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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.602933, antiderivative size = 348, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 72} \[ -\frac{B g^2 n x^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )\right )}{10 b^3 d^3}+\frac{B g n x \left (-10 a^2 b^2 d^4 f^2 g+5 a^3 b d^4 f g^2-a^4 d^4 g^3+10 a b^3 d^4 f^3+b^4 (-c) \left (5 c^2 d f g^2-c^3 g^3-10 c d^2 f^2 g+10 d^3 f^3\right )\right )}{5 b^4 d^4}+\frac{(f+g x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 g}-\frac{B g^3 n x^3 (b c-a d) (-a d g-b c g+5 b d f)}{15 b^2 d^2}-\frac{B n (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac{B g^4 n x^4 (b c-a d)}{20 b d}+\frac{B n (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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))*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 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]

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]

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 = 0.649985, size = 285, normalized size = 0.78 \[ \frac{\frac{B g^2 n x (a d-b c) \left (6 a^2 b d^2 g^2 (-2 c g+10 d f+d g x)-12 a^3 d^3 g^3-2 a b^2 d g \left (6 c^2 g^2-3 c d g (10 f+g x)+d^2 \left (60 f^2+15 f g x+2 g^2 x^2\right )\right )+b^3 \left (6 c^2 d g^2 (10 f+g x)-12 c^3 g^3-2 c d^2 g \left (60 f^2+15 f g x+2 g^2 x^2\right )+d^3 \left (60 f^2 g x+120 f^3+20 f g^2 x^2+3 g^3 x^3\right )\right )\right )}{12 b^4 d^4}+(f+g x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n (b f-a g)^5 \log (a+b x)}{b^5}+\frac{B n (d f-c g)^5 \log (c+d x)}{d^5}}{5 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4*(A + B*Log[e*((a + b*x)/(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 = 0.438, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{4} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

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 = 1.26038, size = 852, normalized size = 2.34 \begin{align*} \frac{1}{5} \, B g^{4} x^{5} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{1}{5} \, A g^{4} x^{5} + B f g^{3} x^{4} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f g^{3} x^{4} + 2 \, B f^{2} g^{2} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + 2 \, A f^{2} g^{2} x^{3} + 2 \, B f^{3} g x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\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 (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f^{4} x \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 3.13323, size = 1488, normalized size = 4.09 \begin{align*} \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} n\right )} x^{4} + 4 \,{\left (30 \, A 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 \, A 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 \, A 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} 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 (e\right ) + 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{align*}

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^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*n)*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)*n)*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)*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^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*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(e) + 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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out