3.3.32 \(\int (f+g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [232]
Optimal. Leaf size=150 \[ -\frac {B (b c-a d) g (3 b d f-b c g-a d g) x}{3 b^2 d^2}-\frac {B (b c-a d) g^2 x^2}{6 b d}-\frac {B (b f-a g)^3 \log (a+b x)}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 g}+\frac {B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \]
[Out]
-1/3*B*(-a*d+b*c)*g*(-a*d*g-b*c*g+3*b*d*f)*x/b^2/d^2-1/6*B*(-a*d+b*c)*g^2*x^2/b/d-1/3*B*(-a*g+b*f)^3*ln(b*x+a)
/b^3/g+1/3*(g*x+f)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/g+1/3*B*(-c*g+d*f)^3*ln(d*x+c)/d^3/g
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Rubi [A]
time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2548, 84}
\begin {gather*} \frac {(f+g x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g}-\frac {B (b f-a g)^3 \log (a+b x)}{3 b^3 g}-\frac {B g x (b c-a d) (-a d g-b c g+3 b d f)}{3 b^2 d^2}-\frac {B g^2 x^2 (b c-a d)}{6 b d}+\frac {B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
-1/3*(B*(b*c - a*d)*g*(3*b*d*f - b*c*g - a*d*g)*x)/(b^2*d^2) - (B*(b*c - a*d)*g^2*x^2)/(6*b*d) - (B*(b*f - a*g
)^3*Log[a + b*x])/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*g) + (B*(d*f - c*g)^3*Log[
c + d*x])/(3*d^3*g)
Rule 84
Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Rule 2548
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Rubi steps
\begin {align*} \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 g}-\frac {B \int \frac {(b c-a d) (f+g x)^3}{(a+b x) (c+d x)} \, dx}{3 g}\\ &=\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 g}-\frac {(B (b c-a d)) \int \frac {(f+g x)^3}{(a+b x) (c+d x)} \, dx}{3 g}\\ &=\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 g}-\frac {(B (b c-a d)) \int \left (\frac {g^2 (3 b d f-b c g-a d g)}{b^2 d^2}+\frac {g^3 x}{b d}+\frac {(b f-a g)^3}{b^2 (b c-a d) (a+b x)}+\frac {(d f-c g)^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{3 g}\\ &=-\frac {B (b c-a d) g (3 b d f-b c g-a d g) x}{3 b^2 d^2}-\frac {B (b c-a d) g^2 x^2}{6 b d}-\frac {B (b f-a g)^3 \log (a+b x)}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 g}+\frac {B (d f-c g)^3 \log (c+d x)}{3 d^3 g}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 142, normalized size = 0.95 \begin {gather*} \frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {B \left (2 b d (b c-a d) g^2 (3 b d f-b c g-a d g) x+b^2 d^2 (b c-a d) g^3 x^2+2 d^3 (b f-a g)^3 \log (a+b x)-2 b^3 (d f-c g)^3 \log (c+d x)\right )}{2 b^3 d^3}}{3 g} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - (B*(2*b*d*(b*c - a*d)*g^2*(3*b*d*f - b*c*g - a*d*g)*x + b^
2*d^2*(b*c - a*d)*g^3*x^2 + 2*d^3*(b*f - a*g)^3*Log[a + b*x] - 2*b^3*(d*f - c*g)^3*Log[c + d*x]))/(2*b^3*d^3))
/(3*g)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2997\) vs.
\(2(140)=280\).
time = 0.39, size = 2998, normalized size = 19.99
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {\left (g x +f \right )^{3} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g}+\frac {g^{2} A \,x^{3}}{3}+g A f \,x^{2}+\frac {g^{2} B a \,x^{2}}{6 b}-\frac {g^{2} B c \,x^{2}}{6 d}+A \,f^{2} x +\frac {g^{2} B \ln \left (-b x -a \right ) a^{3}}{3 b^{3}}-\frac {g B \ln \left (-b x -a \right ) a^{2} f}{b^{2}}+\frac {B \ln \left (-b x -a \right ) a \,f^{2}}{b}-\frac {B \ln \left (-b x -a \right ) f^{3}}{3 g}-\frac {g^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}+\frac {g B \ln \left (d x +c \right ) c^{2} f}{d^{2}}-\frac {B \ln \left (d x +c \right ) c \,f^{2}}{d}+\frac {B \ln \left (d x +c \right ) f^{3}}{3 g}-\frac {g^{2} B \,a^{2} x}{3 b^{2}}+\frac {g B a f x}{b}+\frac {g^{2} B \,c^{2} x}{3 d^{2}}-\frac {g B c f x}{d}\) |
\(265\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(2998\) |
| default |
\(\text {Expression too large to display}\) |
\(2998\) |
| | |
|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURNVERBOSE)
[Out]
-1/d^2*e*(a*d-b*c)*(-B*d^3*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b^2/(b*e-(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*d)^3*a^2+B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/(b*e-(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*d)*c^2*g^2+2/3*B/d*g^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c^2-B*g/(b*e-(b*e/d+(a*
d-b*c)*e/d/(d*x+c))*d)*c*f+B*d^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/(b*e-(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*d)*f^2+B*e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b/(b*e-(b
*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*c^2+2*B*d*g*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*
e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c*f-B*d/e*g/b^2*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a*f-2/3*B*d^3/e*g
^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/b^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3
*a*c-B*d^2/e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b^2/(b*e-(b*e/d+(a*d-b*c)*e/d
/(d*x+c))*d)^2*a*c+B*d^3/e*g*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b^2/(b*e-(b*e/d+(
a*d-b*c)*e/d/(d*x+c))*d)^2*a*f-B*d^2/e*g*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b/(b*
e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c*f-2*B*d*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b
/e/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c*f*g+A*d^2*(1/3*e^2*g^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3/(b*e-(b*e/d+
(a*d-b*c)*e/d/(d*x+c))*d)^3+e*g*(a*c*d*g-a*d^2*f-b*c^2*g+b*c*d*f)/d^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2+
(c^2*g^2-2*c*d*f*g+d^2*f^2)/d^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d))+1/3*B*d/e*g^2/b^3*ln(b*e-(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*d)*a^2-1/6*B*d*e*g^2/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a^2-1/6*B/d*e*g^2*b/(b*e-(b*e/d
+(a*d-b*c)*e/d/(d*x+c))*d)^2*c^2+1/3*B/e*g^2/b^2*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a*c+1/3*B/d/e*g^2/b*l
n(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c^2-B/e*g/b*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*c*f-B*d*g^2*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*c^2+1/3*B*d^4
/e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/b^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*
d)^3*a^2+1/3*B*d^2/e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3/b/(b*e-(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*d)^3*c^2+B*d/e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/b/(b*e-(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*c^2-2*B*d*e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(
b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*c+2*B*d*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x
+c))/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*c-2*B*d^2*g*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*
e/d/(d*x+c))/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2*a*f+2*B*d^2*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+
(a*d-b*c)*e/d/(d*x+c))^2/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3*a*c+B*d*g/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+
c))*d)*a*f+B*d^2*e*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/(b*e-(b*e/d+(a*d-b*c)*e
/d/(d*x+c))*d)^3*a^2+B*d/b/e*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*f^2+1/3*B*e*g^2/(b*e-(b*e/d+(a*d-b*c)*e/d
/(d*x+c))*d)^2*a*c-1/3*B*g^2/b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)*a*c-1/3*B*d*g^2/b^2/(b*e-(b*e/d+(a*d-b*c)
*e/d/(d*x+c))*d)*a^2-2*B*g^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*d)^2*c^2)
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Maxima [A]
time = 0.28, size = 268, normalized size = 1.79 \begin {gather*} \frac {1}{3} \, A g^{2} x^{3} + A f g x^{2} + {\left (x \log \left (\frac {b x e}{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 f^{2} + {\left (x^{2} \log \left (\frac {b x e}{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 f g + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b x e}{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 g^{2} + A f^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")
[Out]
1/3*A*g^2*x^3 + A*f*g*x^2 + (x*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*f
^2 + (x^2*log(b*x*e/(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*f*g + 1/6*(2*x^3*log(b*x*e/(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*g^2 + A*f^2*x
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Fricas [A]
time = 0.39, size = 279, normalized size = 1.86 \begin {gather*} \frac {2 \, A b^{3} d^{3} g^{2} x^{3} + {\left (6 \, A b^{3} d^{3} f g - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g^{2}\right )} x^{2} + 2 \, {\left (3 \, A b^{3} d^{3} f^{2} - 3 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} f g + {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} g^{2}\right )} x + 2 \, {\left (3 \, B a b^{2} d^{3} f^{2} - 3 \, B a^{2} b d^{3} f g + B a^{3} d^{3} g^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, B b^{3} c d^{2} f^{2} - 3 \, B b^{3} c^{2} d f g + B b^{3} c^{3} g^{2}\right )} \log \left (d x + c\right ) + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B b^{3} d^{3} f g x^{2} + 3 \, B b^{3} d^{3} f^{2} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{6 \, b^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^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 + (6*A*b^3*d^3*f*g - (B*b^3*c*d^2 - B*a*b^2*d^3)*g^2)*x^2 + 2*(3*A*b^3*d^3*f^2 - 3*(B
*b^3*c*d^2 - B*a*b^2*d^3)*f*g + (B*b^3*c^2*d - B*a^2*b*d^3)*g^2)*x + 2*(3*B*a*b^2*d^3*f^2 - 3*B*a^2*b*d^3*f*g
+ B*a^3*d^3*g^2)*log(b*x + a) - 2*(3*B*b^3*c*d^2*f^2 - 3*B*b^3*c^2*d*f*g + B*b^3*c^3*g^2)*log(d*x + c) + 2*(B*
b^3*d^3*g^2*x^3 + 3*B*b^3*d^3*f*g*x^2 + 3*B*b^3*d^3*f^2*x)*log((b*x + a)*e/(d*x + c)))/(b^3*d^3)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 658 vs.
\(2 (131) = 262\).
time = 8.04, size = 658, normalized size = 4.39 \begin {gather*} \frac {A g^{2} x^{3}}{3} + \frac {B a \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right ) \log {\left (x + \frac {B a^{3} c d^{2} g^{2} - 3 B a^{2} b c d^{2} f g + \frac {B a^{2} d^{3} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{b} + B a b^{2} c^{3} g^{2} - 3 B a b^{2} c^{2} d f g + 6 B a b^{2} c d^{2} f^{2} - B a c d^{2} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{B a^{3} d^{3} g^{2} - 3 B a^{2} b d^{3} f g + 3 B a b^{2} d^{3} f^{2} + B b^{3} c^{3} g^{2} - 3 B b^{3} c^{2} d f g + 3 B b^{3} c d^{2} f^{2}} \right )}}{3 b^{3}} - \frac {B c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) \log {\left (x + \frac {B a^{3} c d^{2} g^{2} - 3 B a^{2} b c d^{2} f g + B a b^{2} c^{3} g^{2} - 3 B a b^{2} c^{2} d f g + 6 B a b^{2} c d^{2} f^{2} - B a b^{2} c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) + \frac {B b^{3} c^{2} \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right )}{d}}{B a^{3} d^{3} g^{2} - 3 B a^{2} b d^{3} f g + 3 B a b^{2} d^{3} f^{2} + B b^{3} c^{3} g^{2} - 3 B b^{3} c^{2} d f g + 3 B b^{3} c d^{2} f^{2}} \right )}}{3 d^{3}} + x^{2} \left (A f g + \frac {B a g^{2}}{6 b} - \frac {B c g^{2}}{6 d}\right ) + x \left (A f^{2} - \frac {B a^{2} g^{2}}{3 b^{2}} + \frac {B a f g}{b} + \frac {B c^{2} g^{2}}{3 d^{2}} - \frac {B c f g}{d}\right ) + \left (B f^{2} x + B f g x^{2} + \frac {B g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
A*g**2*x**3/3 + B*a*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2)*log(x + (B*a**3*c*d**2*g**2 - 3*B*a**2*b*c*d**2*f*g
+ B*a**2*d**3*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2)/b + B*a*b**2*c**3*g**2 - 3*B*a*b**2*c**2*d*f*g + 6*B*a*b**
2*c*d**2*f**2 - B*a*c*d**2*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2))/(B*a**3*d**3*g**2 - 3*B*a**2*b*d**3*f*g + 3*
B*a*b**2*d**3*f**2 + B*b**3*c**3*g**2 - 3*B*b**3*c**2*d*f*g + 3*B*b**3*c*d**2*f**2))/(3*b**3) - B*c*(c**2*g**2
- 3*c*d*f*g + 3*d**2*f**2)*log(x + (B*a**3*c*d**2*g**2 - 3*B*a**2*b*c*d**2*f*g + B*a*b**2*c**3*g**2 - 3*B*a*b
**2*c**2*d*f*g + 6*B*a*b**2*c*d**2*f**2 - B*a*b**2*c*(c**2*g**2 - 3*c*d*f*g + 3*d**2*f**2) + B*b**3*c**2*(c**2
*g**2 - 3*c*d*f*g + 3*d**2*f**2)/d)/(B*a**3*d**3*g**2 - 3*B*a**2*b*d**3*f*g + 3*B*a*b**2*d**3*f**2 + B*b**3*c*
*3*g**2 - 3*B*b**3*c**2*d*f*g + 3*B*b**3*c*d**2*f**2))/(3*d**3) + x**2*(A*f*g + B*a*g**2/(6*b) - B*c*g**2/(6*d
)) + x*(A*f**2 - B*a**2*g**2/(3*b**2) + B*a*f*g/b + B*c**2*g**2/(3*d**2) - B*c*f*g/d) + (B*f**2*x + B*f*g*x**2
+ B*g**2*x**3/3)*log(e*(a + b*x)/(c + d*x))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5950 vs.
\(2 (141) = 282\).
time = 5.43, size = 5950, normalized size = 39.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
[Out]
1/6*(6*B*b^7*c^2*d^2*f^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 12*B*a*b^6*c*d^3*f^2*e^4*log(-b*e + (b*x*
e + a*e)*d/(d*x + c)) + 6*B*a^2*b^5*d^4*f^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6*B*b^7*c^3*d*f*g*e^4*
log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 6*B*a*b^6*c^2*d^2*f*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 6*B*
a^2*b^5*c*d^3*f*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 6*B*a^3*b^4*d^4*f*g*e^4*log(-b*e + (b*x*e + a*e)
*d/(d*x + c)) + 2*B*b^7*c^4*g^2*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 2*B*a*b^6*c^3*d*g^2*e^4*log(-b*e +
(b*x*e + a*e)*d/(d*x + c)) - 2*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)) - 18*(b*x*e + a*e)*B*b^6*c^2*d^3*f^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*b^5*c*d^4*f^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c) - 18*(b*x*e + a*e)*B*a^2*b^4*d^5*f^2*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 18*(b*x*e + a
*e)*B*b^6*c^3*d^2*f*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 18*(b*x*e + a*e)*B*a*b^5*c^2*d^3*f
*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 18*(b*x*e + a*e)*B*a^2*b^4*c*d^4*f*g*e^3*log(-b*e + (
b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 18*(b*x*e + a*e)*B*a^3*b^3*d^5*f*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x +
c))/(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) + 6*(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) + 6*(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) + 18*(b*x*e + a*e)^2*B*b^5*c^2*d^4*f^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*b^4*c*d^5*f^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^2 + 18*(b*x*e + a*e)^2*B*a^2*b^3*d^6*f^2*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 18*(b*x*e
+ a*e)^2*B*b^5*c^3*d^3*f*g*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 18*(b*x*e + a*e)^2*B*a*b^4*
c^2*d^4*f*g*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 18*(b*x*e + a*e)^2*B*a^2*b^3*c*d^5*f*g*e^2
*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 18*(b*x*e + a*e)^2*B*a^3*b^2*d^6*f*g*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*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 - 6*(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
- 6*(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 - 6*(b*x*e + a*e)^3*B*b^4*c^2*d^5*f^
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*b^3*c*d^6*f^2*e*log(-b*e + (b*x
*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 6*(b*x*e + a*e)^3*B*a^2*b^2*d^7*f^2*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c
))/(d*x + c)^3 + 6*(b*x*e + a*e)^3*B*b^4*c^3*d^4*f*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 6*(
b*x*e + a*e)^3*B*a*b^3*c^2*d^5*f*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 6*(b*x*e + a*e)^3*B*a
^2*b^2*c*d^6*f*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 6*(b*x*e + a*e)^3*B*a^3*b*d^7*f*g*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*e + (b*x*e + a*e
)*d/(d*x + c))/(d*x + c)^3 + 2*(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 + 2*(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 + 6*(b*x*e + a*e)*B*b^6*c^2*d^3*f^2
*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 12*(b*x*e + a*e)*B*a*b^5*c*d^4*f^2*e^3*log((b*x*e + a*e)/(d*x +
c))/(d*x + c) + 6*(b*x*e + a*e)*B*a^2*b^4*d^5*f^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 12*(b*x*e + a*e
)*B*a*b^5*c^2*d^3*f*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a^2*b^4*c*d^4*f*g*e^3*lo
g((b*x*e + a*e)/(d*x + c))/(d*x + c) - 12*(b*x*e + a*e)*B*a^3*b^3*d^5*f*g*e^3*log((b*x*e + a*e)/(d*x + c))/(d*
x + c) + 6*(b*x*e + a*e)*B*a^2*b^4*c^2*d^3*g^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 12*(b*x*e + a*e)*B
*a^3*b^3*c*d^4*g^2*e^3*log((b*x*e + a*e)/(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*x
*e + a*e)/(d*x + c))/(d*x + c) - 12*(b*x*e + a*e)^2*B*b^5*c^2*d^4*f^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x +
c)^2 + 24*(b*x*e + a*e)^2*B*a*b^4*c*d^5*f^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 12*(b*x*e + a*e)^2*
B*a^2*b^3*d^6*f^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 6*(b*x*e + a*e)^2*B*b^5*c^3*d^3*f*g*e^2*log((
b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 6*(b*x*e + a*e)^2*B*a*b^4*c^2*d^4*f*g*e^2*log((b*x*e + a*e)/(d*x + c))/(
d*x + c)^2 - 30*(b*x*e + a*e)^2*B*a^2*b^3*c*d^5*f*g*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 18*(b*x*e +
a*e)^2*B*a^3*b^2*d^6*f*g*e^2*log((b*x*e + a*e)...
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Mupad [B]
time = 4.73, size = 356, normalized size = 2.37 \begin {gather*} x^2\,\left (\frac {3\,A\,a\,d\,g^2+3\,A\,b\,c\,g^2+B\,a\,d\,g^2-B\,b\,c\,g^2+6\,A\,b\,d\,f\,g}{6\,b\,d}-\frac {A\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,b\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,f^2\,x+B\,f\,g\,x^2+\frac {B\,g^2\,x^3}{3}\right )-x\,\left (\frac {\left (\frac {3\,A\,a\,d\,g^2+3\,A\,b\,c\,g^2+B\,a\,d\,g^2-B\,b\,c\,g^2+6\,A\,b\,d\,f\,g}{3\,b\,d}-\frac {A\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}\right )\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}-\frac {3\,A\,a\,c\,g^2+3\,A\,b\,d\,f^2+6\,A\,a\,d\,f\,g+6\,A\,b\,c\,f\,g+3\,B\,a\,d\,f\,g-3\,B\,b\,c\,f\,g}{3\,b\,d}+\frac {A\,a\,c\,g^2}{b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^3\,g^2-3\,B\,a^2\,b\,f\,g+3\,B\,a\,b^2\,f^2\right )}{3\,b^3}-\frac {\ln \left (c+d\,x\right )\,\left (B\,c^3\,g^2-3\,B\,c^2\,d\,f\,g+3\,B\,c\,d^2\,f^2\right )}{3\,d^3}+\frac {A\,g^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))),x)
[Out]
x^2*((3*A*a*d*g^2 + 3*A*b*c*g^2 + B*a*d*g^2 - B*b*c*g^2 + 6*A*b*d*f*g)/(6*b*d) - (A*g^2*(3*a*d + 3*b*c))/(6*b*
d)) + log((e*(a + b*x))/(c + d*x))*((B*g^2*x^3)/3 + B*f^2*x + B*f*g*x^2) - x*((((3*A*a*d*g^2 + 3*A*b*c*g^2 + B
*a*d*g^2 - B*b*c*g^2 + 6*A*b*d*f*g)/(3*b*d) - (A*g^2*(3*a*d + 3*b*c))/(3*b*d))*(3*a*d + 3*b*c))/(3*b*d) - (3*A
*a*c*g^2 + 3*A*b*d*f^2 + 6*A*a*d*f*g + 6*A*b*c*f*g + 3*B*a*d*f*g - 3*B*b*c*f*g)/(3*b*d) + (A*a*c*g^2)/(b*d)) +
(log(a + b*x)*(B*a^3*g^2 + 3*B*a*b^2*f^2 - 3*B*a^2*b*f*g))/(3*b^3) - (log(c + d*x)*(B*c^3*g^2 + 3*B*c*d^2*f^2
- 3*B*c^2*d*f*g))/(3*d^3) + (A*g^2*x^3)/3
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