3.1.57 \(\int (f+g x)^4 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [57]

Optimal. Leaf size=364 \[ \frac {B (b c-a d) g \left (a^3 d^3 g^3-a^2 b d^2 g^2 (5 d f-c g)+a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )-b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right ) n x}{5 b^4 d^4}-\frac {B (b c-a d) g^2 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) n x^2}{10 b^3 d^3}-\frac {B (b c-a d) g^3 (5 b d f-b c g-a d g) n x^3}{15 b^2 d^2}-\frac {B (b c-a d) g^4 n x^4}{20 b d}-\frac {B (b f-a g)^5 n \log (a+b x)}{5 b^5 g}+\frac {(f+g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 g}+\frac {B (d f-c g)^5 n \log (c+d x)}{5 d^5 g} \]

[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))*n*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))*n*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)*n*x^3/b^2/d
^2-1/20*B*(-a*d+b*c)*g^4*n*x^4/b/d-1/5*B*(-a*g+b*f)^5*n*ln(b*x+a)/b^5/g+1/5*(g*x+f)^5*(A+B*ln(e*((b*x+a)/(d*x+
c))^n))/g+1/5*B*(-c*g+d*f)^5*n*ln(d*x+c)/d^5/g

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 364, 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 g^2 n 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 n 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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 g}-\frac {B n (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac {B g^3 n 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 n x^4 (b c-a d)}{20 b d}+\frac {B n (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))^n]),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))*n*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))*n*x^2)/(10*b^3*d^3) - (B*(b*c - a*d)*g^3*(5
*b*d*f - b*c*g - a*d*g)*n*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*n*x^4)/(20*b*d) - (B*(b*f - a*g)^5*n*Log[a +
b*x])/(5*b^5*g) + ((f + g*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*g) + (B*(d*f - c*g)^5*n*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 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 (f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac {(f+g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 g}-\frac {(B n) \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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 g}-\frac {(B (b c-a d) n) \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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 g}-\frac {(B (b c-a d) n) \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 ) n 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 ) n 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) n x^3}{15 b^2 d^2}-\frac {B (b c-a d) g^4 n x^4}{20 b d}-\frac {B (b f-a g)^5 n \log (a+b x)}{5 b^5 g}+\frac {(f+g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 g}+\frac {B (d f-c g)^5 n \log (c+d x)}{5 d^5 g}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.44, size = 285, normalized size = 0.78 \begin {gather*} \frac {\frac {B (-b c+a d) g^2 n 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 n \log (a+b x)}{b^5}+(f+g x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {B (d f-c g)^5 n \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))^n]),x]

[Out]

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

________________________________________________________________________________________

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

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))^n)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 636, normalized size = 1.75 \begin {gather*} \frac {1}{5} \, B g^{4} x^{5} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{5} \, A g^{4} x^{5} + B f g^{3} x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + A f g^{3} x^{4} + 2 \, B f^{2} g^{2} x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + 2 \, A f^{2} g^{2} x^{3} + 2 \, B f^{3} g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + 2 \, A f^{3} g x^{2} + \frac {1}{60} \, B g^{4} n {\left (\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 )} - \frac {1}{6} \, B f g^{3} n {\left (\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^{2} g^{2} n {\left (\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 )} - 2 \, B f^{3} g n {\left (\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^{4} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B f^{4} x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + 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))^n)),x, algorithm="maxima")

[Out]

1/5*B*g^4*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/5*A*g^4*x^5 + B*f*g^3*x^4*log((b*x/(d*x + c) + a/(d*x
+ c))^n*e) + A*f*g^3*x^4 + 2*B*f^2*g^2*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 2*A*f^2*g^2*x^3 + 2*B*f^3
*g*x^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 2*A*f^3*g*x^2 + 1/60*B*g^4*n*(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)) - 1/6*B*f*g^3*n*(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^2*g^2*n*(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)) - 2*B*f^3*g*n*(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^4*n*(a*l
og(b*x + a)/b - c*log(d*x + c)/d) + B*f^4*x*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + A*f^4*x

________________________________________________________________________________________

Fricas [A]
time = 0.78, size = 664, normalized size = 1.82 \begin {gather*} \frac {12 \, {\left (A + B\right )} b^{5} d^{5} g^{4} x^{5} + 3 \, {\left (20 \, {\left (A + B\right )} b^{5} d^{5} f g^{3} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \, {\left (30 \, {\left (A + B\right )} b^{5} d^{5} f^{2} g^{2} - {\left (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 )} n\right )} x^{3} + 6 \, {\left (20 \, {\left (A + B\right )} b^{5} d^{5} f^{3} g - {\left (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 )} n\right )} x^{2} + 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 )} n \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 )} n \log \left (d x + c\right ) + 12 \, {\left (5 \, {\left (A + B\right )} b^{5} d^{5} f^{4} - {\left (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 )} n\right )} x + 12 \, {\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B b^{5} d^{5} f g^{3} n x^{4} + 10 \, B b^{5} d^{5} f^{2} g^{2} n x^{3} + 10 \, B b^{5} d^{5} f^{3} g n x^{2} + 5 \, B b^{5} d^{5} f^{4} n x\right )} \log \left (\frac {b x + a}{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))^n)),x, algorithm="fricas")

[Out]

1/60*(12*(A + B)*b^5*d^5*g^4*x^5 + 3*(20*(A + B)*b^5*d^5*f*g^3 - (B*b^5*c*d^4 - B*a*b^4*d^5)*g^4*n)*x^4 + 4*(3
0*(A + B)*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)*n)*x^3
+ 6*(20*(A + B)*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)*n)*x^2 + 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)*n*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)*n*log(d*x + c) + 12*(5*(A + B)*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)*n)*x + 12*(B*b^5*d^5*g^4*n*x^5 + 5*B*b^5*d^5*f*g^
3*n*x^4 + 10*B*b^5*d^5*f^2*g^2*n*x^3 + 10*B*b^5*d^5*f^3*g*n*x^2 + 5*B*b^5*d^5*f^4*n*x)*log((b*x + a)/(d*x + c)
))/(b^5*d^5)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11806 vs. \(2 (351) = 702\).
time = 7.13, size = 11806, normalized size = 32.43 \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))^n)),x, algorithm="giac")

[Out]

1/60*(12*(5*B*b^6*c^2*d^4*f^4*n - 10*B*a*b^5*c*d^5*f^4*n - 20*(b*x + a)*B*b^5*c^2*d^5*f^4*n/(d*x + c) + 5*B*a^
2*b^4*d^6*f^4*n + 40*(b*x + a)*B*a*b^4*c*d^6*f^4*n/(d*x + c) + 30*(b*x + a)^2*B*b^4*c^2*d^6*f^4*n/(d*x + c)^2
- 20*(b*x + a)*B*a^2*b^3*d^7*f^4*n/(d*x + c) - 60*(b*x + a)^2*B*a*b^3*c*d^7*f^4*n/(d*x + c)^2 - 20*(b*x + a)^3
*B*b^3*c^2*d^7*f^4*n/(d*x + c)^3 + 30*(b*x + a)^2*B*a^2*b^2*d^8*f^4*n/(d*x + c)^2 + 40*(b*x + a)^3*B*a*b^2*c*d
^8*f^4*n/(d*x + c)^3 + 5*(b*x + a)^4*B*b^2*c^2*d^8*f^4*n/(d*x + c)^4 - 20*(b*x + a)^3*B*a^2*b*d^9*f^4*n/(d*x +
c)^3 - 10*(b*x + a)^4*B*a*b*c*d^9*f^4*n/(d*x + c)^4 + 5*(b*x + a)^4*B*a^2*d^10*f^4*n/(d*x + c)^4 - 10*B*b^6*c
^3*d^3*f^3*g*n + 10*B*a*b^5*c^2*d^4*f^3*g*n + 50*(b*x + a)*B*b^5*c^3*d^4*f^3*g*n/(d*x + c) + 10*B*a^2*b^4*c*d^
5*f^3*g*n - 70*(b*x + a)*B*a*b^4*c^2*d^5*f^3*g*n/(d*x + c) - 90*(b*x + a)^2*B*b^4*c^3*d^5*f^3*g*n/(d*x + c)^2
- 10*B*a^3*b^3*d^6*f^3*g*n - 10*(b*x + a)*B*a^2*b^3*c*d^6*f^3*g*n/(d*x + c) + 150*(b*x + a)^2*B*a*b^3*c^2*d^6*
f^3*g*n/(d*x + c)^2 + 70*(b*x + a)^3*B*b^3*c^3*d^6*f^3*g*n/(d*x + c)^3 + 30*(b*x + a)*B*a^3*b^2*d^7*f^3*g*n/(d
*x + c) - 30*(b*x + a)^2*B*a^2*b^2*c*d^7*f^3*g*n/(d*x + c)^2 - 130*(b*x + a)^3*B*a*b^2*c^2*d^7*f^3*g*n/(d*x +
c)^3 - 20*(b*x + a)^4*B*b^2*c^3*d^7*f^3*g*n/(d*x + c)^4 - 30*(b*x + a)^2*B*a^3*b*d^8*f^3*g*n/(d*x + c)^2 + 50*
(b*x + a)^3*B*a^2*b*c*d^8*f^3*g*n/(d*x + c)^3 + 40*(b*x + a)^4*B*a*b*c^2*d^8*f^3*g*n/(d*x + c)^4 + 10*(b*x + a
)^3*B*a^3*d^9*f^3*g*n/(d*x + c)^3 - 20*(b*x + a)^4*B*a^2*c*d^9*f^3*g*n/(d*x + c)^4 + 10*B*b^6*c^4*d^2*f^2*g^2*
n - 10*B*a*b^5*c^3*d^3*f^2*g^2*n - 50*(b*x + a)*B*b^5*c^4*d^3*f^2*g^2*n/(d*x + c) + 50*(b*x + a)*B*a*b^4*c^3*d
^4*f^2*g^2*n/(d*x + c) + 100*(b*x + a)^2*B*b^4*c^4*d^4*f^2*g^2*n/(d*x + c)^2 - 10*B*a^3*b^3*c*d^5*f^2*g^2*n +
30*(b*x + a)*B*a^2*b^3*c^2*d^5*f^2*g^2*n/(d*x + c) - 130*(b*x + a)^2*B*a*b^3*c^3*d^5*f^2*g^2*n/(d*x + c)^2 - 9
0*(b*x + a)^3*B*b^3*c^4*d^5*f^2*g^2*n/(d*x + c)^3 + 10*B*a^4*b^2*d^6*f^2*g^2*n - 10*(b*x + a)*B*a^3*b^2*c*d^6*
f^2*g^2*n/(d*x + c) - 30*(b*x + a)^2*B*a^2*b^2*c^2*d^6*f^2*g^2*n/(d*x + c)^2 + 150*(b*x + a)^3*B*a*b^2*c^3*d^6
*f^2*g^2*n/(d*x + c)^3 + 30*(b*x + a)^4*B*b^2*c^4*d^6*f^2*g^2*n/(d*x + c)^4 - 20*(b*x + a)*B*a^4*b*d^7*f^2*g^2
*n/(d*x + c) + 50*(b*x + a)^2*B*a^3*b*c*d^7*f^2*g^2*n/(d*x + c)^2 - 30*(b*x + a)^3*B*a^2*b*c^2*d^7*f^2*g^2*n/(
d*x + c)^3 - 60*(b*x + a)^4*B*a*b*c^3*d^7*f^2*g^2*n/(d*x + c)^4 + 10*(b*x + a)^2*B*a^4*d^8*f^2*g^2*n/(d*x + c)
^2 - 30*(b*x + a)^3*B*a^3*c*d^8*f^2*g^2*n/(d*x + c)^3 + 30*(b*x + a)^4*B*a^2*c^2*d^8*f^2*g^2*n/(d*x + c)^4 - 5
*B*b^6*c^5*d*f*g^3*n + 5*B*a*b^5*c^4*d^2*f*g^3*n + 25*(b*x + a)*B*b^5*c^5*d^2*f*g^3*n/(d*x + c) - 25*(b*x + a)
*B*a*b^4*c^4*d^3*f*g^3*n/(d*x + c) - 50*(b*x + a)^2*B*b^4*c^5*d^3*f*g^3*n/(d*x + c)^2 + 50*(b*x + a)^2*B*a*b^3
*c^4*d^4*f*g^3*n/(d*x + c)^2 + 50*(b*x + a)^3*B*b^3*c^5*d^4*f*g^3*n/(d*x + c)^3 + 5*B*a^4*b^2*c*d^5*f*g^3*n -
20*(b*x + a)*B*a^3*b^2*c^2*d^5*f*g^3*n/(d*x + c) + 30*(b*x + a)^2*B*a^2*b^2*c^3*d^5*f*g^3*n/(d*x + c)^2 - 70*(
b*x + a)^3*B*a*b^2*c^4*d^5*f*g^3*n/(d*x + c)^3 - 20*(b*x + a)^4*B*b^2*c^5*d^5*f*g^3*n/(d*x + c)^4 - 5*B*a^5*b*
d^6*f*g^3*n + 15*(b*x + a)*B*a^4*b*c*d^6*f*g^3*n/(d*x + c) - 10*(b*x + a)^2*B*a^3*b*c^2*d^6*f*g^3*n/(d*x + c)^
2 - 10*(b*x + a)^3*B*a^2*b*c^3*d^6*f*g^3*n/(d*x + c)^3 + 40*(b*x + a)^4*B*a*b*c^4*d^6*f*g^3*n/(d*x + c)^4 + 5*
(b*x + a)*B*a^5*d^7*f*g^3*n/(d*x + c) - 20*(b*x + a)^2*B*a^4*c*d^7*f*g^3*n/(d*x + c)^2 + 30*(b*x + a)^3*B*a^3*
c^2*d^7*f*g^3*n/(d*x + c)^3 - 20*(b*x + a)^4*B*a^2*c^3*d^7*f*g^3*n/(d*x + c)^4 + B*b^6*c^6*g^4*n - B*a*b^5*c^5
*d*g^4*n - 5*(b*x + a)*B*b^5*c^6*d*g^4*n/(d*x + c) + 5*(b*x + a)*B*a*b^4*c^5*d^2*g^4*n/(d*x + c) + 10*(b*x + a
)^2*B*b^4*c^6*d^2*g^4*n/(d*x + c)^2 - 10*(b*x + a)^2*B*a*b^3*c^5*d^3*g^4*n/(d*x + c)^2 - 10*(b*x + a)^3*B*b^3*
c^6*d^3*g^4*n/(d*x + c)^3 + 10*(b*x + a)^3*B*a*b^2*c^5*d^4*g^4*n/(d*x + c)^3 + 5*(b*x + a)^4*B*b^2*c^6*d^4*g^4
*n/(d*x + c)^4 - B*a^5*b*c*d^5*g^4*n + 5*(b*x + a)*B*a^4*b*c^2*d^5*g^4*n/(d*x + c) - 10*(b*x + a)^2*B*a^3*b*c^
3*d^5*g^4*n/(d*x + c)^2 + 10*(b*x + a)^3*B*a^2*b*c^4*d^5*g^4*n/(d*x + c)^3 - 10*(b*x + a)^4*B*a*b*c^5*d^5*g^4*
n/(d*x + c)^4 + B*a^6*d^6*g^4*n - 5*(b*x + a)*B*a^5*c*d^6*g^4*n/(d*x + c) + 10*(b*x + a)^2*B*a^4*c^2*d^6*g^4*n
/(d*x + c)^2 - 10*(b*x + a)^3*B*a^3*c^3*d^6*g^4*n/(d*x + c)^3 + 5*(b*x + a)^4*B*a^2*c^4*d^6*g^4*n/(d*x + c)^4)
*log((b*x + a)/(d*x + c))/(b^5*d^5 - 5*(b*x + a)*b^4*d^6/(d*x + c) + 10*(b*x + a)^2*b^3*d^7/(d*x + c)^2 - 10*(
b*x + a)^3*b^2*d^8/(d*x + c)^3 + 5*(b*x + a)^4*b*d^9/(d*x + c)^4 - (b*x + a)^5*d^10/(d*x + c)^5) - (120*B*b^10
*c^3*d^3*f^3*g*n - 360*B*a*b^9*c^2*d^4*f^3*g*n - 480*(b*x + a)*B*b^9*c^3*d^4*f^3*g*n/(d*x + c) + 360*B*a^2*b^8
*c*d^5*f^3*g*n + 1440*(b*x + a)*B*a*b^8*c^2*d^5*f^3*g*n/(d*x + c) + 720*(b*x + a)^2*B*b^8*c^3*d^5*f^3*g*n/(d*x
+ c)^2 - 120*B*a^3*b^7*d^6*f^3*g*n - 1440*(b*x + a)*B*a^2*b^7*c*d^6*f^3*g*n/(d*x + c) - 2160*(b*x + a)^2*B*a*
b^7*c^2*d^6*f^3*g*n/(d*x + c)^2 - 480*(b*x + a)^3*B*b^7*c^3*d^6*f^3*g*n/(d*x + c)^3 + 480*(b*x + a)*B*a^3*b^6*
d^7*f^3*g*n/(d*x + c) + 2160*(b*x + a)^2*B*a^2*...

________________________________________________________________________________________

Mupad [B]
time = 4.68, size = 1433, normalized size = 3.94 \begin {gather*} x^4\,\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{20\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,b\,d}\right )+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\,n-10\,B\,b\,c\,f^2\,g^2\,n}{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+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{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+30\,A\,b\,d\,f^2\,g^2+5\,B\,a\,d\,f\,g^3\,n-5\,B\,b\,c\,f\,g^3\,n}{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+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{5\,b\,d}-\frac {A\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}\right )}{2\,b\,d}\right )-x^3\,\left (\frac {\left (\frac {5\,A\,a\,d\,g^4+5\,A\,b\,c\,g^4+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{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+30\,A\,b\,d\,f^2\,g^2+5\,B\,a\,d\,f\,g^3\,n-5\,B\,b\,c\,f\,g^3\,n}{15\,b\,d}+\frac {A\,a\,c\,g^4}{3\,b\,d}\right )+x\,\left (\frac {5\,A\,b\,d\,f^4+20\,A\,a\,d\,f^3\,g+20\,A\,b\,c\,f^3\,g+30\,A\,a\,c\,f^2\,g^2+10\,B\,a\,d\,f^3\,g\,n-10\,B\,b\,c\,f^3\,g\,n}{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\,n-10\,B\,b\,c\,f^2\,g^2\,n}{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+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{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+30\,A\,b\,d\,f^2\,g^2+5\,B\,a\,d\,f\,g^3\,n-5\,B\,b\,c\,f\,g^3\,n}{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+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{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+20\,A\,b\,d\,f\,g^3+B\,a\,d\,g^4\,n-B\,b\,c\,g^4\,n}{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+30\,A\,b\,d\,f^2\,g^2+5\,B\,a\,d\,f\,g^3\,n-5\,B\,b\,c\,f\,g^3\,n}{5\,b\,d}+\frac {A\,a\,c\,g^4}{b\,d}\right )}{b\,d}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\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 )+\frac {A\,g^4\,x^5}{5}+\frac {\ln \left (a+b\,x\right )\,\left (\frac {B\,n\,a^5\,g^4}{5}-B\,n\,a^4\,b\,f\,g^3+2\,B\,n\,a^3\,b^2\,f^2\,g^2-2\,B\,n\,a^2\,b^3\,f^3\,g+B\,n\,a\,b^4\,f^4\right )}{b^5}-\frac {\ln \left (c+d\,x\right )\,\left (B\,n\,c^5\,g^4-5\,B\,n\,c^4\,d\,f\,g^3+10\,B\,n\,c^3\,d^2\,f^2\,g^2-10\,B\,n\,c^2\,d^3\,f^3\,g+5\,B\,n\,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))^n)),x)

[Out]

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

________________________________________________________________________________________