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

Optimal. Leaf size=182 \[ \frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}+\frac{2 B g^4 x (b c-a d)^4}{5 d^4}-\frac{B g^4 (a+b x)^2 (b c-a d)^3}{5 b d^3}+\frac{2 B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{2 B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 (a+b x)^4 (b c-a d)}{10 b d} \]

[Out]

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

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Rubi [A]  time = 0.109519, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2525, 12, 43} \[ \frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}+\frac{2 B g^4 x (b c-a d)^4}{5 d^4}-\frac{B g^4 (a+b x)^2 (b c-a d)^3}{5 b d^3}+\frac{2 B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{2 B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}-\frac{B g^4 (a+b x)^4 (b c-a d)}{10 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0956361, size = 144, normalized size = 0.79 \[ \frac{g^4 \left ((a+b x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac{B (b c-a d) \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )}{6 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.423, size = 1606, normalized size = 8.8 \begin{align*} \text{result too large to display} \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)^2/(d*x+c)^2)),x)

[Out]

1/10*B*g^4*b^3*a*x^4-113/30/d^4*B*g^4*b^3*a*c^4+2/5/d^5*B*g^4*b^4*c^5*ln(1/(d*x+c))+8/d*B*g^4*a^4*ln(1/(d*x+c)
*a*d-b*c/(d*x+c)+b)*c+2/d*B*g^4*a^4*ln(1/(d*x+c))*c+1/d*B*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*a^4*c*g^4+
1/5/d^5*B*g^4*b^4*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*c^5+8/5/d^5*B*g^4*b^4*c^5*ln(1/(d*x+c)*a*d-b*c/(d*
x+c)+b)+2*B*g^4*b*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*a^3*x^2-12*B*g^4/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d
*x+c)+b)*a^5*c+2*B*g^4*b^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*a^2*x^3+B*g^4*b^3*ln(e*(1/(d*x+c)*a*d-b*c
/(d*x+c)+b)^2/d^2)*a*x^4-1/10/d*B*g^4*b^4*c*x^4+5/6/d^5*B*g^4*b^4*c^5+1/5/d^5*A*g^4*b^4*c^5+1/d*A*g^4*a^4*c+8/
5/d*B*g^4*a^4*c+8/15*B*g^4*b^2*a^2*x^3+1/5*B*g^4*b^4*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*x^5+B*ln(e*(1/(
d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*x*a^4*g^4-2/5*B*g^4/b*a^5*ln(1/(d*x+c))-8/5*B*g^4/b*a^5*ln(1/(d*x+c)*a*d-b*c/
(d*x+c)+b)+A*g^4*x^4*a*b^3+2*A*g^4*x^3*a^2*b^2+2*A*g^4*x^2*a^3*b+6/5*B*g^4*b*a^3*x^2-1/d^4*A*g^4*a*b^3*c^4+2/d
^3*A*g^4*a^2*b^2*c^3-2/d^2*A*g^4*a^3*b*c^2-26/5/d^2*B*g^4*b*a^3*c^2+98/15/d^3*B*g^4*b^2*a^2*c^3+8/5*B*x*a^4*g^
4+1/5*A*g^4*x^5*b^4+A*g^4*x*a^4-12/d^4*B*g^4/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a*c^5*b^4-40/d^2*B*g^4*
b^2/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^3*c^3+30/d*B*g^4*b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a
^4*c^2+30/d^3*B*g^4*b^3/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*c^4+2/15/d^2*B*g^4*b^4*c^2*x^3-1/5/d^3*B
*g^4*b^4*c^3*x^2+2/5/d^4*B*g^4*b^4*c^4*x-2/d^3*B*g^4*b^3*a*x*c^3-4/d*B*g^4*b*a^3*x*c-1/d^4*B*g^4*b^3*ln(e*(1/(
d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*a*c^4+2/d^3*B*g^4*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*a^2*b^2*c^3-16/d^
2*B*g^4*b*a^3*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^2+16/d^3*B*g^4*b^2*a^2*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^3-8/d
^4*B*g^4*b^3*a*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^4-4/d^2*B*g^4*b*a^3*ln(1/(d*x+c))*c^2+2*d*B*g^4/b/(a*d-b*c)*l
n(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^6+4/d^3*B*g^4*b^2*a^2*ln(1/(d*x+c))*c^3-2/d^4*B*g^4*b^3*a*ln(1/(d*x+c))*c^4+2
/d^5*B*g^4/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^6*b^5-2/d^2*B*g^4*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/
d^2)*a^3*b*c^2-2/3/d*B*g^4*b^3*a*x^3*c+1/d^2*B*g^4*b^3*a*x^2*c^2-2/d*B*g^4*b^2*a^2*x^2*c+4/d^2*B*g^4*b^2*a^2*x
*c^2

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Maxima [B]  time = 1.38107, size = 1195, normalized size = 6.57 \begin{align*} \text{result too large to display} \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)^2/(d*x+c)^2)),x, algorithm="maxima")

[Out]

1/5*A*b^4*g^4*x^5 + A*a*b^3*g^4*x^4 + 2*A*a^2*b^2*g^4*x^3 + 2*A*a^3*b*g^4*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*a^4*g^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*a^3*b*g^4 + 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*a^2*b^2*g^4 + 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*
a*b^3*g^4 + 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
*b^4*g^4 + A*a^4*g^4*x

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Fricas [B]  time = 1.30286, size = 952, normalized size = 5.23 \begin{align*} \frac{6 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \,{\left (B b^{5} c d^{4} -{\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \,{\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} +{\left (15 \, A + 4 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \,{\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} - 2 \,{\left (5 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 6 \,{\left (2 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 20 \, B a^{2} b^{3} c^{2} d^{3} - 20 \, B a^{3} b^{2} c d^{4} +{\left (5 \, A + 8 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 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} \log \left (d x + c\right ) + 6 \,{\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 (\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 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)^2/(d*x+c)^2)),x, algorithm="fricas")

[Out]

1/30*(6*A*b^5*d^5*g^4*x^5 + 12*B*a^5*d^5*g^4*log(b*x + a) - 3*(B*b^5*c*d^4 - (10*A + B)*a*b^4*d^5)*g^4*x^4 + 4
*(B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + (15*A + 4*B)*a^2*b^3*d^5)*g^4*x^3 - 6*(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 - 2*(5*A + 3*B)*a^3*b^2*d^5)*g^4*x^2 + 6*(2*B*b^5*c^4*d - 10*B*a*b^4*c^3*d^2 + 20*B*a^2*b^
3*c^2*d^3 - 20*B*a^3*b^2*c*d^4 + (5*A + 8*B)*a^4*b*d^5)*g^4*x - 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*log(d*x + c) + 6*(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((b^2*e*x^2 + 2*a*b*e*x +
a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/(b*d^5)

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Sympy [B]  time = 8.4322, size = 1018, normalized size = 5.59 \begin{align*} \text{result too large to display} \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)**2/(d*x+c)**2)),x)

[Out]

A*b**4*g**4*x**5/5 + 2*B*a**5*g**4*log(x + (2*B*a**6*d**5*g**4/b + 10*B*a**5*c*d**4*g**4 - 20*B*a**4*b*c**2*d*
*3*g**4 + 20*B*a**3*b**2*c**3*d**2*g**4 - 10*B*a**2*b**3*c**4*d*g**4 + 2*B*a*b**4*c**5*g**4)/(2*B*a**5*d**5*g*
*4 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a**3*b**2*c**2*d**3*g**4 + 20*B*a**2*b**3*c**3*d**2*g**4 - 10*B*a*b**4*c**
4*d*g**4 + 2*B*b**5*c**5*g**4))/(5*b) - 2*B*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 -
5*a*b**3*c**3*d + b**4*c**4)*log(x + (12*B*a**5*c*d**4*g**4 - 20*B*a**4*b*c**2*d**3*g**4 + 20*B*a**3*b**2*c**3
*d**2*g**4 - 10*B*a**2*b**3*c**4*d*g**4 + 2*B*a*b**4*c**5*g**4 - 2*B*a*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3
+ 10*a**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b**4*c**4) + 2*B*b*c**2*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10
*a**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b**4*c**4)/d)/(2*B*a**5*d**5*g**4 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a*
*3*b**2*c**2*d**3*g**4 + 20*B*a**2*b**3*c**3*d**2*g**4 - 10*B*a*b**4*c**4*d*g**4 + 2*B*b**5*c**5*g**4))/(5*d**
5) + (B*a**4*g**4*x + 2*B*a**3*b*g**4*x**2 + 2*B*a**2*b**2*g**4*x**3 + B*a*b**3*g**4*x**4 + B*b**4*g**4*x**5/5
)*log(e*(a + b*x)**2/(c + d*x)**2) + x**4*(10*A*a*b**3*d*g**4 + B*a*b**3*d*g**4 - B*b**4*c*g**4)/(10*d) + x**3
*(30*A*a**2*b**2*d**2*g**4 + 8*B*a**2*b**2*d**2*g**4 - 10*B*a*b**3*c*d*g**4 + 2*B*b**4*c**2*g**4)/(15*d**2) +
x**2*(10*A*a**3*b*d**3*g**4 + 6*B*a**3*b*d**3*g**4 - 10*B*a**2*b**2*c*d**2*g**4 + 5*B*a*b**3*c**2*d*g**4 - B*b
**4*c**3*g**4)/(5*d**3) + x*(5*A*a**4*d**4*g**4 + 8*B*a**4*d**4*g**4 - 20*B*a**3*b*c*d**3*g**4 + 20*B*a**2*b**
2*c**2*d**2*g**4 - 10*B*a*b**3*c**3*d*g**4 + 2*B*b**4*c**4*g**4)/(5*d**4)

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Giac [B]  time = 162.17, size = 670, normalized size = 3.68 \begin{align*} \frac{2 \, B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} + \frac{1}{5} \,{\left (A b^{4} g^{4} + B b^{4} g^{4}\right )} x^{5} - \frac{{\left (B b^{4} c g^{4} - 10 \, A a b^{3} d g^{4} - 11 \, B a b^{3} d g^{4}\right )} x^{4}}{10 \, d} + \frac{2 \,{\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} + 15 \, A a^{2} b^{2} d^{2} g^{4} + 19 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac{1}{5} \,{\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac{{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} - 10 \, A a^{3} b d^{3} g^{4} - 16 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{5 \, d^{3}} + \frac{{\left (2 \, B b^{4} c^{4} g^{4} - 10 \, B a b^{3} c^{3} d g^{4} + 20 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 20 \, B a^{3} b c d^{3} g^{4} + 5 \, A a^{4} d^{4} g^{4} + 13 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} - \frac{2 \,{\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (-d x - c\right )}{5 \, 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)^2/(d*x+c)^2)),x, algorithm="giac")

[Out]

2/5*B*a^5*g^4*log(b*x + a)/b + 1/5*(A*b^4*g^4 + B*b^4*g^4)*x^5 - 1/10*(B*b^4*c*g^4 - 10*A*a*b^3*d*g^4 - 11*B*a
*b^3*d*g^4)*x^4/d + 2/15*(B*b^4*c^2*g^4 - 5*B*a*b^3*c*d*g^4 + 15*A*a^2*b^2*d^2*g^4 + 19*B*a^2*b^2*d^2*g^4)*x^3
/d^2 + 1/5*(B*b^4*g^4*x^5 + 5*B*a*b^3*g^4*x^4 + 10*B*a^2*b^2*g^4*x^3 + 10*B*a^3*b*g^4*x^2 + 5*B*a^4*g^4*x)*log
((b^2*x^2 + 2*a*b*x + a^2)/(d^2*x^2 + 2*c*d*x + c^2)) - 1/5*(B*b^4*c^3*g^4 - 5*B*a*b^3*c^2*d*g^4 + 10*B*a^2*b^
2*c*d^2*g^4 - 10*A*a^3*b*d^3*g^4 - 16*B*a^3*b*d^3*g^4)*x^2/d^3 + 1/5*(2*B*b^4*c^4*g^4 - 10*B*a*b^3*c^3*d*g^4 +
20*B*a^2*b^2*c^2*d^2*g^4 - 20*B*a^3*b*c*d^3*g^4 + 5*A*a^4*d^4*g^4 + 13*B*a^4*d^4*g^4)*x/d^4 - 2/5*(B*b^4*c^5*
g^4 - 5*B*a*b^3*c^4*d*g^4 + 10*B*a^2*b^2*c^3*d^2*g^4 - 10*B*a^3*b*c^2*d^3*g^4 + 5*B*a^4*c*d^4*g^4)*log(-d*x -
c)/d^5