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

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

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 182, normalized size of antiderivative = 1.00, 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} \[ \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 x (b c-a d)^4}{5 d^4}+\frac {B g^4 (a+b x)^2 (b c-a d)^3}{5 b d^3}-\frac {2 B g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}+\frac {2 B g^4 (b c-a d)^5 \log (c+d x)}{5 b d^5}+\frac {B g^4 (a+b x)^4 (b c-a d)}{10 b d} \]

Antiderivative was successfully verified.

[In]

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

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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 144, normalized size = 0.79 \[ \frac {g^4 \left ((a+b x)^5 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )-\frac {B (a d-b c) \left (4 d^3 (a+b x)^3 (a d-b c)+6 d^2 (a+b x)^2 (b c-a d)^2-12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (c+d x)+3 d^4 (a+b x)^4\right )}{6 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^4*(-1/6*(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]))/d^5 + (a + b*x)^5*(A + B*Log[(e*(c + d*x)^2
)/(a + b*x)^2])))/(5*b)

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fricas [B]  time = 0.70, size = 457, normalized size = 2.51 \[ \frac {6 \, A b^{5} d^{5} g^{4} x^{5} - 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) + 3 \, {\left (B b^{5} c d^{4} + {\left (10 \, A - B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} - 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} - {\left (15 \, A - 4 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} + 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} + 2 \, {\left (5 \, A - 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} - 6 \, {\left (2 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 20 \, B a^{2} b^{3} c^{2} d^{3} - 20 \, B a^{3} b^{2} c d^{4} - {\left (5 \, A - 8 \, B\right )} a^{4} b d^{5}\right )} g^{4} x + 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 6 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{30 \, b d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^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((d^2*e*x^2 + 2*c*d*e*x +
c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)))/(b*d^5)

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giac [B]  time = 78.00, size = 493, normalized size = 2.71 \[ -\frac {2 \, B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} + \frac {1}{5} \, {\left (A b^{4} g^{4} + B b^{4} g^{4}\right )} x^{5} + \frac {{\left (B b^{4} c g^{4} + 10 \, A a b^{3} d g^{4} + 9 \, B a b^{3} d g^{4}\right )} x^{4}}{10 \, d} - \frac {2 \, {\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} - 15 \, A a^{2} b^{2} d^{2} g^{4} - 11 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} + 10 \, A a^{3} b d^{3} g^{4} + 4 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{5 \, d^{3}} - \frac {{\left (2 \, B b^{4} c^{4} g^{4} - 10 \, B a b^{3} c^{3} d g^{4} + 20 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 20 \, B a^{3} b c d^{3} g^{4} - 5 \, A a^{4} d^{4} g^{4} + 3 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} + \frac {2 \, {\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (d x + c\right )}{5 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^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 + 9*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 - 11*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
((d^2*x^2 + 2*c*d*x + c^2)/(b^2*x^2 + 2*a*b*x + a^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 + 4*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 + 3*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

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maple [B]  time = 0.13, size = 1030, normalized size = 5.66 \[ \frac {B \,b^{4} g^{4} x^{5} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )}{5}+\frac {A \,b^{4} g^{4} x^{5}}{5}+B a \,b^{3} g^{4} x^{4} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )+A a \,b^{3} g^{4} x^{4}+2 B \,a^{2} b^{2} g^{4} x^{3} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )-\frac {B a \,b^{3} g^{4} x^{4}}{10}+\frac {B \,b^{4} c \,g^{4} x^{4}}{10 d}+2 A \,a^{2} b^{2} g^{4} x^{3}+2 B \,a^{3} b \,g^{4} x^{2} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )-\frac {8 B \,a^{2} b^{2} g^{4} x^{3}}{15}+\frac {2 B a \,b^{3} c \,g^{4} x^{3}}{3 d}-\frac {2 B \,b^{4} c^{2} g^{4} x^{3}}{15 d^{2}}+2 A \,a^{3} b \,g^{4} x^{2}+B \,a^{4} g^{4} x \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )-\frac {6 B \,a^{3} b \,g^{4} x^{2}}{5}+\frac {2 B \,a^{2} b^{2} c \,g^{4} x^{2}}{d}-\frac {B a \,b^{3} c^{2} g^{4} x^{2}}{d^{2}}+\frac {B \,b^{4} c^{3} g^{4} x^{2}}{5 d^{3}}+A \,a^{4} g^{4} x +\frac {2 B \,a^{5} g^{4} \ln \left (\frac {1}{b x +a}\right )}{5 b}+\frac {B \,a^{5} g^{4} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )}{5 b}-\frac {2 B \,a^{5} g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{5 b}-\frac {2 B \,a^{4} c \,g^{4} \ln \left (\frac {1}{b x +a}\right )}{d}+\frac {2 B \,a^{4} c \,g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d}-\frac {8 B \,a^{4} g^{4} x}{5}+\frac {4 B \,a^{3} b \,c^{2} g^{4} \ln \left (\frac {1}{b x +a}\right )}{d^{2}}-\frac {4 B \,a^{3} b \,c^{2} g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{2}}+\frac {4 B \,a^{3} b c \,g^{4} x}{d}-\frac {4 B \,a^{2} b^{2} c^{3} g^{4} \ln \left (\frac {1}{b x +a}\right )}{d^{3}}+\frac {4 B \,a^{2} b^{2} c^{3} g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{3}}-\frac {4 B \,a^{2} b^{2} c^{2} g^{4} x}{d^{2}}+\frac {2 B a \,b^{3} c^{4} g^{4} \ln \left (\frac {1}{b x +a}\right )}{d^{4}}-\frac {2 B a \,b^{3} c^{4} g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{4}}+\frac {2 B a \,b^{3} c^{3} g^{4} x}{d^{3}}-\frac {2 B \,b^{4} c^{5} g^{4} \ln \left (\frac {1}{b x +a}\right )}{5 d^{5}}+\frac {2 B \,b^{4} c^{5} g^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{5 d^{5}}-\frac {2 B \,b^{4} c^{4} g^{4} x}{5 d^{4}}+\frac {A \,a^{5} g^{4}}{5 b}-\frac {5 B \,a^{5} g^{4}}{6 b}+\frac {77 B \,a^{4} c \,g^{4}}{30 d}-\frac {47 B \,a^{3} b \,c^{2} g^{4}}{15 d^{2}}+\frac {9 B \,a^{2} b^{2} c^{3} g^{4}}{5 d^{3}}-\frac {2 B a \,b^{3} c^{4} g^{4}}{5 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2)),x)

[Out]

-5/6/b*g^4*B*a^5+1/5/b*A*a^5*g^4+1/5*b^4*A*x^5*g^4-8/5*B*x*a^4*g^4-2/5/b*g^4*B*a^5*ln(a/(b*x+a)*d-1/(b*x+a)*b*
c-d)+1/5*b^4*B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/b^2)*x^5*g^4+1/5/b*B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/
b^2)*a^5*g^4+B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/b^2)*x*a^4*g^4+2/5/b*g^4*B*a^5*ln(1/(b*x+a))-8/15*b^2*B*x^
3*a^2*g^4-6/5*b*B*x^2*a^3*g^4+b^3*A*x^4*a*g^4+2*b^2*A*x^3*a^2*g^4+2*b*A*x^2*a^3*g^4-1/10*b^3*B*x^4*a*g^4+2*b^2
*B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/b^2)*x^3*a^2*g^4+2*b*B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/b^2)*x^2*a
^3*g^4+2*g^4*B*a^4/d*ln(a/(b*x+a)*d-1/(b*x+a)*b*c-d)*c-2*g^4*B*a^4/d*ln(1/(b*x+a))*c+A*a^4*g^4*x+9/5*b^2*g^4*B
*c^3/d^3*a^2-47/15*b*g^4*B*c^2/d^2*a^3-2/5*b^3*g^4*B*c^4/d^4*a+77/30*g^4*B*c/d*a^4+1/5*b^4*g^4*B*c^3/d^3*x^2-2
/15*b^4*g^4*B*c^2/d^2*x^3-2/5*b^4*g^4*B*c^4/d^4*x+1/10*b^4*g^4*B*c/d*x^4-2/5*b^4*g^4*B*c^5/d^5*ln(1/(b*x+a))+2
/5*b^4*g^4*B*c^5/d^5*ln(a/(b*x+a)*d-1/(b*x+a)*b*c-d)+b^3*B*ln(e*(a/(b*x+a)*d-1/(b*x+a)*b*c-d)^2/b^2)*x^4*a*g^4
-4*b*g^4*B*a^3/d^2*ln(a/(b*x+a)*d-1/(b*x+a)*b*c-d)*c^2+4*b*g^4*B*a^3/d^2*ln(1/(b*x+a))*c^2+4*b*g^4*B*c/d*x*a^3
+2*b^3*g^4*B*a/d^4*ln(1/(b*x+a))*c^4-4*b^2*g^4*B*a^2/d^3*ln(1/(b*x+a))*c^3+4*b^2*g^4*B*a^2/d^3*ln(a/(b*x+a)*d-
1/(b*x+a)*b*c-d)*c^3-2*b^3*g^4*B*a/d^4*ln(a/(b*x+a)*d-1/(b*x+a)*b*c-d)*c^4-b^3*g^4*B*c^2/d^2*x^2*a-4*b^2*g^4*B
*c^2/d^2*x*a^2+2*b^3*g^4*B*c^3/d^3*x*a+2/3*b^3*g^4*B*c/d*x^3*a+2*b^2*g^4*B*c/d*x^2*a^2

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maxima [B]  time = 1.40, size = 882, normalized size = 4.85 \[ \frac {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} + {\left (x \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{4} g^{4} + 2 \, {\left (x^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a^{3} b g^{4} + 2 \, {\left (x^{3} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\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 a^{2} b^{2} g^{4} + \frac {1}{3} \, {\left (3 \, x^{4} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\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 a b^{3} g^{4} + \frac {1}{30} \, {\left (6 \, x^{5} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\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 b^{4} g^{4} + A a^{4} g^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^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(d^2*e*x^2/(b^2*x^2 + 2*
a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 2*a*log(b*x + a)/b + 2
*c*log(d*x + c)/d)*B*a^4*g^4 + 2*(x^2*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x +
a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^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(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2
*e/(b^2*x^2 + 2*a*b*x + a^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(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*
e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^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(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/
(b^2*x^2 + 2*a*b*x + a^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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mupad [B]  time = 4.79, size = 1024, normalized size = 5.63 \[ x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{10\,b\,d}+\frac {a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}\right )-x^3\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{3\,d}+\frac {A\,a\,b^3\,c\,g^4}{3\,d}\right )+x\,\left (\frac {a^3\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c-4\,B\,a\,d+4\,B\,b\,c\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{5\,b\,d}+\frac {2\,a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{b\,d}\right )+\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (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+\frac {B\,b^4\,g^4\,x^5}{5}\right )+x^4\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{20\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (2\,B\,a^4\,c\,d^4\,g^4-4\,B\,a^3\,b\,c^2\,d^3\,g^4+4\,B\,a^2\,b^2\,c^3\,d^2\,g^4-2\,B\,a\,b^3\,c^4\,d\,g^4+\frac {2\,B\,b^4\,c^5\,g^4}{5}\right )}{d^5}+\frac {A\,b^4\,g^4\,x^5}{5}-\frac {2\,B\,a^5\,g^4\,\ln \left (a+b\,x\right )}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^4*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2)),x)

[Out]

x^2*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c)
)/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/d + (A*a*b^3*c*g^4)/d
))/(10*b*d) + (a^2*b*g^4*(5*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/d - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c - 2*B
*a*d + 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(2*b*d)) - x^3*((((b^3*g^4*(25*A*a*d + 5*A*b*c -
2*B*a*d + 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(15*b*d) - (a*b^2*g^4*(10*A*a*
d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(3*d) + (A*a*b^3*c*g^4)/(3*d)) + x*((a^3*g^4*(5*A*a*d + 10*A*b*c - 4*B*a*d +
4*B*b*c))/d - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(5*d)
- (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c - 2*B*a*d + 2*B
*b*c))/d + (A*a*b^3*c*g^4)/d))/(5*b*d) + (2*a^2*b*g^4*(5*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/d - (a*c*((b^3*
g^4*(25*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(b*d)))/(5*b*d) + (a
*c*((((b^3*g^4*(25*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5
*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/d + (A*a*b^3*c*g^4)/d))/(b*d)) + log((e*
(c + d*x)^2)/(a + b*x)^2)*((B*b^4*g^4*x^5)/5 + B*a^4*g^4*x + 2*B*a^3*b*g^4*x^2 + B*a*b^3*g^4*x^4 + 2*B*a^2*b^2
*g^4*x^3) + x^4*((b^3*g^4*(25*A*a*d + 5*A*b*c - 2*B*a*d + 2*B*b*c))/(20*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(20*d
)) + (log(c + d*x)*((2*B*b^4*c^5*g^4)/5 + 2*B*a^4*c*d^4*g^4 - 4*B*a^3*b*c^2*d^3*g^4 + 4*B*a^2*b^2*c^3*d^2*g^4
- 2*B*a*b^3*c^4*d*g^4))/d^5 + (A*b^4*g^4*x^5)/5 - (2*B*a^5*g^4*log(a + b*x))/(5*b)

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sympy [B]  time = 6.70, size = 998, normalized size = 5.48 \[ \frac {A b^{4} g^{4} x^{5}}{5} - \frac {2 B a^{5} g^{4} \log {\left (x + \frac {\frac {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}} \right )}}{5 b} + \frac {2 B c g^{4} \left (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}\right ) \log {\left (x + \frac {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} \left (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}\right ) + \frac {2 B b c^{2} g^{4} \left (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}\right )}{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}} \right )}}{5 d^{5}} + x^{4} \left (A a b^{3} g^{4} - \frac {B a b^{3} g^{4}}{10} + \frac {B b^{4} c g^{4}}{10 d}\right ) + x^{3} \left (2 A a^{2} b^{2} g^{4} - \frac {8 B a^{2} b^{2} g^{4}}{15} + \frac {2 B a b^{3} c g^{4}}{3 d} - \frac {2 B b^{4} c^{2} g^{4}}{15 d^{2}}\right ) + x^{2} \left (2 A a^{3} b g^{4} - \frac {6 B a^{3} b g^{4}}{5} + \frac {2 B a^{2} b^{2} c g^{4}}{d} - \frac {B a b^{3} c^{2} g^{4}}{d^{2}} + \frac {B b^{4} c^{3} g^{4}}{5 d^{3}}\right ) + x \left (A a^{4} g^{4} - \frac {8 B a^{4} g^{4}}{5} + \frac {4 B a^{3} b c g^{4}}{d} - \frac {4 B a^{2} b^{2} c^{2} g^{4}}{d^{2}} + \frac {2 B a b^{3} c^{3} g^{4}}{d^{3}} - \frac {2 B b^{4} c^{4} g^{4}}{5 d^{4}}\right ) + \left (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} + \frac {B b^{4} g^{4} x^{5}}{5}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**4*(A+B*ln(e*(d*x+c)**2/(b*x+a)**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) + x**4*(A*a*b**3*g**4 - B*a*b**3*g**4/10 + B*b**4*c*g**4/(10*d)) + x**3*(2*A*a**2*b**2*g**4 - 8*B*a**2*b**2
*g**4/15 + 2*B*a*b**3*c*g**4/(3*d) - 2*B*b**4*c**2*g**4/(15*d**2)) + x**2*(2*A*a**3*b*g**4 - 6*B*a**3*b*g**4/5
+ 2*B*a**2*b**2*c*g**4/d - B*a*b**3*c**2*g**4/d**2 + B*b**4*c**3*g**4/(5*d**3)) + x*(A*a**4*g**4 - 8*B*a**4*g
**4/5 + 4*B*a**3*b*c*g**4/d - 4*B*a**2*b**2*c**2*g**4/d**2 + 2*B*a*b**3*c**3*g**4/d**3 - 2*B*b**4*c**4*g**4/(5
*d**4)) + (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*(c + d*x)**2/(a + b*x)**2)

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