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

Optimal. Leaf size=149 \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{4 b}-\frac {B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac {B g^3 x (b c-a d)^3}{4 d^3}-\frac {B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac {B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]

[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*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)/(b*x+a)))/b

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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} \[ \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{4 b}+\frac {B g^3 x (b c-a d)^3}{4 d^3}-\frac {B g^3 (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac {B g^3 (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac {B g^3 (a+b x)^3 (b c-a d)}{12 b d} \]

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.08, size = 120, normalized size = 0.81 \[ \frac {g^3 \left ((a+b x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+\frac {B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{6 d^4}\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 1.12, size = 320, normalized size = 2.15 \[ \frac {6 \, A b^{4} d^{4} g^{3} x^{4} - 6 \, B a^{4} d^{4} g^{3} \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c d^{3} + {\left (12 \, A - B\right )} a b^{3} d^{4}\right )} g^{3} x^{3} - 3 \, {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} - 3 \, {\left (4 \, A - B\right )} a^{2} b^{2} d^{4}\right )} g^{3} x^{2} + 6 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} + {\left (4 \, A - 3 \, B\right )} a^{3} b d^{4}\right )} g^{3} x - 6 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} \log \left (d x + c\right ) + 6 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{24 \, b d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(d*x+c)/(b*x+a))),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((d*e*x + c*e)/(b*x + a)))/(b*d^4)

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giac [B]  time = 1.19, size = 4137, normalized size = 27.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*B*b^5*c^5*d^4*g^3*e^5*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) - 30*B*a*b^4*c^4*d^5*g^3*e^5*log(-d*e + (d
*x*e + c*e)*b/(b*x + a)) + 60*B*a^2*b^3*c^3*d^6*g^3*e^5*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) - 60*B*a^3*b^2*c
^2*d^7*g^3*e^5*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) + 30*B*a^4*b*c*d^8*g^3*e^5*log(-d*e + (d*x*e + c*e)*b/(b*
x + a)) - 6*B*a^5*d^9*g^3*e^5*log(-d*e + (d*x*e + c*e)*b/(b*x + a)) - 24*(d*x*e + c*e)*B*b^6*c^5*d^3*g^3*e^4*l
og(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) + 120*(d*x*e + c*e)*B*a*b^5*c^4*d^4*g^3*e^4*log(-d*e + (d*x*e +
c*e)*b/(b*x + a))/(b*x + a) - 240*(d*x*e + c*e)*B*a^2*b^4*c^3*d^5*g^3*e^4*log(-d*e + (d*x*e + c*e)*b/(b*x + a
))/(b*x + a) + 240*(d*x*e + c*e)*B*a^3*b^3*c^2*d^6*g^3*e^4*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) - 1
20*(d*x*e + c*e)*B*a^4*b^2*c*d^7*g^3*e^4*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) + 24*(d*x*e + c*e)*B*
a^5*b*d^8*g^3*e^4*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a) + 36*(d*x*e + c*e)^2*B*b^7*c^5*d^2*g^3*e^3*l
og(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 - 180*(d*x*e + c*e)^2*B*a*b^6*c^4*d^3*g^3*e^3*log(-d*e + (d*x
*e + c*e)*b/(b*x + a))/(b*x + a)^2 + 360*(d*x*e + c*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3*log(-d*e + (d*x*e + c*e)*b/
(b*x + a))/(b*x + a)^2 - 360*(d*x*e + c*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(
b*x + a)^2 + 180*(d*x*e + c*e)^2*B*a^4*b^3*c*d^6*g^3*e^3*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 - 3
6*(d*x*e + c*e)^2*B*a^5*b^2*d^7*g^3*e^3*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^2 - 24*(d*x*e + c*e)^3
*B*b^8*c^5*d*g^3*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^3 + 120*(d*x*e + c*e)^3*B*a*b^7*c^4*d^2*g
^3*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^3 - 240*(d*x*e + c*e)^3*B*a^2*b^6*c^3*d^3*g^3*e^2*log(-
d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^3 + 240*(d*x*e + c*e)^3*B*a^3*b^5*c^2*d^4*g^3*e^2*log(-d*e + (d*x*e
+ c*e)*b/(b*x + a))/(b*x + a)^3 - 120*(d*x*e + c*e)^3*B*a^4*b^4*c*d^5*g^3*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x
+ a))/(b*x + a)^3 + 24*(d*x*e + c*e)^3*B*a^5*b^3*d^6*g^3*e^2*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^
3 + 6*(d*x*e + c*e)^4*B*b^9*c^5*g^3*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^4 - 30*(d*x*e + c*e)^4*B
*a*b^8*c^4*d*g^3*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^4 + 60*(d*x*e + c*e)^4*B*a^2*b^7*c^3*d^2*g^
3*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^4 - 60*(d*x*e + c*e)^4*B*a^3*b^6*c^2*d^3*g^3*e*log(-d*e +
(d*x*e + c*e)*b/(b*x + a))/(b*x + a)^4 + 30*(d*x*e + c*e)^4*B*a^4*b^5*c*d^4*g^3*e*log(-d*e + (d*x*e + c*e)*b/(
b*x + a))/(b*x + a)^4 - 6*(d*x*e + c*e)^4*B*a^5*b^4*d^5*g^3*e*log(-d*e + (d*x*e + c*e)*b/(b*x + a))/(b*x + a)^
4 + 24*(d*x*e + c*e)*B*b^6*c^5*d^3*g^3*e^4*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 120*(d*x*e + c*e)*B*a*b^5*
c^4*d^4*g^3*e^4*log((d*x*e + c*e)/(b*x + a))/(b*x + a) + 240*(d*x*e + c*e)*B*a^2*b^4*c^3*d^5*g^3*e^4*log((d*x*
e + c*e)/(b*x + a))/(b*x + a) - 240*(d*x*e + c*e)*B*a^3*b^3*c^2*d^6*g^3*e^4*log((d*x*e + c*e)/(b*x + a))/(b*x
+ a) + 120*(d*x*e + c*e)*B*a^4*b^2*c*d^7*g^3*e^4*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 24*(d*x*e + c*e)*B*a
^5*b*d^8*g^3*e^4*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 36*(d*x*e + c*e)^2*B*b^7*c^5*d^2*g^3*e^3*log((d*x*e
+ c*e)/(b*x + a))/(b*x + a)^2 + 180*(d*x*e + c*e)^2*B*a*b^6*c^4*d^3*g^3*e^3*log((d*x*e + c*e)/(b*x + a))/(b*x
+ a)^2 - 360*(d*x*e + c*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 + 360*(d*x*e +
c*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 - 180*(d*x*e + c*e)^2*B*a^4*b^3*c*d
^6*g^3*e^3*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 + 36*(d*x*e + c*e)^2*B*a^5*b^2*d^7*g^3*e^3*log((d*x*e + c*
e)/(b*x + a))/(b*x + a)^2 + 24*(d*x*e + c*e)^3*B*b^8*c^5*d*g^3*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 -
120*(d*x*e + c*e)^3*B*a*b^7*c^4*d^2*g^3*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 + 240*(d*x*e + c*e)^3*B*a
^2*b^6*c^3*d^3*g^3*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 - 240*(d*x*e + c*e)^3*B*a^3*b^5*c^2*d^4*g^3*e^
2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 + 120*(d*x*e + c*e)^3*B*a^4*b^4*c*d^5*g^3*e^2*log((d*x*e + c*e)/(b*
x + a))/(b*x + a)^3 - 24*(d*x*e + c*e)^3*B*a^5*b^3*d^6*g^3*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 - 6*(d
*x*e + c*e)^4*B*b^9*c^5*g^3*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^4 + 30*(d*x*e + c*e)^4*B*a*b^8*c^4*d*g^3*
e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^4 - 60*(d*x*e + c*e)^4*B*a^2*b^7*c^3*d^2*g^3*e*log((d*x*e + c*e)/(b*x
+ a))/(b*x + a)^4 + 60*(d*x*e + c*e)^4*B*a^3*b^6*c^2*d^3*g^3*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^4 - 30*
(d*x*e + c*e)^4*B*a^4*b^5*c*d^4*g^3*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^4 + 6*(d*x*e + c*e)^4*B*a^5*b^4*d
^5*g^3*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^4 + 6*A*b^5*c^5*d^4*g^3*e^5 - 11*B*b^5*c^5*d^4*g^3*e^5 - 30*A*
a*b^4*c^4*d^5*g^3*e^5 + 55*B*a*b^4*c^4*d^5*g^3*e^5 + 60*A*a^2*b^3*c^3*d^6*g^3*e^5 - 110*B*a^2*b^3*c^3*d^6*g^3*
e^5 - 60*A*a^3*b^2*c^2*d^7*g^3*e^5 + 110*B*a^3*b^2*c^2*d^7*g^3*e^5 + 30*A*a^4*b*c*d^8*g^3*e^5 - 55*B*a^4*b*c*d
^8*g^3*e^5 - 6*A*a^5*d^9*g^3*e^5 + 11*B*a^5*d^9*g^3*e^5 + 26*(d*x*e + c*e)*B*b^6*c^5*d^3*g^3*e^4/(b*x + a) - 1
30*(d*x*e + c*e)*B*a*b^5*c^4*d^4*g^3*e^4/(b*x + a) + 260*(d*x*e + c*e)*B*a^2*b^4*c^3*d^5*g^3*e^4/(b*x + a) - 2
60*(d*x*e + c*e)*B*a^3*b^3*c^2*d^6*g^3*e^4/(b*x + a) + 130*(d*x*e + c*e)*B*a^4*b^2*c*d^7*g^3*e^4/(b*x + a) - 2
6*(d*x*e + c*e)*B*a^5*b*d^8*g^3*e^4/(b*x + a) - 21*(d*x*e + c*e)^2*B*b^7*c^5*d^2*g^3*e^3/(b*x + a)^2 + 105*(d*
x*e + c*e)^2*B*a*b^6*c^4*d^3*g^3*e^3/(b*x + a)^2 - 210*(d*x*e + c*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3/(b*x + a)^2 +
210*(d*x*e + c*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3/(b*x + a)^2 - 105*(d*x*e + c*e)^2*B*a^4*b^3*c*d^6*g^3*e^3/(b*x
+ a)^2 + 21*(d*x*e + c*e)^2*B*a^5*b^2*d^7*g^3*e^3/(b*x + a)^2 + 6*(d*x*e + c*e)^3*B*b^8*c^5*d*g^3*e^2/(b*x + a
)^3 - 30*(d*x*e + c*e)^3*B*a*b^7*c^4*d^2*g^3*e^2/(b*x + a)^3 + 60*(d*x*e + c*e)^3*B*a^2*b^6*c^3*d^3*g^3*e^2/(b
*x + a)^3 - 60*(d*x*e + c*e)^3*B*a^3*b^5*c^2*d^4*g^3*e^2/(b*x + a)^3 + 30*(d*x*e + c*e)^3*B*a^4*b^4*c*d^5*g^3*
e^2/(b*x + a)^3 - 6*(d*x*e + c*e)^3*B*a^5*b^3*d^6*g^3*e^2/(b*x + a)^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*d^8*e^4 - 4*(d*x*e + c*e)*b^2*d^7*e^3/(b*x + a) + 6*(d*x*e + c*e)^2*b^3*d^
6*e^2/(b*x + a)^2 - 4*(d*x*e + c*e)^3*b^4*d^5*e/(b*x + a)^3 + (d*x*e + c*e)^4*b^5*d^4/(b*x + a)^4)

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maple [B]  time = 0.14, size = 2191, normalized size = 14.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*e^2*B*g^3/d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a*c^3-e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+
a)*a*d+b*e/(b*x+a)*c)^4*a^3*d^3*c+1/4/b*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*
c)^4*a^4*d^4+3/2*b*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*a^2*c^2*d^2-35/2
*b^3*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*a^4/(b*x+a)^4*c^4-b^2*e^4*B*g^
3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^3*a*d-1/4*b^7*e^4*B*g^3*ln(d*e/b-e*(a*d-b
*c)/b/(b*x+a))/d^4/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^8/(b*x+a)^4+1/4/b*B*g^3*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a
))-d*e)*a^4-e*B*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)*a^3*c-B*g^3/d*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a^3*c
+1/4*b^3*B*g^3/d^4*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*c^4+1/4*b^3*e^4*A*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c
)^4*c^4-b^2*e^4*A*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^3*a*d+1/2*b*e^3*B*g^3*d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c
)^3*a^2*c^2+3/2*b*e^4*A*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*a^2*c^2*d^2+3/2*b*e*B*g^3/d/(-e/(b*x+a)*a*d+b*e/(
b*x+a)*c)*a^2*c^2-b^2*e*B*g^3/d^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)*c^3*a+1/4*b^3*e*B*g^3/d^3/(-e/(b*x+a)*a*d+b*e
/(b*x+a)*c)*c^4+1/4*b^3*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^4+3/2*b*B
*g^3/d^2*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a^2*c^2-1/8*b^3*e^2*B*g^3/d^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^
2*c^4-3/4*b*e^2*B*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a^2*c^2+2*b^6*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))
/d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^7/(b*x+a)^4*a-7*b^5*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/d^2/(-e/
(b*x+a)*a*d+b*e/(b*x+a)*c)^4*c^6/(b*x+a)^4*a^2+14*b^4*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/d/(-e/(b*x+a)*
a*d+b*e/(b*x+a)*c)^4*a^3/(b*x+a)^4*c^5-7*b*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^2/(-e/(b*x+a)*a*d+b*e/(
b*x+a)*c)^4*a^6/(b*x+a)^4*c^2+14*b^2*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c
)^4*a^5/(b*x+a)^4*c^3-1/3*e^3*B*g^3*d^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^3*c-e^4*A*g^3/(-e/(b*x+a)*a*d+b*e/(
b*x+a)*c)^4*a^3*d^3*c-1/4/b*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^4/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*a^8
/(b*x+a)^4+2*e^4*B*g^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^4*a^7/(b*x+a)^4*c-1/
3*b^2*e^3*B*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a*c^3-1/8/b*e^2*B*g^3*d^2/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a^
4+1/12/b*e^3*B*g^3*d^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*a^4+1/2*e^2*B*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^2*a^3
*c*d-b^2*B*g^3/d^3*ln(b*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-d*e)*a*c^3+1/4/b*e*B*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)*
a^4*d+1/12*b^3*e^3*B*g^3/d/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)^3*c^4+1/4/b*e^4*A*g^3/(-e/(b*x+a)*a*d+b*e/(b*x+a)*c)
^4*a^4*d^4

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maxima [B]  time = 1.30, size = 436, normalized size = 2.93 \[ \frac {1}{4} \, A b^{3} g^{3} x^{4} + A a b^{2} g^{3} x^{3} + \frac {3}{2} \, A a^{2} b g^{3} x^{2} + {\left (x \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {a \log \left (b x + a\right )}{b} + \frac {c \log \left (d x + c\right )}{d}\right )} B a^{3} g^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{2} b g^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b^{2} g^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} g^{3} + A a^{3} g^{3} x \]

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)/(b*x+a))),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*e*x/(b*x + a) + c*e/(b*x + a)) - a*log(b*
x + a)/b + c*log(d*x + c)/d)*B*a^3*g^3 + 3/2*(x^2*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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(d*e*x/(b*x + a) + c*e/(b*x + a)) -
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(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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 = 4.69, size = 566, normalized size = 3.80 \[ x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{d}\right )}{4\,b\,d}+\frac {a^2\,g^3\,\left (8\,A\,a\,d+12\,A\,b\,c-3\,B\,a\,d+3\,B\,b\,c\right )}{2\,d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{4\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {a\,b\,g^3\,\left (6\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}+\frac {A\,a\,b^2\,c\,g^3}{2\,d}\right )+\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (B\,a^3\,g^3\,x+\frac {3\,B\,a^2\,b\,g^3\,x^2}{2}+B\,a\,b^2\,g^3\,x^3+\frac {B\,b^3\,g^3\,x^4}{4}\right )+x^3\,\left (\frac {b^2\,g^3\,\left (16\,A\,a\,d+4\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{12\,d}-\frac {A\,b^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,a^3\,c\,d^3\,g^3+6\,B\,a^2\,b\,c^2\,d^2\,g^3-4\,B\,a\,b^2\,c^3\,d\,g^3+B\,b^3\,c^4\,g^3\right )}{4\,d^4}+\frac {A\,b^3\,g^3\,x^4}{4}-\frac {B\,a^4\,g^3\,\ln \left (a+b\,x\right )}{4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^3*(A + B*log((e*(c + d*x))/(a + b*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*(c + d*x))/(a + b*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 = 4.35, size = 706, normalized size = 4.74 \[ \frac {A b^{3} g^{3} x^{4}}{4} - \frac {B a^{4} g^{3} \log {\left (x + \frac {\frac {B a^{5} d^{4} g^{3}}{b} + 4 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 b} + \frac {B c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {5 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3} - B a c g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{3} \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{4 d^{4}} + x^{3} \left (A a b^{2} g^{3} - \frac {B a b^{2} g^{3}}{12} + \frac {B b^{3} c g^{3}}{12 d}\right ) + x^{2} \left (\frac {3 A a^{2} b g^{3}}{2} - \frac {3 B a^{2} b g^{3}}{8} + \frac {B a b^{2} c g^{3}}{2 d} - \frac {B b^{3} c^{2} g^{3}}{8 d^{2}}\right ) + x \left (A a^{3} g^{3} - \frac {3 B a^{3} g^{3}}{4} + \frac {3 B a^{2} b c g^{3}}{2 d} - \frac {B a b^{2} c^{2} g^{3}}{d^{2}} + \frac {B b^{3} c^{3} g^{3}}{4 d^{3}}\right ) + \left (B a^{3} g^{3} x + \frac {3 B a^{2} b g^{3} x^{2}}{2} + B a b^{2} g^{3} x^{3} + \frac {B b^{3} g^{3} x^{4}}{4}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )} \]

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)/(b*x+a))),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*(c + d
*x)/(a + b*x))

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