2} - 10 B a^{2} b^{3} d^{5} f^{3} g + 5 B a b^{4} d^{5} f^{4} + B b^{5} c^{5} g^{4} - 5 B b^{5} c^{4} d f g^{3} + 10 B b^{5} c^{3} d^{2} f^{2} g^{2} - 10 B b^{5} c^{2} d^{3} f^{3} g + 5 B b^{5} c d^{4} f^{4}} \right )}}{5 d^{5}} + x^{4} \left (A f g^{3} + \frac {B a g^{4}}{20 b} - \frac {B c g^{4}}{20 d}\right ) + x^{3} \cdot \left (2 A f^{2} g^{2} - \frac {B a^{2} g^{4}}{15 b^{2}} + \frac {B a f g^{3}}{3 b} + \frac {B c^{2} g^{4}}{15 d^{2}} - \frac {B c f g^{3}}{3 d}\right ) + x^{2} \cdot \left (2 A f^{3} g + \frac {B a^{3} g^{4}}{10 b^{3}} - \frac {B a^{2} f g^{3}}{2 b^{2}} + \frac {B a f^{2} g^{2}}{b} - \frac {B c^{3} g^{4}}{10 d^{3}} + \frac {B c^{2} f g^{3}}{2 d^{2}} - \frac {B c f^{2} g^{2}}{d}\right ) + x \left (A f^{4} - \frac {B a^{4} g^{4}}{5 b^{4}} + \frac {B a^{3} f g^{3}}{b^{3}} - \frac {2 B a^{2} f^{2} g^{2}}{b^{2}} + \frac {2 B a f^{3} g}{b} + \frac {B c^{4} g^{4}}{5 d^{4}} - \frac {B c^{3} f g^{3}}{d^{3}} + \frac {2 B c^{2} f^{2} g^{2}}{d^{2}} - \frac {2 B c f^{3} g}{d}\right ) + \left (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} + \frac {B g^{4} x^{5}}{5}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19084 vs. \(2 (342) = 684\).
time = 4.90, size = 19084, normalized size = 53.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*B*b^11*c^2*d^4*f^4*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 120*B*a*b^10*c*d^5*f^4*e^6*log(-b*e +
(b*x*e + a*e)*d/(d*x + c)) + 60*B*a^2*b^9*d^6*f^4*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 120*B*b^11*c^3*d
^3*f^3*g*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 120*B*a*b^10*c^2*d^4*f^3*g*e^6*log(-b*e + (b*x*e + a*e)*d
/(d*x + c)) + 120*B*a^2*b^9*c*d^5*f^3*g*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 120*B*a^3*b^8*d^6*f^3*g*e^
6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 120*B*b^11*c^4*d^2*f^2*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))
- 120*B*a*b^10*c^3*d^3*f^2*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 120*B*a^3*b^8*c*d^5*f^2*g^2*e^6*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 120*B*a^4*b^7*d^6*f^2*g^2*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 60
*B*b^11*c^5*d*f*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 60*B*a*b^10*c^4*d^2*f*g^3*e^6*log(-b*e + (b*x*
e + a*e)*d/(d*x + c)) + 60*B*a^4*b^7*c*d^5*f*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 60*B*a^5*b^6*d^6*
f*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 12*B*b^11*c^6*g^4*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))
- 12*B*a*b^10*c^5*d*g^4*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 12*B*a^5*b^6*c*d^5*g^4*e^6*log(-b*e + (b*x
*e + a*e)*d/(d*x + c)) + 12*B*a^6*b^5*d^6*g^4*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 300*(b*x*e + a*e)*B*
b^10*c^2*d^5*f^4*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*B*a*b^9*c*d^6*f^4*e^5
*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 300*(b*x*e + a*e)*B*a^2*b^8*d^7*f^4*e^5*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*B*b^10*c^3*d^4*f^3*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c) - 600*(b*x*e + a*e)*B*a*b^9*c^2*d^5*f^3*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 60
0*(b*x*e + a*e)*B*a^2*b^8*c*d^6*f^3*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*
B*a^3*b^7*d^7*f^3*g*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 600*(b*x*e + a*e)*B*b^10*c^4*d^3*f^2
*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*B*a*b^9*c^3*d^4*f^2*g^2*e^5*log(-
b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)*B*a^3*b^7*c*d^6*f^2*g^2*e^5*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c) - 600*(b*x*e + a*e)*B*a^4*b^6*d^7*f^2*g^2*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c
))/(d*x + c) + 300*(b*x*e + a*e)*B*b^10*c^5*d^2*f*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 30
0*(b*x*e + a*e)*B*a*b^9*c^4*d^3*f*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 300*(b*x*e + a*e)*
B*a^4*b^6*c*d^6*f*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 300*(b*x*e + a*e)*B*a^5*b^5*d^7*f*
g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 60*(b*x*e + a*e)*B*b^10*c^6*d*g^4*e^5*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c) + 60*(b*x*e + a*e)*B*a*b^9*c^5*d^2*g^4*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x +
c))/(d*x + c) + 60*(b*x*e + a*e)*B*a^5*b^5*c*d^6*g^4*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 60
*(b*x*e + a*e)*B*a^6*b^4*d^7*g^4*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 600*(b*x*e + a*e)^2*B*b
^9*c^2*d^6*f^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1200*(b*x*e + a*e)^2*B*a*b^8*c*d^7*f^4*
e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 600*(b*x*e + a*e)^2*B*a^2*b^7*d^8*f^4*e^4*log(-b*e + (
b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1200*(b*x*e + a*e)^2*B*b^9*c^3*d^5*f^3*g*e^4*log(-b*e + (b*x*e + a*e)*
d/(d*x + c))/(d*x + c)^2 + 1200*(b*x*e + a*e)^2*B*a*b^8*c^2*d^6*f^3*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c)^2 + 1200*(b*x*e + a*e)^2*B*a^2*b^7*c*d^7*f^3*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^2 - 1200*(b*x*e + a*e)^2*B*a^3*b^6*d^8*f^3*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 1200*(b*
x*e + a*e)^2*B*b^9*c^4*d^4*f^2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1200*(b*x*e + a*e)^
2*B*a*b^8*c^3*d^5*f^2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1200*(b*x*e + a*e)^2*B*a^3*b
^6*c*d^7*f^2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 1200*(b*x*e + a*e)^2*B*a^4*b^5*d^8*f^
2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 600*(b*x*e + a*e)^2*B*b^9*c^5*d^3*f*g^3*e^4*log(
-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 600*(b*x*e + a*e)^2*B*a*b^8*c^4*d^4*f*g^3*e^4*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c)^2 + 600*(b*x*e + a*e)^2*B*a^4*b^5*c*d^7*f*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(
d*x + c))/(d*x + c)^2 - 600*(b*x*e + a*e)^2*B*a^5*b^4*d^8*f*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^2 + 120*(b*x*e + a*e)^2*B*b^9*c^6*d^2*g^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*(b
*x*e + a*e)^2*B*a*b^8*c^5*d^3*g^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 120*(b*x*e + a*e)^2*
B*a^5*b^4*c*d^7*g^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 120*(b*x*e + a*e)^2*B*a^6*b^3*d^8*
g^4*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 600*(b*x*e + a*e)^3*B*b^8*c^2*d^7*f^4*e^3*log(-b*e
+ (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120...

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Mupad [B]
time = 5.34, size = 1392, normalized size = 3.92 \begin {gather*} x^2\,\left (\frac {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+10\,B\,a\,d\,f^2\,g^2-10\,B\,b\,c\,f^2\,g^2}{10\,b\,d}+\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {5\,A\,a\,c\,g^4+20\,A\,a\,d\,f\,g^3+20\,A\,b\,c\,f\,g^3+5\,B\,a\,d\,f\,g^3-5\,B\,b\,c\,f\,g^3+30\,A\,b\,d\,f^2\,g^2}{5\,b\,d}+\frac {A\,a\,c\,g^4}{b\,d}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )}{2\,b\,d}\right )+x^4\,\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{20\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,b\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (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+\frac {B\,g^4\,x^5}{5}\right )+x\,\left (\frac {5\,A\,b\,d\,f^4+20\,A\,a\,d\,f^3\,g+20\,A\,b\,c\,f^3\,g+10\,B\,a\,d\,f^3\,g-10\,B\,b\,c\,f^3\,g+30\,A\,a\,c\,f^2\,g^2}{5\,b\,d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {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+10\,B\,a\,d\,f^2\,g^2-10\,B\,b\,c\,f^2\,g^2}{5\,b\,d}+\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {5\,A\,a\,c\,g^4+20\,A\,a\,d\,f\,g^3+20\,A\,b\,c\,f\,g^3+5\,B\,a\,d\,f\,g^3-5\,B\,b\,c\,f\,g^3+30\,A\,b\,d\,f^2\,g^2}{5\,b\,d}+\frac {A\,a\,c\,g^4}{b\,d}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {5\,A\,a\,c\,g^4+20\,A\,a\,d\,f\,g^3+20\,A\,b\,c\,f\,g^3+5\,B\,a\,d\,f\,g^3-5\,B\,b\,c\,f\,g^3+30\,A\,b\,d\,f^2\,g^2}{5\,b\,d}+\frac {A\,a\,c\,g^4}{b\,d}\right )}{b\,d}\right )-x^3\,\left (\frac {\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+B\,a\,d\,g^4-B\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {5\,A\,a\,c\,g^4+20\,A\,a\,d\,f\,g^3+20\,A\,b\,c\,f\,g^3+5\,B\,a\,d\,f\,g^3-5\,B\,b\,c\,f\,g^3+30\,A\,b\,d\,f^2\,g^2}{15\,b\,d}+\frac {A\,a\,c\,g^4}{3\,b\,d}\right )+\frac {A\,g^4\,x^5}{5}+\frac {\ln \left (a+b\,x\right )\,\left (\frac {B\,a^5\,g^4}{5}-B\,a^4\,b\,f\,g^3+2\,B\,a^3\,b^2\,f^2\,g^2-2\,B\,a^2\,b^3\,f^3\,g+B\,a\,b^4\,f^4\right )}{b^5}-\frac {\ln \left (c+d\,x\right )\,\left (B\,c^5\,g^4-5\,B\,c^4\,d\,f\,g^3+10\,B\,c^3\,d^2\,f^2\,g^2-10\,B\,c^2\,d^3\,f^3\,g+5\,B\,c\,d^4\,f^4\right )}{5\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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