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

Optimal. Leaf size=118 \[ \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {B g^2 x (b c-a d)^2}{3 d^2}-\frac {B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 118, 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} \[ \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac {B g^2 x (b c-a d)^2}{3 d^2}-\frac {B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac {B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 99, normalized size = 0.84 \[ \frac {g^2 \left (\frac {B (a d-b c) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 222, normalized size = 1.88 \[ \frac {2 \, A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) - {\left (B b^{3} c d^{2} - {\left (6 \, A + B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} + 2 \, {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + {\left (3 \, A + 2 \, B\right )} a^{2} b d^{3}\right )} g^{2} x - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{6 \, b d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.29, size = 2450, normalized size = 20.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*b^7*c^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*B*a*b^6*c^3*d*g^2*e^4*log(-b*e + (b*x*e + a
*e)*d/(d*x + c)) + 12*B*a^2*b^5*c^2*d^2*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 8*B*a^3*b^4*c*d^3*g^2*
e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 2*B*a^4*b^3*d^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6*
(b*x*e + a*e)*B*b^6*c^4*d*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a*b^5*c
^3*d^2*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 36*(b*x*e + a*e)*B*a^2*b^4*c^2*d^3*g^2*e^3*lo
g(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a^3*b^3*c*d^4*g^2*e^3*log(-b*e + (b*x*e + a
*e)*d/(d*x + c))/(d*x + c) - 6*(b*x*e + a*e)*B*a^4*b^2*d^5*g^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c) + 6*(b*x*e + a*e)^2*B*b^5*c^4*d^2*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e +
a*e)^2*B*a*b^4*c^3*d^3*g^2*e^2*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^2*b
^3*c^2*d^4*g^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e + a*e)^2*B*a^3*b^2*c*d^5*g^2*
e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 6*(b*x*e + a*e)^2*B*a^4*b*d^6*g^2*e^2*log(-b*e + (b*x*
e + a*e)*d/(d*x + c))/(d*x + c)^2 - 2*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c)
)/(d*x + c)^3 + 8*(b*x*e + a*e)^3*B*a*b^3*c^3*d^4*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 12
*(b*x*e + a*e)^3*B*a^2*b^2*c^2*d^5*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 8*(b*x*e + a*e)^3
*B*a^3*b*c*d^6*g^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 2*(b*x*e + a*e)^3*B*a^4*d^7*g^2*e*log
(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 2*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g^2*e*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^3 - 8*(b*x*e + a*e)^3*B*a*b^3*c^3*d^4*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 12*(b*x
*e + a*e)^3*B*a^2*b^2*c^2*d^5*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 8*(b*x*e + a*e)^3*B*a^3*b*c*d^6
*g^2*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 2*(b*x*e + a*e)^3*B*a^4*d^7*g^2*e*log((b*x*e + a*e)/(d*x + c
))/(d*x + c)^3 + 2*A*b^7*c^4*g^2*e^4 + 3*B*b^7*c^4*g^2*e^4 - 8*A*a*b^6*c^3*d*g^2*e^4 - 12*B*a*b^6*c^3*d*g^2*e^
4 + 12*A*a^2*b^5*c^2*d^2*g^2*e^4 + 18*B*a^2*b^5*c^2*d^2*g^2*e^4 - 8*A*a^3*b^4*c*d^3*g^2*e^4 - 12*B*a^3*b^4*c*d
^3*g^2*e^4 + 2*A*a^4*b^3*d^4*g^2*e^4 + 3*B*a^4*b^3*d^4*g^2*e^4 - 6*(b*x*e + a*e)*A*b^6*c^4*d*g^2*e^3/(d*x + c)
- 7*(b*x*e + a*e)*B*b^6*c^4*d*g^2*e^3/(d*x + c) + 24*(b*x*e + a*e)*A*a*b^5*c^3*d^2*g^2*e^3/(d*x + c) + 28*(b*
x*e + a*e)*B*a*b^5*c^3*d^2*g^2*e^3/(d*x + c) - 36*(b*x*e + a*e)*A*a^2*b^4*c^2*d^3*g^2*e^3/(d*x + c) - 42*(b*x*
e + a*e)*B*a^2*b^4*c^2*d^3*g^2*e^3/(d*x + c) + 24*(b*x*e + a*e)*A*a^3*b^3*c*d^4*g^2*e^3/(d*x + c) + 28*(b*x*e
+ a*e)*B*a^3*b^3*c*d^4*g^2*e^3/(d*x + c) - 6*(b*x*e + a*e)*A*a^4*b^2*d^5*g^2*e^3/(d*x + c) - 7*(b*x*e + a*e)*B
*a^4*b^2*d^5*g^2*e^3/(d*x + c) + 6*(b*x*e + a*e)^2*A*b^5*c^4*d^2*g^2*e^2/(d*x + c)^2 + 4*(b*x*e + a*e)^2*B*b^5
*c^4*d^2*g^2*e^2/(d*x + c)^2 - 24*(b*x*e + a*e)^2*A*a*b^4*c^3*d^3*g^2*e^2/(d*x + c)^2 - 16*(b*x*e + a*e)^2*B*a
*b^4*c^3*d^3*g^2*e^2/(d*x + c)^2 + 36*(b*x*e + a*e)^2*A*a^2*b^3*c^2*d^4*g^2*e^2/(d*x + c)^2 + 24*(b*x*e + a*e)
^2*B*a^2*b^3*c^2*d^4*g^2*e^2/(d*x + c)^2 - 24*(b*x*e + a*e)^2*A*a^3*b^2*c*d^5*g^2*e^2/(d*x + c)^2 - 16*(b*x*e
+ a*e)^2*B*a^3*b^2*c*d^5*g^2*e^2/(d*x + c)^2 + 6*(b*x*e + a*e)^2*A*a^4*b*d^6*g^2*e^2/(d*x + c)^2 + 4*(b*x*e +
a*e)^2*B*a^4*b*d^6*g^2*e^2/(d*x + c)^2)*(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^4*d^3*e^3 - 3*(b*x*e + a*e)*b^3*d^4*e^2/(d*x + c) + 3*(b*x*e + a*e)^2*b^2*d^5*e/(d*x + c)^2 - (b*x*e + a*
e)^3*b*d^6/(d*x + c)^3)

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maple [B]  time = 0.15, size = 3283, normalized size = 27.82 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.24, size = 280, normalized size = 2.37 \[ \frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*A*b^2*g^2*x^3 + A*a*b*g^2*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^2*g^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*b*g^2 + 1/6*(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*b^2*g^2 + A*a^2*g^2*x

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mupad [B]  time = 4.48, size = 290, normalized size = 2.46 \[ x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,a^2\,c\,d^2\,g^2-3\,B\,a\,b\,c^2\,d\,g^2+B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(6*d) - (A*b*g^2*(3*a*d + 3*b*c))/(6*d)) - x*(((3*a*d + 3*b*c
)*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(3*d) - (A*b*g^2*(3*a*d + 3*b*c))/(3*d)))/(3*b*d) - (a*g^2*(3*A
*a*d + 3*A*b*c + B*a*d - B*b*c))/d + (A*a*b*c*g^2)/d) + log((e*(a + b*x))/(c + d*x))*((B*b^2*g^2*x^3)/3 + B*a^
2*g^2*x + B*a*b*g^2*x^2) - (log(c + d*x)*(B*b^2*c^3*g^2 + 3*B*a^2*c*d^2*g^2 - 3*B*a*b*c^2*d*g^2))/(3*d^3) + (A
*b^2*g^2*x^3)/3 + (B*a^3*g^2*log(a + b*x))/(3*b)

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sympy [B]  time = 2.92, size = 491, normalized size = 4.16 \[ \frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} \log {\left (x + \frac {\frac {B a^{4} d^{3} g^{2}}{b} + 3 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 b} - \frac {B c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {4 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2} - B a c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} + \frac {B a b g^{2}}{6} - \frac {B b^{2} c g^{2}}{6 d}\right ) + x \left (A a^{2} g^{2} + \frac {2 B a^{2} g^{2}}{3} - \frac {B a b c g^{2}}{d} + \frac {B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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