3.202 \(\int (a g+b g x)^3 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2})) \, dx\)

Optimal. Leaf size=151 \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}-\frac {B g^3 (b c-a d)^4 \log (c+d x)}{2 b d^4}+\frac {B g^3 x (b c-a d)^3}{2 d^3}-\frac {B g^3 (a+b x)^2 (b c-a d)^2}{4 b d^2}+\frac {B g^3 (a+b x)^3 (b c-a d)}{6 b d} \]

[Out]

1/2*B*(-a*d+b*c)^3*g^3*x/d^3-1/4*B*(-a*d+b*c)^2*g^3*(b*x+a)^2/b/d^2+1/6*B*(-a*d+b*c)*g^3*(b*x+a)^3/b/d-1/2*B*(
-a*d+b*c)^4*g^3*ln(d*x+c)/b/d^4+1/4*g^3*(b*x+a)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b

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Rubi [A]  time = 0.10, antiderivative size = 151, 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} \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}+\frac {B g^3 x (b c-a d)^3}{2 d^3}-\frac {B g^3 (a+b x)^2 (b c-a d)^2}{4 b d^2}-\frac {B g^3 (b c-a d)^4 \log (c+d x)}{2 b d^4}+\frac {B g^3 (a+b x)^3 (b c-a d)}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]),x]

[Out]

(B*(b*c - a*d)^3*g^3*x)/(2*d^3) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(4*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3
)/(6*b*d) - (B*(b*c - a*d)^4*g^3*Log[c + d*x])/(2*b*d^4) + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*
x)^2]))/(4*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)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx &=\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 b}-\frac {B \int \frac {2 (-b c+a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b g}\\ &=\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 b}+\frac {\left (B (b c-a d) g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{2 b}\\ &=\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 b}+\frac {\left (B (b c-a d) g^3\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{2 b}\\ &=\frac {B (b c-a d)^3 g^3 x}{2 d^3}-\frac {B (b c-a d)^2 g^3 (a+b x)^2}{4 b d^2}+\frac {B (b c-a d) g^3 (a+b x)^3}{6 b d}-\frac {B (b c-a d)^4 g^3 \log (c+d x)}{2 b d^4}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 122, normalized size = 0.81 \[ \frac {g^3 \left ((a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )+\frac {B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{3 d^4}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]),x]

[Out]

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

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fricas [B]  time = 0.53, size = 343, normalized size = 2.27 \[ \frac {3 \, A b^{4} d^{4} g^{3} x^{4} - 6 \, B a^{4} d^{4} g^{3} \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c d^{3} + {\left (6 \, A - B\right )} a b^{3} d^{4}\right )} g^{3} x^{3} - 3 \, {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} - 3 \, {\left (2 \, A - B\right )} a^{2} b^{2} d^{4}\right )} g^{3} x^{2} + 6 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} + {\left (2 \, A - 3 \, B\right )} a^{3} b d^{4}\right )} g^{3} x - 6 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} \log \left (d x + c\right ) + 3 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{12 \, b d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(d*x+c)^2/(b*x+a)^2)),x, algorithm="fricas")

[Out]

1/12*(3*A*b^4*d^4*g^3*x^4 - 6*B*a^4*d^4*g^3*log(b*x + a) + 2*(B*b^4*c*d^3 + (6*A - B)*a*b^3*d^4)*g^3*x^3 - 3*(
B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 - 3*(2*A - B)*a^2*b^2*d^4)*g^3*x^2 + 6*(B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 6*B*
a^2*b^2*c*d^3 + (2*A - 3*B)*a^3*b*d^4)*g^3*x - 6*(B*b^4*c^4 - 4*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2 - 4*B*a^3*
b*c*d^3)*g^3*log(d*x + c) + 3*(B*b^4*d^4*g^3*x^4 + 4*B*a*b^3*d^4*g^3*x^3 + 6*B*a^2*b^2*d^4*g^3*x^2 + 4*B*a^3*b
*d^4*g^3*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^4)

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giac [B]  time = 18.85, size = 364, normalized size = 2.41 \[ -\frac {B a^{4} g^{3} \log \left (b x + a\right )}{2 \, b} + \frac {1}{4} \, {\left (A b^{3} g^{3} + B b^{3} g^{3}\right )} x^{4} + \frac {{\left (B b^{3} c g^{3} + 6 \, A a b^{2} d g^{3} + 5 \, B a b^{2} d g^{3}\right )} x^{3}}{6 \, d} + \frac {1}{4} \, {\left (B b^{3} g^{3} x^{4} + 4 \, B a b^{2} g^{3} x^{3} + 6 \, B a^{2} b g^{3} x^{2} + 4 \, B a^{3} g^{3} x\right )} \log \left (\frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {{\left (B b^{3} c^{2} g^{3} - 4 \, B a b^{2} c d g^{3} - 6 \, A a^{2} b d^{2} g^{3} - 3 \, B a^{2} b d^{2} g^{3}\right )} x^{2}}{4 \, d^{2}} + \frac {{\left (B b^{3} c^{3} g^{3} - 4 \, B a b^{2} c^{2} d g^{3} + 6 \, B a^{2} b c d^{2} g^{3} + 2 \, A a^{3} d^{3} g^{3} - B a^{3} d^{3} g^{3}\right )} x}{2 \, d^{3}} - \frac {{\left (B b^{3} c^{4} g^{3} - 4 \, B a b^{2} c^{3} d g^{3} + 6 \, B a^{2} b c^{2} d^{2} g^{3} - 4 \, B a^{3} c d^{3} g^{3}\right )} \log \left (-d x - c\right )}{2 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(d*x+c)^2/(b*x+a)^2)),x, algorithm="giac")

[Out]

-1/2*B*a^4*g^3*log(b*x + a)/b + 1/4*(A*b^3*g^3 + B*b^3*g^3)*x^4 + 1/6*(B*b^3*c*g^3 + 6*A*a*b^2*d*g^3 + 5*B*a*b
^2*d*g^3)*x^3/d + 1/4*(B*b^3*g^3*x^4 + 4*B*a*b^2*g^3*x^3 + 6*B*a^2*b*g^3*x^2 + 4*B*a^3*g^3*x)*log((d^2*x^2 + 2
*c*d*x + c^2)/(b^2*x^2 + 2*a*b*x + a^2)) - 1/4*(B*b^3*c^2*g^3 - 4*B*a*b^2*c*d*g^3 - 6*A*a^2*b*d^2*g^3 - 3*B*a^
2*b*d^2*g^3)*x^2/d^2 + 1/2*(B*b^3*c^3*g^3 - 4*B*a*b^2*c^2*d*g^3 + 6*B*a^2*b*c*d^2*g^3 + 2*A*a^3*d^3*g^3 - B*a^
3*d^3*g^3)*x/d^3 - 1/2*(B*b^3*c^4*g^3 - 4*B*a*b^2*c^3*d*g^3 + 6*B*a^2*b*c^2*d^2*g^3 - 4*B*a^3*c*d^3*g^3)*log(-
d*x - c)/d^4

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maple [B]  time = 0.07, size = 788, normalized size = 5.22 \[ \frac {B \,b^{3} g^{3} 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 )}{4}+\frac {A \,b^{3} g^{3} x^{4}}{4}+B a \,b^{2} g^{3} 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 )+A a \,b^{2} g^{3} x^{3}+\frac {3 B \,a^{2} b \,g^{3} 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 )}{2}-\frac {B a \,b^{2} g^{3} x^{3}}{6}+\frac {B \,b^{3} c \,g^{3} x^{3}}{6 d}+\frac {3 A \,a^{2} b \,g^{3} x^{2}}{2}+B \,a^{3} g^{3} 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 {3 B \,a^{2} b \,g^{3} x^{2}}{4}+\frac {B a \,b^{2} c \,g^{3} x^{2}}{d}-\frac {B \,b^{3} c^{2} g^{3} x^{2}}{4 d^{2}}+A \,a^{3} g^{3} x +\frac {B \,a^{4} g^{3} \ln \left (\frac {1}{b x +a}\right )}{2 b}+\frac {B \,a^{4} g^{3} \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )}{4 b}-\frac {B \,a^{4} g^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{2 b}-\frac {2 B \,a^{3} c \,g^{3} \ln \left (\frac {1}{b x +a}\right )}{d}+\frac {2 B \,a^{3} c \,g^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d}-\frac {3 B \,a^{3} g^{3} x}{2}+\frac {3 B \,a^{2} b \,c^{2} g^{3} \ln \left (\frac {1}{b x +a}\right )}{d^{2}}-\frac {3 B \,a^{2} b \,c^{2} g^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{2}}+\frac {3 B \,a^{2} b c \,g^{3} x}{d}-\frac {2 B a \,b^{2} c^{3} g^{3} \ln \left (\frac {1}{b x +a}\right )}{d^{3}}+\frac {2 B a \,b^{2} c^{3} g^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{3}}-\frac {2 B a \,b^{2} c^{2} g^{3} x}{d^{2}}+\frac {B \,b^{3} c^{4} g^{3} \ln \left (\frac {1}{b x +a}\right )}{2 d^{4}}-\frac {B \,b^{3} c^{4} g^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{2 d^{4}}+\frac {B \,b^{3} c^{3} g^{3} x}{2 d^{3}}+\frac {A \,a^{4} g^{3}}{4 b}-\frac {11 B \,a^{4} g^{3}}{12 b}+\frac {13 B \,a^{3} c \,g^{3}}{6 d}-\frac {7 B \,a^{2} b \,c^{2} g^{3}}{4 d^{2}}+\frac {B a \,b^{2} c^{3} g^{3}}{2 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2)),x)

[Out]

-3/2*B*x*a^3*g^3+1/4/b*A*a^4*g^3-11/12/b*B*a^4*g^3+1/4*b^3*A*x^4*g^3+A*x*a^3*g^3+B*ln((1/(b*x+a)*a*d-1/(b*x+a)
*b*c-d)^2/b^2*e)*x*a^3*g^3-1/6*b^2*B*x^3*a*g^3-3/4*b*B*x^2*a^2*g^3+3/2*b*A*x^2*a^2*g^3+1/4*b^3*B*ln((1/(b*x+a)
*a*d-1/(b*x+a)*b*c-d)^2/b^2*e)*x^4*g^3-1/2/b*g^3*B*a^4*ln(1/(b*x+a)*a*d-1/(b*x+a)*b*c-d)+1/4/b*B*ln((1/(b*x+a)
*a*d-1/(b*x+a)*b*c-d)^2/b^2*e)*a^4*g^3+1/2/b*g^3*B*a^4*ln(1/(b*x+a))+b^2*A*x^3*a*g^3+3/2*b*B*ln((1/(b*x+a)*a*d
-1/(b*x+a)*b*c-d)^2/b^2*e)*x^2*a^2*g^3+13/6*g^3*B*c/d*a^3-7/4*b*g^3*B*c^2/d^2*a^2+1/2*b^2*g^3*B*c^3/d^3*a+1/6*
b^3*g^3*B*c/d*x^3-1/4*b^3*g^3*B*c^2/d^2*x^2+1/2*b^3*g^3*B*c^3/d^3*x-1/2*b^3*g^3*B*c^4/d^4*ln(1/(b*x+a)*a*d-1/(
b*x+a)*b*c-d)+1/2*b^3*g^3*B*c^4/d^4*ln(1/(b*x+a))+b^2*B*ln((1/(b*x+a)*a*d-1/(b*x+a)*b*c-d)^2/b^2*e)*x^3*a*g^3+
2*g^3*B*a^3/d*ln(1/(b*x+a)*a*d-1/(b*x+a)*b*c-d)*c-2*g^3*B*a^3/d*ln(1/(b*x+a))*c-2*b^2*g^3*B*a/d^3*ln(1/(b*x+a)
)*c^3+3*b*g^3*B*a^2/d^2*ln(1/(b*x+a))*c^2-3*b*g^3*B*a^2/d^2*ln(1/(b*x+a)*a*d-1/(b*x+a)*b*c-d)*c^2+2*b^2*g^3*B*
a/d^3*ln(1/(b*x+a)*a*d-1/(b*x+a)*b*c-d)*c^3+b^2*g^3*B*c/d*x^2*a+3*b*g^3*B*c/d*x*a^2-2*b^2*g^3*B*c^2/d^2*x*a

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maxima [B]  time = 1.31, size = 645, normalized size = 4.27 \[ \frac {1}{4} \, A b^{3} g^{3} x^{4} + A a b^{2} g^{3} x^{3} + \frac {3}{2} \, A a^{2} b g^{3} x^{2} + {\left (x \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{3} g^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a^{2} b g^{3} + {\left (x^{3} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b^{2} g^{3} + \frac {1}{12} \, {\left (3 \, x^{4} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} g^{3} + A a^{3} g^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(d*x+c)^2/(b*x+a)^2)),x, algorithm="maxima")

[Out]

1/4*A*b^3*g^3*x^4 + A*a*b^2*g^3*x^3 + 3/2*A*a^2*b*g^3*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^3*g^3 + 3/2*(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^2*b*g^3
+ (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*b^2*g^3 + 1/12*(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*b^3*g^3 + A*a^3*g^3*x

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mupad [B]  time = 4.85, size = 567, normalized size = 3.75 \[ \ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (B\,a^3\,g^3\,x+\frac {3\,B\,a^2\,b\,g^3\,x^2}{2}+B\,a\,b^2\,g^3\,x^3+\frac {B\,b^3\,g^3\,x^4}{4}\right )-x^2\,\left (\frac {\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,g^3\,\left (3\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{2\,d}\right )+x\,\left (\frac {\left (2\,a\,d+2\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}-\frac {2\,a\,b\,g^3\,\left (3\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{d}\right )}{2\,b\,d}+\frac {a^2\,g^3\,\left (4\,A\,a\,d+6\,A\,b\,c-3\,B\,a\,d+3\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )}{b\,d}\right )+x^3\,\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{6\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{6\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,a^3\,c\,d^3\,g^3+6\,B\,a^2\,b\,c^2\,d^2\,g^3-4\,B\,a\,b^2\,c^3\,d\,g^3+B\,b^3\,c^4\,g^3\right )}{2\,d^4}+\frac {A\,b^3\,g^3\,x^4}{4}-\frac {B\,a^4\,g^3\,\ln \left (a+b\,x\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^3*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2)),x)

[Out]

log((e*(c + d*x)^2)/(a + b*x)^2)*((B*b^3*g^3*x^4)/4 + B*a^3*g^3*x + (3*B*a^2*b*g^3*x^2)/2 + B*a*b^2*g^3*x^3) -
x^2*((((b^2*g^3*(8*A*a*d + 2*A*b*c - B*a*d + B*b*c))/(2*d) - (A*b^2*g^3*(2*a*d + 2*b*c))/(2*d))*(2*a*d + 2*b*
c))/(4*b*d) - (a*b*g^3*(3*A*a*d + 2*A*b*c - B*a*d + B*b*c))/d + (A*a*b^2*c*g^3)/(2*d)) + x*(((2*a*d + 2*b*c)*(
(((b^2*g^3*(8*A*a*d + 2*A*b*c - B*a*d + B*b*c))/(2*d) - (A*b^2*g^3*(2*a*d + 2*b*c))/(2*d))*(2*a*d + 2*b*c))/(2
*b*d) - (2*a*b*g^3*(3*A*a*d + 2*A*b*c - B*a*d + B*b*c))/d + (A*a*b^2*c*g^3)/d))/(2*b*d) + (a^2*g^3*(4*A*a*d +
6*A*b*c - 3*B*a*d + 3*B*b*c))/d - (a*c*((b^2*g^3*(8*A*a*d + 2*A*b*c - B*a*d + B*b*c))/(2*d) - (A*b^2*g^3*(2*a*
d + 2*b*c))/(2*d)))/(b*d)) + x^3*((b^2*g^3*(8*A*a*d + 2*A*b*c - B*a*d + B*b*c))/(6*d) - (A*b^2*g^3*(2*a*d + 2*
b*c))/(6*d)) - (log(c + d*x)*(B*b^3*c^4*g^3 - 4*B*a^3*c*d^3*g^3 + 6*B*a^2*b*c^2*d^2*g^3 - 4*B*a*b^2*c^3*d*g^3)
)/(2*d^4) + (A*b^3*g^3*x^4)/4 - (B*a^4*g^3*log(a + b*x))/(2*b)

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sympy [B]  time = 4.35, size = 707, normalized size = 4.68 \[ \frac {A b^{3} g^{3} x^{4}}{4} - \frac {B a^{4} g^{3} \log {\left (x + \frac {\frac {B a^{5} d^{4} g^{3}}{b} + 4 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{2 b} + \frac {B c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {5 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3} - B a c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{2 d^{4}} + x^{3} \left (A a b^{2} g^{3} - \frac {B a b^{2} g^{3}}{6} + \frac {B b^{3} c g^{3}}{6 d}\right ) + x^{2} \left (\frac {3 A a^{2} b g^{3}}{2} - \frac {3 B a^{2} b g^{3}}{4} + \frac {B a b^{2} c g^{3}}{d} - \frac {B b^{3} c^{2} g^{3}}{4 d^{2}}\right ) + x \left (A a^{3} g^{3} - \frac {3 B a^{3} g^{3}}{2} + \frac {3 B a^{2} b c g^{3}}{d} - \frac {2 B a b^{2} c^{2} g^{3}}{d^{2}} + \frac {B b^{3} c^{3} g^{3}}{2 d^{3}}\right ) + \left (B a^{3} g^{3} x + \frac {3 B a^{2} b g^{3} x^{2}}{2} + B a b^{2} g^{3} x^{3} + \frac {B b^{3} g^{3} x^{4}}{4}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(A+B*ln(e*(d*x+c)**2/(b*x+a)**2)),x)

[Out]

A*b**3*g**3*x**4/4 - B*a**4*g**3*log(x + (B*a**5*d**4*g**3/b + 4*B*a**4*c*d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**
3 + 4*B*a**2*b**2*c**3*d*g**3 - B*a*b**3*c**4*g**3)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2
*c**2*d**2*g**3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(2*b) + B*c*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*
a*b*c*d + b**2*c**2)*log(x + (5*B*a**4*c*d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**3 + 4*B*a**2*b**2*c**3*d*g**3 - B
*a*b**3*c**4*g**3 - B*a*c*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2) + B*b*c**2*g**3*(2*a*d - b*
c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2)/d)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2*c**2*d*
*2*g**3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(2*d**4) + x**3*(A*a*b**2*g**3 - B*a*b**2*g**3/6 + B*b**
3*c*g**3/(6*d)) + x**2*(3*A*a**2*b*g**3/2 - 3*B*a**2*b*g**3/4 + B*a*b**2*c*g**3/d - B*b**3*c**2*g**3/(4*d**2))
+ x*(A*a**3*g**3 - 3*B*a**3*g**3/2 + 3*B*a**2*b*c*g**3/d - 2*B*a*b**2*c**2*g**3/d**2 + B*b**3*c**3*g**3/(2*d*
*3)) + (B*a**3*g**3*x + 3*B*a**2*b*g**3*x**2/2 + B*a*b**2*g**3*x**3 + B*b**3*g**3*x**4/4)*log(e*(c + d*x)**2/(
a + b*x)**2)

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