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

Optimal. Leaf size=188 \[ \frac{g^4 (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}+\frac{B g^4 n x (b c-a d)^4}{5 d^4}-\frac{B g^4 n (a+b x)^2 (b c-a d)^3}{10 b d^3}+\frac{B g^4 n (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{B g^4 n (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 n (a+b x)^4 (b c-a d)}{20 b d} \]

[Out]

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

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Rubi [A]  time = 0.1431, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 43} \[ \frac{g^4 (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}+\frac{B g^4 n x (b c-a d)^4}{5 d^4}-\frac{B g^4 n (a+b x)^2 (b c-a d)^3}{10 b d^3}+\frac{B g^4 n (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{B g^4 n (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 n (a+b x)^4 (b c-a d)}{20 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b}-\frac{(B n) \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b g}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b}-\frac{\left (B (b c-a d) g^4 n\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b}-\frac{\left (B (b c-a d) g^4 n\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac{B (b c-a d)^4 g^4 n x}{5 d^4}-\frac{B (b c-a d)^3 g^4 n (a+b x)^2}{10 b d^3}+\frac{B (b c-a d)^2 g^4 n (a+b x)^3}{15 b d^2}-\frac{B (b c-a d) g^4 n (a+b x)^4}{20 b d}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b}-\frac{B (b c-a d)^5 g^4 n \log (c+d x)}{5 b d^5}\\ \end{align*}

Mathematica [A]  time = 0.128648, size = 146, normalized size = 0.78 \[ \frac{g^4 \left ((a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n (b c-a d) \left (6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (a d-b c)-12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (c+d x)+3 d^4 (a+b x)^4\right )}{12 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^4*((a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - (B*(b*c - a*d)*n*(-12*b*d*(b*c - a*d)^3*x + 6*d^2*(
b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 3*d^4*(a + b*x)^4 + 12*(b*c - a*d)^4*Log[c + d*x
]))/(12*d^5)))/(5*b)

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Maple [F]  time = 0.546, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \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((b*g*x+a*g)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

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

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Maxima [B]  time = 1.28014, size = 913, normalized size = 4.86 \begin{align*} \frac{1}{5} \, B b^{4} 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 b^{4} g^{4} x^{5} + B a b^{3} g^{4} x^{4} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A a b^{3} g^{4} x^{4} + 2 \, B a^{2} b^{2} g^{4} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, B a^{3} b g^{4} x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + 2 \, A a^{3} b g^{4} x^{2} + \frac{1}{60} \, B b^{4} 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 a b^{3} g^{4} 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 a^{2} b^{2} g^{4} 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 a^{3} b g^{4} 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 a^{4} g^{4} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B a^{4} g^{4} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A a^{4} g^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.1829, size = 1188, normalized size = 6.32 \begin{align*} \frac{12 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} n \log \left (b x + a\right ) - 12 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} n \log \left (d x + c\right ) + 3 \,{\left (20 \, A a b^{4} d^{5} g^{4} -{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \,{\left (30 \, A a^{2} b^{3} d^{5} g^{4} +{\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + 4 \, B a^{2} b^{3} d^{5}\right )} g^{4} n\right )} x^{3} + 6 \,{\left (20 \, A a^{3} b^{2} d^{5} g^{4} -{\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 6 \, B a^{3} b^{2} d^{5}\right )} g^{4} n\right )} x^{2} + 12 \,{\left (5 \, A a^{4} b d^{5} g^{4} +{\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} + 4 \, B a^{4} b d^{5}\right )} g^{4} n\right )} x + 12 \,{\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (e\right ) + 12 \,{\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B a b^{4} d^{5} g^{4} n x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} n x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} n x^{2} + 5 \, B a^{4} b d^{5} g^{4} n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{60 \, b d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

Timed out

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

[Out]

Timed out