3.89 \(\int (a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\)
Optimal. Leaf size=149 \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}+\frac {B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B g^3 x (b c-a d)^3}{4 d^3}+\frac {B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac {B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]
[Out]
-1/4*B*(-a*d+b*c)^3*g^3*x/d^3+1/8*B*(-a*d+b*c)^2*g^3*(b*x+a)^2/b/d^2-1/12*B*(-a*d+b*c)*g^3*(b*x+a)^3/b/d+1/4*g
^3*(b*x+a)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/b+1/4*B*(-a*d+b*c)^4*g^3*ln(d*x+c)/b/d^4
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Rubi [A] time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used =
{2525, 12, 43} \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}-\frac {B g^3 x (b c-a d)^3}{4 d^3}+\frac {B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac {B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
-(B*(b*c - a*d)^3*g^3*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*(a + b*x)^
3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + (B*(b*c - a*d)^4*g^3*Log[c + d*x]
)/(4*b*d^4)
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 (a+b x)}{c+d x}\right )\right ) \, dx &=\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b}-\frac {B \int \frac {(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 (a+b x)}{c+d x}\right )\right )}{4 b}-\frac {\left (B (b c-a d) g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{4 b}\\ &=\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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}{4 b}\\ &=-\frac {B (b c-a d)^3 g^3 x}{4 d^3}+\frac {B (b c-a d)^2 g^3 (a+b x)^2}{8 b d^2}-\frac {B (b c-a d) g^3 (a+b x)^3}{12 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b}+\frac {B (b c-a d)^4 g^3 \log (c+d x)}{4 b d^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 120, normalized size = 0.81 \[ \frac {g^3 \left ((a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{6 d^4}\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
(g^3*((a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - (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]))/(6*d^4)))/(4*b)
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fricas [B] time = 1.01, size = 318, normalized size = 2.13 \[ \frac {6 \, A b^{4} d^{4} g^{3} x^{4} + 6 \, B a^{4} d^{4} g^{3} \log \left (b x + a\right ) - 2 \, {\left (B b^{4} c d^{3} - {\left (12 \, A + B\right )} a b^{3} d^{4}\right )} g^{3} x^{3} + 3 \, {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \, {\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} g^{3} x^{2} - 6 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - {\left (4 \, A + 3 \, B\right )} a^{3} b d^{4}\right )} g^{3} x + 6 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} \log \left (d x + c\right ) + 6 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")
[Out]
1/24*(6*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 - (12*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*(4*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 - (4*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) + 6*(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((b*e*x + a*e)/(d*x + c)))/(b*d^4)
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giac [B] time = 1.81, size = 3795, normalized size = 25.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
[Out]
-1/24*(6*B*b^9*c^5*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 30*B*a*b^8*c^4*d*g^3*e^5*log(-b*e + (b*x*e
+ a*e)*d/(d*x + c)) + 60*B*a^2*b^7*c^3*d^2*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 60*B*a^3*b^6*c^2*d^
3*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 30*B*a^4*b^5*c*d^4*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x +
c)) - 6*B*a^5*b^4*d^5*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 24*(b*x*e + a*e)*B*b^8*c^5*d*g^3*e^4*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 120*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^3*e^4*log(-b*e + (b*x*e +
a*e)*d/(d*x + c))/(d*x + c) - 240*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c) + 240*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 12
0*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a
^5*b^3*d^6*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 36*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g^3*e^3*
log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 180*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^3*e^3*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 360*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^2 - 360*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/
(d*x + c)^2 + 180*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 -
36*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e + a*e)^
3*B*b^6*c^5*d^3*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120*(b*x*e + a*e)^3*B*a*b^5*c^4*d^
4*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 240*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^3*e^2*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 240*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*g^3*e^2*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 120*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(
d*x + c))/(d*x + c)^3 + 24*(b*x*e + a*e)^3*B*a^5*b*d^8*g^3*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^3 + 6*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 30*(b*x*e + a*e
)^4*B*a*b^4*c^4*d^5*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3
*d^6*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^3*e*log(
-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^3*e*log(-b*e + (b*x*e + a*e
)*d/(d*x + c))/(d*x + c)^4 - 6*(b*x*e + a*e)^4*B*a^5*d^9*g^3*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
^4 - 6*(b*x*e + a*e)^4*B*b^5*c^5*d^4*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B*a*b
^4*c^4*d^5*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^3*e*log((b*
x*e + a*e)/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d
*x + c)^4 - 30*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*(b*x*e + a*e)^
4*B*a^5*d^9*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*A*b^9*c^5*g^3*e^5 + 11*B*b^9*c^5*g^3*e^5 - 30*A
*a*b^8*c^4*d*g^3*e^5 - 55*B*a*b^8*c^4*d*g^3*e^5 + 60*A*a^2*b^7*c^3*d^2*g^3*e^5 + 110*B*a^2*b^7*c^3*d^2*g^3*e^5
- 60*A*a^3*b^6*c^2*d^3*g^3*e^5 - 110*B*a^3*b^6*c^2*d^3*g^3*e^5 + 30*A*a^4*b^5*c*d^4*g^3*e^5 + 55*B*a^4*b^5*c*
d^4*g^3*e^5 - 6*A*a^5*b^4*d^5*g^3*e^5 - 11*B*a^5*b^4*d^5*g^3*e^5 - 24*(b*x*e + a*e)*A*b^8*c^5*d*g^3*e^4/(d*x +
c) - 38*(b*x*e + a*e)*B*b^8*c^5*d*g^3*e^4/(d*x + c) + 120*(b*x*e + a*e)*A*a*b^7*c^4*d^2*g^3*e^4/(d*x + c) + 1
90*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^3*e^4/(d*x + c) - 240*(b*x*e + a*e)*A*a^2*b^6*c^3*d^3*g^3*e^4/(d*x + c) - 3
80*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^3*e^4/(d*x + c) + 240*(b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g^3*e^4/(d*x + c) +
380*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^3*e^4/(d*x + c) - 120*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g^3*e^4/(d*x + c) -
190*(b*x*e + a*e)*B*a^4*b^4*c*d^5*g^3*e^4/(d*x + c) + 24*(b*x*e + a*e)*A*a^5*b^3*d^6*g^3*e^4/(d*x + c) + 38*(
b*x*e + a*e)*B*a^5*b^3*d^6*g^3*e^4/(d*x + c) + 36*(b*x*e + a*e)^2*A*b^7*c^5*d^2*g^3*e^3/(d*x + c)^2 + 45*(b*x*
e + a*e)^2*B*b^7*c^5*d^2*g^3*e^3/(d*x + c)^2 - 180*(b*x*e + a*e)^2*A*a*b^6*c^4*d^3*g^3*e^3/(d*x + c)^2 - 225*(
b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^3*e^3/(d*x + c)^2 + 360*(b*x*e + a*e)^2*A*a^2*b^5*c^3*d^4*g^3*e^3/(d*x + c)^2
+ 450*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3/(d*x + c)^2 - 360*(b*x*e + a*e)^2*A*a^3*b^4*c^2*d^5*g^3*e^3/(
d*x + c)^2 - 450*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3/(d*x + c)^2 + 180*(b*x*e + a*e)^2*A*a^4*b^3*c*d^6*g
^3*e^3/(d*x + c)^2 + 225*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^3*e^3/(d*x + c)^2 - 36*(b*x*e + a*e)^2*A*a^5*b^2*d^
7*g^3*e^3/(d*x + c)^2 - 45*(b*x*e + a*e)^2*B*a^5*b^2*d^7*g^3*e^3/(d*x + c)^2 - 24*(b*x*e + a*e)^3*A*b^6*c^5*d^
3*g^3*e^2/(d*x + c)^3 - 18*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^3*e^2/(d*x + c)^3 + 120*(b*x*e + a*e)^3*A*a*b^5*c^4
*d^4*g^3*e^2/(d*x + c)^3 + 90*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^3*e^2/(d*x + c)^3 - 240*(b*x*e + a*e)^3*A*a^2*
b^4*c^3*d^5*g^3*e^2/(d*x + c)^3 - 180*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^3*e^2/(d*x + c)^3 + 240*(b*x*e + a*e
)^3*A*a^3*b^3*c^2*d^6*g^3*e^2/(d*x + c)^3 + 180*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*g^3*e^2/(d*x + c)^3 - 120*(b
*x*e + a*e)^3*A*a^4*b^2*c*d^7*g^3*e^2/(d*x + c)^3 - 90*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^3*e^2/(d*x + c)^3 + 2
4*(b*x*e + a*e)^3*A*a^5*b*d^8*g^3*e^2/(d*x + c)^3 + 18*(b*x*e + a*e)^3*B*a^5*b*d^8*g^3*e^2/(d*x + c)^3)*(b*c/(
(b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/(b^5*d^4*e^4 - 4*(b*x*e + a*e)*b^4*d^5*e^3/(
d*x + c) + 6*(b*x*e + a*e)^2*b^3*d^6*e^2/(d*x + c)^2 - 4*(b*x*e + a*e)^3*b^2*d^7*e/(d*x + c)^3 + (b*x*e + a*e)
^4*b*d^8/(d*x + c)^4)
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maple [B] time = 0.16, size = 5556, normalized size = 37.29 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*g*x+a*g)^3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
result too large to display
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maxima [B] time = 1.33, size = 439, normalized size = 2.95 \[ \frac {1}{4} \, A b^{3} g^{3} x^{4} + A a b^{2} g^{3} x^{3} + \frac {3}{2} \, A a^{2} b g^{3} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{3} g^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{2} b g^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b^{2} g^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} g^{3} + A a^{3} g^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c))),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(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*
x + a)/b - c*log(d*x + c)/d)*B*a^3*g^3 + 3/2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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^2*b*g^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) +
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 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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 = 4.64, size = 566, normalized size = 3.80 \[ x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{d}\right )}{4\,b\,d}+\frac {a^2\,g^3\,\left (8\,A\,a\,d+12\,A\,b\,c+3\,B\,a\,d-3\,B\,b\,c\right )}{2\,d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{2\,d}+\frac {A\,a\,b^2\,c\,g^3}{2\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^3\,g^3\,x+\frac {3\,B\,a^2\,b\,g^3\,x^2}{2}+B\,a\,b^2\,g^3\,x^3+\frac {B\,b^3\,g^3\,x^4}{4}\right )+x^3\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,a^3\,c\,d^3\,g^3+6\,B\,a^2\,b\,c^2\,d^2\,g^3-4\,B\,a\,b^2\,c^3\,d\,g^3+B\,b^3\,c^4\,g^3\right )}{4\,d^4}+\frac {A\,b^3\,g^3\,x^4}{4}+\frac {B\,a^4\,g^3\,\ln \left (a+b\,x\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))),x)
[Out]
x*(((4*a*d + 4*b*c)*((((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d - B*b*c))/(4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d
))*(4*a*d + 4*b*c))/(4*b*d) - (a*b*g^3*(6*A*a*d + 4*A*b*c + B*a*d - B*b*c))/d + (A*a*b^2*c*g^3)/d))/(4*b*d) +
(a^2*g^3*(8*A*a*d + 12*A*b*c + 3*B*a*d - 3*B*b*c))/(2*d) - (a*c*((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d - B*b*c)
)/(4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d)))/(b*d)) - x^2*((((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d - B*b*c))/(
4*d) - (A*b^2*g^3*(4*a*d + 4*b*c))/(4*d))*(4*a*d + 4*b*c))/(8*b*d) - (a*b*g^3*(6*A*a*d + 4*A*b*c + B*a*d - B*b
*c))/(2*d) + (A*a*b^2*c*g^3)/(2*d)) + log((e*(a + b*x))/(c + d*x))*((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^3*((b^2*g^3*(16*A*a*d + 4*A*b*c + B*a*d - B*b*c))/(12*d) - (A*b^2*g^3*(4*
a*d + 4*b*c))/(12*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))/(4*d^4) + (A*b^3*g^3*x^4)/4 + (B*a^4*g^3*log(a + b*x))/(4*b)
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sympy [B] time = 4.30, size = 706, normalized size = 4.74 \[ \frac {A b^{3} g^{3} x^{4}}{4} + \frac {B a^{4} g^{3} \log {\left (x + \frac {\frac {B a^{5} d^{4} g^{3}}{b} + 4 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 b} - \frac {B c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {5 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3} - B a c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 d^{4}} + x^{3} \left (A a b^{2} g^{3} + \frac {B a b^{2} g^{3}}{12} - \frac {B b^{3} c g^{3}}{12 d}\right ) + x^{2} \left (\frac {3 A a^{2} b g^{3}}{2} + \frac {3 B a^{2} b g^{3}}{8} - \frac {B a b^{2} c g^{3}}{2 d} + \frac {B b^{3} c^{2} g^{3}}{8 d^{2}}\right ) + x \left (A a^{3} g^{3} + \frac {3 B a^{3} g^{3}}{4} - \frac {3 B a^{2} b c g^{3}}{2 d} + \frac {B a b^{2} c^{2} g^{3}}{d^{2}} - \frac {B b^{3} c^{3} g^{3}}{4 d^{3}}\right ) + \left (B a^{3} g^{3} x + \frac {3 B a^{2} b g^{3} x^{2}}{2} + B a b^{2} g^{3} x^{3} + \frac {B b^{3} g^{3} x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),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))/(4*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))/(4*d**4) + x**3*(A*a*b**2*g**3 + B*a*b**2*g**3/12 - B*b*
*3*c*g**3/(12*d)) + x**2*(3*A*a**2*b*g**3/2 + 3*B*a**2*b*g**3/8 - B*a*b**2*c*g**3/(2*d) + B*b**3*c**2*g**3/(8*
d**2)) + x*(A*a**3*g**3 + 3*B*a**3*g**3/4 - 3*B*a**2*b*c*g**3/(2*d) + B*a*b**2*c**2*g**3/d**2 - B*b**3*c**3*g*
*3/(4*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*(a + b
*x)/(c + d*x))
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