3.12.35 \(\int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx\) [1135]

Optimal. Leaf size=141 \[ \frac {e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac {B e^3 x^2}{2 b^3}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac {3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac {e^2 x (-3 a B e+A b e+3 b B d)}{b^4}+\frac {B e^3 x^2}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*(3*b*B*d + A*b*e - 3*a*B*e)*x)/b^4 + (B*e^3*x^2)/(2*b^3) - ((A*b - a*B)*(b*d - a*e)^3)/(2*b^5*(a + b*x)^2
) - ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e))/(b^5*(a + b*x)) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*L
og[a + b*x])/b^5

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx &=\int \left (\frac {e^2 (3 b B d+A b e-3 a B e)}{b^4}+\frac {B e^3 x}{b^3}+\frac {(A b-a B) (b d-a e)^3}{b^4 (a+b x)^3}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^4 (a+b x)^2}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac {B e^3 x^2}{2 b^3}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 245, normalized size = 1.74 \begin {gather*} \frac {-A b \left (5 a^3 e^3+a^2 b e^2 (-9 d+4 e x)+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )+B \left (7 a^4 e^3+a^3 b e^2 (-15 d+2 e x)+a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )-a b^3 \left (d^3-12 d^2 e x-12 d e^2 x^2+4 e^3 x^3\right )\right )+6 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(A*b*(5*a^3*e^3 + a^2*b*e^2*(-9*d + 4*e*x) + a*b^2*e*(3*d^2 - 12*d*e*x - 4*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x -
2*e^3*x^3))) + B*(7*a^4*e^3 + a^3*b*e^2*(-15*d + 2*e*x) + a^2*b^2*e*(9*d^2 - 12*d*e*x - 11*e^2*x^2) + b^4*x*(
-2*d^3 + 6*d*e^2*x^2 + e^3*x^3) - a*b^3*(d^3 - 12*d^2*e*x - 12*d*e^2*x^2 + 4*e^3*x^3)) + 6*e*(b*d - a*e)*(b*B*
d + A*b*e - 2*a*B*e)*(a + b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)

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Maple [A]
time = 0.10, size = 268, normalized size = 1.90

method	result	size
default	\(\frac {e^{2} \left (\frac {1}{2} B b e \,x^{2}+A b e x -3 B a e x +3 B b d x \right )}{b^{4}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -4 B \,a^{3} e^{3}+9 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}}{b^{5} \left (b x +a \right )}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {-A \,a^{3} b \,e^{3}+3 A \,b^{2} d \,e^{2} a^{2}-3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}+B \,a^{4} e^{3}-3 B \,a^{3} b d \,e^{2}+3 B \,a^{2} b^{2} d^{2} e -B a \,b^{3} d^{3}}{2 b^{5} \left (b x +a \right )^{2}}\)	\(268\)
norman	\(\frac {\frac {e^{2} \left (A b e -2 B a e +3 B b d \right ) x^{3}}{b^{2}}-\frac {9 A \,a^{3} b \,e^{3}-9 A \,b^{2} d \,e^{2} a^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-18 B \,a^{4} e^{3}+27 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b^{5}}-\frac {\left (6 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -12 B \,a^{3} e^{3}+18 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}\right ) x}{b^{4}}+\frac {B \,e^{3} x^{4}}{2 b}}{\left (b x +a \right )^{2}}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\)	\(268\)
risch	\(\frac {B \,e^{3} x^{2}}{2 b^{3}}+\frac {e^{3} A x}{b^{3}}-\frac {3 e^{3} B a x}{b^{4}}+\frac {3 e^{2} B d x}{b^{3}}+\frac {\left (-3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-9 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x -\frac {5 A \,a^{3} b \,e^{3}-9 A \,b^{2} d \,e^{2} a^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-7 B \,a^{4} e^{3}+15 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b}}{b^{4} \left (b x +a \right )^{2}}-\frac {3 e^{3} \ln \left (b x +a \right ) A a}{b^{4}}+\frac {3 e^{2} \ln \left (b x +a \right ) A d}{b^{3}}+\frac {6 e^{3} \ln \left (b x +a \right ) B \,a^{2}}{b^{5}}-\frac {9 e^{2} \ln \left (b x +a \right ) B a d}{b^{4}}+\frac {3 e \ln \left (b x +a \right ) B \,d^{2}}{b^{3}}\)	\(304\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

e^2/b^4*(1/2*B*b*e*x^2+A*b*e*x-3*B*a*e*x+3*B*b*d*x)-1/b^5*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e-4*B*a^3
*e^3+9*B*a^2*b*d*e^2-6*B*a*b^2*d^2*e+B*b^3*d^3)/(b*x+a)-3/b^5*e*(A*a*b*e^2-A*b^2*d*e-2*B*a^2*e^2+3*B*a*b*d*e-B
*b^2*d^2)*ln(b*x+a)-1/2/b^5*(-A*a^3*b*e^3+3*A*a^2*b^2*d*e^2-3*A*a*b^3*d^2*e+A*b^4*d^3+B*a^4*e^3-3*B*a^3*b*d*e^
2+3*B*a^2*b^2*d^2*e-B*a*b^3*d^3)/(b*x+a)^2

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Maxima [A]
time = 0.30, size = 283, normalized size = 2.01 \begin {gather*} \frac {7 \, B a^{4} e^{3} - 5 \, A a^{3} b e^{3} - {\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \, {\left (3 \, B a^{2} b^{2} e - A a b^{3} e\right )} d^{2} - 3 \, {\left (5 \, B a^{3} b e^{2} - 3 \, A a^{2} b^{2} e^{2}\right )} d - 2 \, {\left (B b^{4} d^{3} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3} - 3 \, {\left (2 \, B a b^{3} e - A b^{4} e\right )} d^{2} + 3 \, {\left (3 \, B a^{2} b^{2} e^{2} - 2 \, A a b^{3} e^{2}\right )} d\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {B b x^{2} e^{3} + 2 \, {\left (3 \, B b d e^{2} - 3 \, B a e^{3} + A b e^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (B b^{2} d^{2} e + 2 \, B a^{2} e^{3} - A a b e^{3} - {\left (3 \, B a b e^{2} - A b^{2} e^{2}\right )} d\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(7*B*a^4*e^3 - 5*A*a^3*b*e^3 - (B*a*b^3 + A*b^4)*d^3 + 3*(3*B*a^2*b^2*e - A*a*b^3*e)*d^2 - 3*(5*B*a^3*b*e^
2 - 3*A*a^2*b^2*e^2)*d - 2*(B*b^4*d^3 - 4*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3 - 3*(2*B*a*b^3*e - A*b^4*e)*d^2 + 3*(3
*B*a^2*b^2*e^2 - 2*A*a*b^3*e^2)*d)*x)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) + 1/2*(B*b*x^2*e^3 + 2*(3*B*b*d*e^2 - 3*
B*a*e^3 + A*b*e^3)*x)/b^4 + 3*(B*b^2*d^2*e + 2*B*a^2*e^3 - A*a*b*e^3 - (3*B*a*b*e^2 - A*b^2*e^2)*d)*log(b*x +
a)/b^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (145) = 290\).
time = 1.76, size = 419, normalized size = 2.97 \begin {gather*} -\frac {2 \, B b^{4} d^{3} x + {\left (B a b^{3} + A b^{4}\right )} d^{3} - {\left (B b^{4} x^{4} + 7 \, B a^{4} - 5 \, A a^{3} b - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{3} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} e^{3} - 3 \, {\left (2 \, B b^{4} d x^{3} + 4 \, B a b^{3} d x^{2} - 4 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d x - {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d\right )} e^{2} - 3 \, {\left (2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} x + {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2}\right )} e - 6 \, {\left ({\left (2 \, B a^{4} - A a^{3} b + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} e^{3} - {\left ({\left (3 \, B a b^{3} - A b^{4}\right )} d x^{2} + 2 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d x + {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d\right )} e^{2} + {\left (B b^{4} d^{2} x^{2} + 2 \, B a b^{3} d^{2} x + B a^{2} b^{2} d^{2}\right )} e\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*B*b^4*d^3*x + (B*a*b^3 + A*b^4)*d^3 - (B*b^4*x^4 + 7*B*a^4 - 5*A*a^3*b - 2*(2*B*a*b^3 - A*b^4)*x^3 - (
11*B*a^2*b^2 - 4*A*a*b^3)*x^2 + 2*(B*a^3*b - 2*A*a^2*b^2)*x)*e^3 - 3*(2*B*b^4*d*x^3 + 4*B*a*b^3*d*x^2 - 4*(B*a
^2*b^2 - A*a*b^3)*d*x - (5*B*a^3*b - 3*A*a^2*b^2)*d)*e^2 - 3*(2*(2*B*a*b^3 - A*b^4)*d^2*x + (3*B*a^2*b^2 - A*a
*b^3)*d^2)*e - 6*((2*B*a^4 - A*a^3*b + (2*B*a^2*b^2 - A*a*b^3)*x^2 + 2*(2*B*a^3*b - A*a^2*b^2)*x)*e^3 - ((3*B*
a*b^3 - A*b^4)*d*x^2 + 2*(3*B*a^2*b^2 - A*a*b^3)*d*x + (3*B*a^3*b - A*a^2*b^2)*d)*e^2 + (B*b^4*d^2*x^2 + 2*B*a
*b^3*d^2*x + B*a^2*b^2*d^2)*e)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (141) = 282\).
time = 3.06, size = 299, normalized size = 2.12 \begin {gather*} \frac {B e^{3} x^{2}}{2 b^{3}} + x \left (\frac {A e^{3}}{b^{3}} - \frac {3 B a e^{3}}{b^{4}} + \frac {3 B d e^{2}}{b^{3}}\right ) + \frac {- 5 A a^{3} b e^{3} + 9 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - A b^{4} d^{3} + 7 B a^{4} e^{3} - 15 B a^{3} b d e^{2} + 9 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x \left (- 6 A a^{2} b^{2} e^{3} + 12 A a b^{3} d e^{2} - 6 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 2 B b^{4} d^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {3 e \left (a e - b d\right ) \left (- A b e + 2 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3/(b*x+a)**3,x)

[Out]

B*e**3*x**2/(2*b**3) + x*(A*e**3/b**3 - 3*B*a*e**3/b**4 + 3*B*d*e**2/b**3) + (-5*A*a**3*b*e**3 + 9*A*a**2*b**2
*d*e**2 - 3*A*a*b**3*d**2*e - A*b**4*d**3 + 7*B*a**4*e**3 - 15*B*a**3*b*d*e**2 + 9*B*a**2*b**2*d**2*e - B*a*b*
*3*d**3 + x*(-6*A*a**2*b**2*e**3 + 12*A*a*b**3*d*e**2 - 6*A*b**4*d**2*e + 8*B*a**3*b*e**3 - 18*B*a**2*b**2*d*e
**2 + 12*B*a*b**3*d**2*e - 2*B*b**4*d**3))/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 3*e*(a*e - b*d)*(-A*b*e
+ 2*B*a*e - B*b*d)*log(a + b*x)/b**5

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Giac [A]
time = 0.92, size = 272, normalized size = 1.93 \begin {gather*} \frac {3 \, {\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {B b^{3} x^{2} e^{3} + 6 \, B b^{3} d x e^{2} - 6 \, B a b^{2} x e^{3} + 2 \, A b^{3} x e^{3}}{2 \, b^{6}} - \frac {B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \, {\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(B*b^2*d^2*e - 3*B*a*b*d*e^2 + A*b^2*d*e^2 + 2*B*a^2*e^3 - A*a*b*e^3)*log(abs(b*x + a))/b^5 + 1/2*(B*b^3*x^2
*e^3 + 6*B*b^3*d*x*e^2 - 6*B*a*b^2*x*e^3 + 2*A*b^3*x*e^3)/b^6 - 1/2*(B*a*b^3*d^3 + A*b^4*d^3 - 9*B*a^2*b^2*d^2
*e + 3*A*a*b^3*d^2*e + 15*B*a^3*b*d*e^2 - 9*A*a^2*b^2*d*e^2 - 7*B*a^4*e^3 + 5*A*a^3*b*e^3 + 2*(B*b^4*d^3 - 6*B
*a*b^3*d^2*e + 3*A*b^4*d^2*e + 9*B*a^2*b^2*d*e^2 - 6*A*a*b^3*d*e^2 - 4*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*x)/((b*x
+ a)^2*b^5)

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Mupad [B]
time = 1.17, size = 290, normalized size = 2.06 \begin {gather*} x\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{b^3}-\frac {3\,B\,a\,e^3}{b^4}\right )-\frac {\frac {-7\,B\,a^4\,e^3+15\,B\,a^3\,b\,d\,e^2+5\,A\,a^3\,b\,e^3-9\,B\,a^2\,b^2\,d^2\,e-9\,A\,a^2\,b^2\,d\,e^2+B\,a\,b^3\,d^3+3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}{2\,b}+x\,\left (-4\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3-6\,B\,a\,b^2\,d^2\,e-6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,B\,a^2\,e^3-9\,B\,a\,b\,d\,e^2-3\,A\,a\,b\,e^3+3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2\right )}{b^5}+\frac {B\,e^3\,x^2}{2\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((A*e^3 + 3*B*d*e^2)/b^3 - (3*B*a*e^3)/b^4) - ((A*b^4*d^3 - 7*B*a^4*e^3 + 5*A*a^3*b*e^3 + B*a*b^3*d^3 - 9*A*
a^2*b^2*d*e^2 - 9*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 15*B*a^3*b*d*e^2)/(2*b) + x*(B*b^3*d^3 - 4*B*a^3*e^3 + 3
*A*a^2*b*e^3 + 3*A*b^3*d^2*e - 6*A*a*b^2*d*e^2 - 6*B*a*b^2*d^2*e + 9*B*a^2*b*d*e^2))/(a^2*b^4 + b^6*x^2 + 2*a*
b^5*x) + (log(a + b*x)*(6*B*a^2*e^3 - 3*A*a*b*e^3 + 3*A*b^2*d*e^2 + 3*B*b^2*d^2*e - 9*B*a*b*d*e^2))/b^5 + (B*e
^3*x^2)/(2*b^3)

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