3.10.21 \(\int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx\)

Optimal. Leaf size=220 \[ -\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}+\frac {b x (b d-a e)^5 (B d-A e)}{e^7}-\frac {(a+b x)^2 (b d-a e)^4 (B d-A e)}{2 e^6}+\frac {(a+b x)^3 (b d-a e)^3 (B d-A e)}{3 e^5}-\frac {(a+b x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {(a+b x)^5 (b d-a e) (B d-A e)}{5 e^3}-\frac {(a+b x)^6 (B d-A e)}{6 e^2}+\frac {B (a+b x)^7}{7 b e} \]

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Rubi [A]  time = 0.19, antiderivative size = 220, 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 = {77} \begin {gather*} -\frac {(a+b x)^6 (B d-A e)}{6 e^2}+\frac {(a+b x)^5 (b d-a e) (B d-A e)}{5 e^3}-\frac {(a+b x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {(a+b x)^3 (b d-a e)^3 (B d-A e)}{3 e^5}-\frac {(a+b x)^2 (b d-a e)^4 (B d-A e)}{2 e^6}+\frac {b x (b d-a e)^5 (B d-A e)}{e^7}-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}+\frac {B (a+b x)^7}{7 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x),x]

[Out]

(b*(b*d - a*e)^5*(B*d - A*e)*x)/e^7 - ((b*d - a*e)^4*(B*d - A*e)*(a + b*x)^2)/(2*e^6) + ((b*d - a*e)^3*(B*d -
A*e)*(a + b*x)^3)/(3*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^4)/(4*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*
x)^5)/(5*e^3) - ((B*d - A*e)*(a + b*x)^6)/(6*e^2) + (B*(a + b*x)^7)/(7*b*e) - ((b*d - a*e)^6*(B*d - A*e)*Log[d
+ e*x])/e^8

Rule 77

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)^6 (A+B x)}{d+e x} \, dx &=\int \left (-\frac {b (b d-a e)^5 (-B d+A e)}{e^7}+\frac {b (b d-a e)^4 (-B d+A e) (a+b x)}{e^6}-\frac {b (b d-a e)^3 (-B d+A e) (a+b x)^2}{e^5}+\frac {b (b d-a e)^2 (-B d+A e) (a+b x)^3}{e^4}-\frac {b (b d-a e) (-B d+A e) (a+b x)^4}{e^3}+\frac {b (-B d+A e) (a+b x)^5}{e^2}+\frac {B (a+b x)^6}{e}+\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {b (b d-a e)^5 (B d-A e) x}{e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^2}{2 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^3}{3 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^4}{4 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^5}{5 e^3}-\frac {(B d-A e) (a+b x)^6}{6 e^2}+\frac {B (a+b x)^7}{7 b e}-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [B]  time = 0.28, size = 501, normalized size = 2.28 \begin {gather*} \frac {e x \left (420 a^6 B e^6+1260 a^5 b e^5 (2 A e-2 B d+B e x)+1050 a^4 b^2 e^4 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+700 a^3 b^3 e^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+105 a^2 b^4 e^2 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+42 a b^5 e \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+b^6 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (b d-a e)^6 (B d-A e) \log (d+e x)}{420 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x),x]

[Out]

(e*x*(420*a^6*B*e^6 + 1260*a^5*b*e^5*(-2*B*d + 2*A*e + B*e*x) + 1050*a^4*b^2*e^4*(3*A*e*(-2*d + e*x) + B*(6*d^
2 - 3*d*e*x + 2*e^2*x^2)) + 700*a^3*b^3*e^3*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*
d*e^2*x^2 + 3*e^3*x^3)) + 105*a^2*b^4*e^2*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 -
30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 42*a*b^5*e*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2
*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 +
10*e^5*x^5)) + b^6*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5
) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)
)) - 420*(b*d - a*e)^6*(B*d - A*e)*Log[d + e*x])/(420*e^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x),x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x), x]

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fricas [B]  time = 0.82, size = 763, normalized size = 3.47 \begin {gather*} \frac {60 \, B b^{6} e^{7} x^{7} - 70 \, {\left (B b^{6} d e^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 84 \, {\left (B b^{6} d^{2} e^{5} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 105 \, {\left (B b^{6} d^{3} e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 140 \, {\left (B b^{6} d^{4} e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - 210 \, {\left (B b^{6} d^{5} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 420 \, {\left (B b^{6} d^{6} e - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x - 420 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="fricas")

[Out]

1/420*(60*B*b^6*e^7*x^7 - 70*(B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 84*(B*b^6*d^2*e^5 - (6*B*a*b^5 + A*
b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 105*(B*b^6*d^3*e^4 - (6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^6 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 140*(B*b^6*d^4*e^3 - (6*B*a*b^5 + A*b^6)
*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*e^7)*x^3 - 210*(B*b^6*d^5*e^2 - (6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 5*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^
2 + 420*(B*b^6*d^6*e - (6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 5*(4*B*a^3*b^3 + 3*
A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*
a^5*b)*e^7)*x - 420*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 -
5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d
^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*log(e*x + d))/e^8

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giac [B]  time = 1.24, size = 858, normalized size = 3.90 \begin {gather*} -{\left (B b^{6} d^{7} - 6 \, B a b^{5} d^{6} e - A b^{6} d^{6} e + 15 \, B a^{2} b^{4} d^{5} e^{2} + 6 \, A a b^{5} d^{5} e^{2} - 20 \, B a^{3} b^{3} d^{4} e^{3} - 15 \, A a^{2} b^{4} d^{4} e^{3} + 15 \, B a^{4} b^{2} d^{3} e^{4} + 20 \, A a^{3} b^{3} d^{3} e^{4} - 6 \, B a^{5} b d^{2} e^{5} - 15 \, A a^{4} b^{2} d^{2} e^{5} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} - A a^{6} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (60 \, B b^{6} x^{7} e^{6} - 70 \, B b^{6} d x^{6} e^{5} + 84 \, B b^{6} d^{2} x^{5} e^{4} - 105 \, B b^{6} d^{3} x^{4} e^{3} + 140 \, B b^{6} d^{4} x^{3} e^{2} - 210 \, B b^{6} d^{5} x^{2} e + 420 \, B b^{6} d^{6} x + 420 \, B a b^{5} x^{6} e^{6} + 70 \, A b^{6} x^{6} e^{6} - 504 \, B a b^{5} d x^{5} e^{5} - 84 \, A b^{6} d x^{5} e^{5} + 630 \, B a b^{5} d^{2} x^{4} e^{4} + 105 \, A b^{6} d^{2} x^{4} e^{4} - 840 \, B a b^{5} d^{3} x^{3} e^{3} - 140 \, A b^{6} d^{3} x^{3} e^{3} + 1260 \, B a b^{5} d^{4} x^{2} e^{2} + 210 \, A b^{6} d^{4} x^{2} e^{2} - 2520 \, B a b^{5} d^{5} x e - 420 \, A b^{6} d^{5} x e + 1260 \, B a^{2} b^{4} x^{5} e^{6} + 504 \, A a b^{5} x^{5} e^{6} - 1575 \, B a^{2} b^{4} d x^{4} e^{5} - 630 \, A a b^{5} d x^{4} e^{5} + 2100 \, B a^{2} b^{4} d^{2} x^{3} e^{4} + 840 \, A a b^{5} d^{2} x^{3} e^{4} - 3150 \, B a^{2} b^{4} d^{3} x^{2} e^{3} - 1260 \, A a b^{5} d^{3} x^{2} e^{3} + 6300 \, B a^{2} b^{4} d^{4} x e^{2} + 2520 \, A a b^{5} d^{4} x e^{2} + 2100 \, B a^{3} b^{3} x^{4} e^{6} + 1575 \, A a^{2} b^{4} x^{4} e^{6} - 2800 \, B a^{3} b^{3} d x^{3} e^{5} - 2100 \, A a^{2} b^{4} d x^{3} e^{5} + 4200 \, B a^{3} b^{3} d^{2} x^{2} e^{4} + 3150 \, A a^{2} b^{4} d^{2} x^{2} e^{4} - 8400 \, B a^{3} b^{3} d^{3} x e^{3} - 6300 \, A a^{2} b^{4} d^{3} x e^{3} + 2100 \, B a^{4} b^{2} x^{3} e^{6} + 2800 \, A a^{3} b^{3} x^{3} e^{6} - 3150 \, B a^{4} b^{2} d x^{2} e^{5} - 4200 \, A a^{3} b^{3} d x^{2} e^{5} + 6300 \, B a^{4} b^{2} d^{2} x e^{4} + 8400 \, A a^{3} b^{3} d^{2} x e^{4} + 1260 \, B a^{5} b x^{2} e^{6} + 3150 \, A a^{4} b^{2} x^{2} e^{6} - 2520 \, B a^{5} b d x e^{5} - 6300 \, A a^{4} b^{2} d x e^{5} + 420 \, B a^{6} x e^{6} + 2520 \, A a^{5} b x e^{6}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="giac")

[Out]

-(B*b^6*d^7 - 6*B*a*b^5*d^6*e - A*b^6*d^6*e + 15*B*a^2*b^4*d^5*e^2 + 6*A*a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^3
- 15*A*a^2*b^4*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 - 6*B*a^5*b*d^2*e^5 - 15*A*a^4*b^2*d^2*e^
5 + B*a^6*d*e^6 + 6*A*a^5*b*d*e^6 - A*a^6*e^7)*e^(-8)*log(abs(x*e + d)) + 1/420*(60*B*b^6*x^7*e^6 - 70*B*b^6*d
*x^6*e^5 + 84*B*b^6*d^2*x^5*e^4 - 105*B*b^6*d^3*x^4*e^3 + 140*B*b^6*d^4*x^3*e^2 - 210*B*b^6*d^5*x^2*e + 420*B*
b^6*d^6*x + 420*B*a*b^5*x^6*e^6 + 70*A*b^6*x^6*e^6 - 504*B*a*b^5*d*x^5*e^5 - 84*A*b^6*d*x^5*e^5 + 630*B*a*b^5*
d^2*x^4*e^4 + 105*A*b^6*d^2*x^4*e^4 - 840*B*a*b^5*d^3*x^3*e^3 - 140*A*b^6*d^3*x^3*e^3 + 1260*B*a*b^5*d^4*x^2*e
^2 + 210*A*b^6*d^4*x^2*e^2 - 2520*B*a*b^5*d^5*x*e - 420*A*b^6*d^5*x*e + 1260*B*a^2*b^4*x^5*e^6 + 504*A*a*b^5*x
^5*e^6 - 1575*B*a^2*b^4*d*x^4*e^5 - 630*A*a*b^5*d*x^4*e^5 + 2100*B*a^2*b^4*d^2*x^3*e^4 + 840*A*a*b^5*d^2*x^3*e
^4 - 3150*B*a^2*b^4*d^3*x^2*e^3 - 1260*A*a*b^5*d^3*x^2*e^3 + 6300*B*a^2*b^4*d^4*x*e^2 + 2520*A*a*b^5*d^4*x*e^2
+ 2100*B*a^3*b^3*x^4*e^6 + 1575*A*a^2*b^4*x^4*e^6 - 2800*B*a^3*b^3*d*x^3*e^5 - 2100*A*a^2*b^4*d*x^3*e^5 + 420
0*B*a^3*b^3*d^2*x^2*e^4 + 3150*A*a^2*b^4*d^2*x^2*e^4 - 8400*B*a^3*b^3*d^3*x*e^3 - 6300*A*a^2*b^4*d^3*x*e^3 + 2
100*B*a^4*b^2*x^3*e^6 + 2800*A*a^3*b^3*x^3*e^6 - 3150*B*a^4*b^2*d*x^2*e^5 - 4200*A*a^3*b^3*d*x^2*e^5 + 6300*B*
a^4*b^2*d^2*x*e^4 + 8400*A*a^3*b^3*d^2*x*e^4 + 1260*B*a^5*b*x^2*e^6 + 3150*A*a^4*b^2*x^2*e^6 - 2520*B*a^5*b*d*
x*e^5 - 6300*A*a^4*b^2*d*x*e^5 + 420*B*a^6*x*e^6 + 2520*A*a^5*b*x*e^6)*e^(-7)

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maple [B]  time = 0.01, size = 989, normalized size = 4.50 \begin {gather*} \frac {B \,b^{6} x^{7}}{7 e}+\frac {A \,b^{6} x^{6}}{6 e}+\frac {B a \,b^{5} x^{6}}{e}-\frac {B \,b^{6} d \,x^{6}}{6 e^{2}}+\frac {6 A a \,b^{5} x^{5}}{5 e}-\frac {A \,b^{6} d \,x^{5}}{5 e^{2}}+\frac {3 B \,a^{2} b^{4} x^{5}}{e}-\frac {6 B a \,b^{5} d \,x^{5}}{5 e^{2}}+\frac {B \,b^{6} d^{2} x^{5}}{5 e^{3}}+\frac {15 A \,a^{2} b^{4} x^{4}}{4 e}-\frac {3 A a \,b^{5} d \,x^{4}}{2 e^{2}}+\frac {A \,b^{6} d^{2} x^{4}}{4 e^{3}}+\frac {5 B \,a^{3} b^{3} x^{4}}{e}-\frac {15 B \,a^{2} b^{4} d \,x^{4}}{4 e^{2}}+\frac {3 B a \,b^{5} d^{2} x^{4}}{2 e^{3}}-\frac {B \,b^{6} d^{3} x^{4}}{4 e^{4}}+\frac {20 A \,a^{3} b^{3} x^{3}}{3 e}-\frac {5 A \,a^{2} b^{4} d \,x^{3}}{e^{2}}+\frac {2 A a \,b^{5} d^{2} x^{3}}{e^{3}}-\frac {A \,b^{6} d^{3} x^{3}}{3 e^{4}}+\frac {5 B \,a^{4} b^{2} x^{3}}{e}-\frac {20 B \,a^{3} b^{3} d \,x^{3}}{3 e^{2}}+\frac {5 B \,a^{2} b^{4} d^{2} x^{3}}{e^{3}}-\frac {2 B a \,b^{5} d^{3} x^{3}}{e^{4}}+\frac {B \,b^{6} d^{4} x^{3}}{3 e^{5}}+\frac {15 A \,a^{4} b^{2} x^{2}}{2 e}-\frac {10 A \,a^{3} b^{3} d \,x^{2}}{e^{2}}+\frac {15 A \,a^{2} b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {3 A a \,b^{5} d^{3} x^{2}}{e^{4}}+\frac {A \,b^{6} d^{4} x^{2}}{2 e^{5}}+\frac {3 B \,a^{5} b \,x^{2}}{e}-\frac {15 B \,a^{4} b^{2} d \,x^{2}}{2 e^{2}}+\frac {10 B \,a^{3} b^{3} d^{2} x^{2}}{e^{3}}-\frac {15 B \,a^{2} b^{4} d^{3} x^{2}}{2 e^{4}}+\frac {3 B a \,b^{5} d^{4} x^{2}}{e^{5}}-\frac {B \,b^{6} d^{5} x^{2}}{2 e^{6}}+\frac {A \,a^{6} \ln \left (e x +d \right )}{e}-\frac {6 A \,a^{5} b d \ln \left (e x +d \right )}{e^{2}}+\frac {6 A \,a^{5} b x}{e}+\frac {15 A \,a^{4} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {15 A \,a^{4} b^{2} d x}{e^{2}}-\frac {20 A \,a^{3} b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {20 A \,a^{3} b^{3} d^{2} x}{e^{3}}+\frac {15 A \,a^{2} b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {15 A \,a^{2} b^{4} d^{3} x}{e^{4}}-\frac {6 A a \,b^{5} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {6 A a \,b^{5} d^{4} x}{e^{5}}+\frac {A \,b^{6} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {A \,b^{6} d^{5} x}{e^{6}}-\frac {B \,a^{6} d \ln \left (e x +d \right )}{e^{2}}+\frac {B \,a^{6} x}{e}+\frac {6 B \,a^{5} b \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 B \,a^{5} b d x}{e^{2}}-\frac {15 B \,a^{4} b^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {15 B \,a^{4} b^{2} d^{2} x}{e^{3}}+\frac {20 B \,a^{3} b^{3} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {20 B \,a^{3} b^{3} d^{3} x}{e^{4}}-\frac {15 B \,a^{2} b^{4} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {15 B \,a^{2} b^{4} d^{4} x}{e^{5}}+\frac {6 B a \,b^{5} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {6 B a \,b^{5} d^{5} x}{e^{6}}-\frac {B \,b^{6} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {B \,b^{6} d^{6} x}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d),x)

[Out]

1/7/e*B*b^6*x^7+1/6/e*A*x^6*b^6+1/e*B*x*a^6+1/e*ln(e*x+d)*A*a^6+3/2/e^3*B*x^4*a*b^5*d^2-5/e^2*A*x^3*a^2*b^4*d+
2/e^3*A*x^3*a*b^5*d^2-20/3/e^2*B*x^3*a^3*b^3*d-10/e^2*A*x^2*a^3*b^3*d+15/2/e^3*A*x^2*a^2*b^4*d^2-3/e^4*A*x^2*a
*b^5*d^3-15/2/e^2*B*x^2*a^4*b^2*d+10/e^3*B*x^2*a^3*b^3*d^2-15/2/e^4*B*x^2*a^2*b^4*d^3+3/e^5*B*x^2*a*b^5*d^4-6/
5/e^2*B*x^5*a*b^5*d-3/2/e^2*A*x^4*a*b^5*d+6/e*A*x*a^5*b-1/e^6*A*x*b^6*d^5+1/e^7*ln(e*x+d)*A*b^6*d^6+5/e^3*B*x^
3*a^2*b^4*d^2+15/e^5*B*x*a^2*b^4*d^4+15/e^3*B*x*a^4*b^2*d^2+6/e^5*A*x*a*b^5*d^4+15/2/e*A*x^2*a^4*b^2+1/2/e^5*A
*x^2*b^6*d^4+5/e*B*x^3*a^4*b^2+1/3/e^5*B*x^3*b^6*d^4-1/3/e^4*A*x^3*b^6*d^3+5/e*B*x^4*a^3*b^3-1/4/e^4*B*x^4*b^6
*d^3+1/4/e^3*A*x^4*b^6*d^2+3/e*B*x^5*a^2*b^4-1/e^2*ln(e*x+d)*B*a^6*d-1/e^8*ln(e*x+d)*B*b^6*d^7+1/e^7*B*x*b^6*d
^6+3/e*B*x^2*a^5*b-1/2/e^6*B*x^2*b^6*d^5+20/3/e*A*x^3*a^3*b^3+1/5/e^3*B*x^5*b^6*d^2+15/4/e*A*x^4*a^2*b^4-1/6/e
^2*B*x^6*b^6*d+6/5/e*A*x^5*a*b^5-1/5/e^2*A*x^5*b^6*d+1/e*B*x^6*a*b^5+6/e^3*ln(e*x+d)*B*a^5*b*d^2-15/e^4*ln(e*x
+d)*B*a^4*b^2*d^3+20/e^5*ln(e*x+d)*B*a^3*b^3*d^4-15/e^6*ln(e*x+d)*B*a^2*b^4*d^5+6/e^7*ln(e*x+d)*B*a*b^5*d^6-2/
e^4*B*x^3*a*b^5*d^3-6/e^6*B*x*a*b^5*d^5-15/4/e^2*B*x^4*a^2*b^4*d-20/e^4*B*x*a^3*b^3*d^3-6/e^2*B*x*a^5*b*d+20/e
^3*A*x*a^3*b^3*d^2-15/e^4*A*x*a^2*b^4*d^3-15/e^2*A*x*a^4*b^2*d-6/e^2*ln(e*x+d)*A*a^5*b*d+15/e^3*ln(e*x+d)*A*a^
4*b^2*d^2-20/e^4*ln(e*x+d)*A*a^3*b^3*d^3+15/e^5*ln(e*x+d)*A*a^2*b^4*d^4-6/e^6*ln(e*x+d)*A*a*b^5*d^5

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maxima [B]  time = 0.61, size = 762, normalized size = 3.46 \begin {gather*} \frac {60 \, B b^{6} e^{6} x^{7} - 70 \, {\left (B b^{6} d e^{5} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{6}\right )} x^{6} + 84 \, {\left (B b^{6} d^{2} e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{6}\right )} x^{5} - 105 \, {\left (B b^{6} d^{3} e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{5} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{6}\right )} x^{4} + 140 \, {\left (B b^{6} d^{4} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{6}\right )} x^{3} - 210 \, {\left (B b^{6} d^{5} e - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{5} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{6}\right )} x^{2} + 420 \, {\left (B b^{6} d^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="maxima")

[Out]

1/420*(60*B*b^6*e^6*x^7 - 70*(B*b^6*d*e^5 - (6*B*a*b^5 + A*b^6)*e^6)*x^6 + 84*(B*b^6*d^2*e^4 - (6*B*a*b^5 + A*
b^6)*d*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^6)*x^5 - 105*(B*b^6*d^3*e^3 - (6*B*a*b^5 + A*b^6)*d^2*e^4 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^6)*x^4 + 140*(B*b^6*d^4*e^2 - (6*B*a*b^5 + A*b^6)
*d^3*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*e^6)*x^3 - 210*(B*b^6*d^5*e - (6*B*a*b^5 + A*b^6)*d^4*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^3 - 5*(4*
B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^6)*x^2
+ 420*(B*b^6*d^6 - (6*B*a*b^5 + A*b^6)*d^5*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*
b^4)*d^3*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)
*e^6)*x)/e^7 - (B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^
5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*log(e*x + d)/e^8

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mupad [B]  time = 1.13, size = 769, normalized size = 3.50 \begin {gather*} x\,\left (\frac {B\,a^6+6\,A\,b\,a^5}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{3\,e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{3\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{4\,e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{5\,e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{5\,e}\right )+x^6\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{6\,e}-\frac {B\,b^6\,d}{6\,e^2}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{2\,e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^6\,d\,e^6+A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5-6\,A\,a^5\,b\,d\,e^6-15\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5+20\,B\,a^3\,b^3\,d^4\,e^3-20\,A\,a^3\,b^3\,d^3\,e^4-15\,B\,a^2\,b^4\,d^5\,e^2+15\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e-6\,A\,a\,b^5\,d^5\,e^2-B\,b^6\,d^7+A\,b^6\,d^6\,e\right )}{e^8}+\frac {B\,b^6\,x^7}{7\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x),x)

[Out]

x*((B*a^6 + 6*A*a^5*b)/e - (d*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5
*B*a))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/e - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/e + (3*a^4*b*(5*A*b + 2*B*a
))/e))/e) - x^3*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/e + (5*
a^2*b^3*(3*A*b + 4*B*a))/e))/(3*e) - (5*a^3*b^2*(4*A*b + 3*B*a))/(3*e)) + x^4*((d*((d*((A*b^6 + 6*B*a*b^5)/e -
(B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/(4*e) + (5*a^2*b^3*(3*A*b + 4*B*a))/(4*e)) - x^5*((d*((A*b^
6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/(5*e) - (3*a*b^4*(2*A*b + 5*B*a))/(5*e)) + x^6*((A*b^6 + 6*B*a*b^5)/(6*e) -
(B*b^6*d)/(6*e^2)) + x^2*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a
))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/e - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/(2*e) + (3*a^4*b*(5*A*b + 2*B*a
))/(2*e)) + (log(d + e*x)*(A*a^6*e^7 - B*b^6*d^7 + A*b^6*d^6*e - B*a^6*d*e^6 - 6*A*a*b^5*d^5*e^2 + 6*B*a^5*b*d
^2*e^5 + 15*A*a^2*b^4*d^4*e^3 - 20*A*a^3*b^3*d^3*e^4 + 15*A*a^4*b^2*d^2*e^5 - 15*B*a^2*b^4*d^5*e^2 + 20*B*a^3*
b^3*d^4*e^3 - 15*B*a^4*b^2*d^3*e^4 - 6*A*a^5*b*d*e^6 + 6*B*a*b^5*d^6*e))/e^8 + (B*b^6*x^7)/(7*e)

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sympy [B]  time = 1.66, size = 736, normalized size = 3.35 \begin {gather*} \frac {B b^{6} x^{7}}{7 e} + x^{6} \left (\frac {A b^{6}}{6 e} + \frac {B a b^{5}}{e} - \frac {B b^{6} d}{6 e^{2}}\right ) + x^{5} \left (\frac {6 A a b^{5}}{5 e} - \frac {A b^{6} d}{5 e^{2}} + \frac {3 B a^{2} b^{4}}{e} - \frac {6 B a b^{5} d}{5 e^{2}} + \frac {B b^{6} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {15 A a^{2} b^{4}}{4 e} - \frac {3 A a b^{5} d}{2 e^{2}} + \frac {A b^{6} d^{2}}{4 e^{3}} + \frac {5 B a^{3} b^{3}}{e} - \frac {15 B a^{2} b^{4} d}{4 e^{2}} + \frac {3 B a b^{5} d^{2}}{2 e^{3}} - \frac {B b^{6} d^{3}}{4 e^{4}}\right ) + x^{3} \left (\frac {20 A a^{3} b^{3}}{3 e} - \frac {5 A a^{2} b^{4} d}{e^{2}} + \frac {2 A a b^{5} d^{2}}{e^{3}} - \frac {A b^{6} d^{3}}{3 e^{4}} + \frac {5 B a^{4} b^{2}}{e} - \frac {20 B a^{3} b^{3} d}{3 e^{2}} + \frac {5 B a^{2} b^{4} d^{2}}{e^{3}} - \frac {2 B a b^{5} d^{3}}{e^{4}} + \frac {B b^{6} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {15 A a^{4} b^{2}}{2 e} - \frac {10 A a^{3} b^{3} d}{e^{2}} + \frac {15 A a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 A a b^{5} d^{3}}{e^{4}} + \frac {A b^{6} d^{4}}{2 e^{5}} + \frac {3 B a^{5} b}{e} - \frac {15 B a^{4} b^{2} d}{2 e^{2}} + \frac {10 B a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 B a^{2} b^{4} d^{3}}{2 e^{4}} + \frac {3 B a b^{5} d^{4}}{e^{5}} - \frac {B b^{6} d^{5}}{2 e^{6}}\right ) + x \left (\frac {6 A a^{5} b}{e} - \frac {15 A a^{4} b^{2} d}{e^{2}} + \frac {20 A a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 A a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 A a b^{5} d^{4}}{e^{5}} - \frac {A b^{6} d^{5}}{e^{6}} + \frac {B a^{6}}{e} - \frac {6 B a^{5} b d}{e^{2}} + \frac {15 B a^{4} b^{2} d^{2}}{e^{3}} - \frac {20 B a^{3} b^{3} d^{3}}{e^{4}} + \frac {15 B a^{2} b^{4} d^{4}}{e^{5}} - \frac {6 B a b^{5} d^{5}}{e^{6}} + \frac {B b^{6} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d),x)

[Out]

B*b**6*x**7/(7*e) + x**6*(A*b**6/(6*e) + B*a*b**5/e - B*b**6*d/(6*e**2)) + x**5*(6*A*a*b**5/(5*e) - A*b**6*d/(
5*e**2) + 3*B*a**2*b**4/e - 6*B*a*b**5*d/(5*e**2) + B*b**6*d**2/(5*e**3)) + x**4*(15*A*a**2*b**4/(4*e) - 3*A*a
*b**5*d/(2*e**2) + A*b**6*d**2/(4*e**3) + 5*B*a**3*b**3/e - 15*B*a**2*b**4*d/(4*e**2) + 3*B*a*b**5*d**2/(2*e**
3) - B*b**6*d**3/(4*e**4)) + x**3*(20*A*a**3*b**3/(3*e) - 5*A*a**2*b**4*d/e**2 + 2*A*a*b**5*d**2/e**3 - A*b**6
*d**3/(3*e**4) + 5*B*a**4*b**2/e - 20*B*a**3*b**3*d/(3*e**2) + 5*B*a**2*b**4*d**2/e**3 - 2*B*a*b**5*d**3/e**4
+ B*b**6*d**4/(3*e**5)) + x**2*(15*A*a**4*b**2/(2*e) - 10*A*a**3*b**3*d/e**2 + 15*A*a**2*b**4*d**2/(2*e**3) -
3*A*a*b**5*d**3/e**4 + A*b**6*d**4/(2*e**5) + 3*B*a**5*b/e - 15*B*a**4*b**2*d/(2*e**2) + 10*B*a**3*b**3*d**2/e
**3 - 15*B*a**2*b**4*d**3/(2*e**4) + 3*B*a*b**5*d**4/e**5 - B*b**6*d**5/(2*e**6)) + x*(6*A*a**5*b/e - 15*A*a**
4*b**2*d/e**2 + 20*A*a**3*b**3*d**2/e**3 - 15*A*a**2*b**4*d**3/e**4 + 6*A*a*b**5*d**4/e**5 - A*b**6*d**5/e**6
+ B*a**6/e - 6*B*a**5*b*d/e**2 + 15*B*a**4*b**2*d**2/e**3 - 20*B*a**3*b**3*d**3/e**4 + 15*B*a**2*b**4*d**4/e**
5 - 6*B*a*b**5*d**5/e**6 + B*b**6*d**6/e**7) - (-A*e + B*d)*(a*e - b*d)**6*log(d + e*x)/e**8

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