3.10.22 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=277 \[ -\frac {b^5 (d+e x)^5 (-6 a B e-A b e+7 b B d)}{5 e^8}+\frac {3 b^4 (d+e x)^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{4 e^8}-\frac {5 b^3 (d+e x)^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {5 b^2 (d+e x)^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8}+\frac {(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}+\frac {(b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)}{e^8}-\frac {3 b x (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^7}+\frac {b^6 B (d+e x)^6}{6 e^8} \]

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Rubi [A]  time = 0.59, antiderivative size = 277, 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 {5 b^2 (d+e x)^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8}-\frac {5 b^3 (d+e x)^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {3 b^4 (d+e x)^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{4 e^8}-\frac {b^5 (d+e x)^5 (-6 a B e-A b e+7 b B d)}{5 e^8}+\frac {(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}-\frac {3 b x (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^7}+\frac {(b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)}{e^8}+\frac {b^6 B (d+e x)^6}{6 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^2,x]

[Out]

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

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Mathematica [B]  time = 0.29, size = 643, normalized size = 2.32 \begin {gather*} \frac {60 a^6 e^6 (B d-A e)+360 a^5 b e^5 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+450 a^4 b^2 e^4 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+200 a^3 b^3 e^3 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+75 a^2 b^4 e^2 \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+6 a b^5 e \left (5 A e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )+60 (d+e x) (b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)+b^6 \left (6 A e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+B \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )}{60 e^8 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^2,x]

[Out]

(60*a^6*e^6*(B*d - A*e) + 360*a^5*b*e^5*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 450*a^4*b^2*e^4*(2*A*e*(-d^2 +
d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 200*a^3*b^3*e^3*(3*A*e*(2*d^3 - 4*d^2*e*x
- 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 75*a^2*b^4*e^2*
(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2
+ 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 6*a*b^5*e*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2
*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x
^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + b^6*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^
4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4
+ 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 60*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)*Log[d
+ e*x])/(60*e^8*(d + e*x))

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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)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^2,x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^2, x]

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fricas [B]  time = 1.56, size = 1067, normalized size = 3.85

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/60*(10*B*b^6*e^7*x^7 + 60*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5 + A*b^6)*d^6*e + 180*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^5*e^2 - 300*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 300*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 180*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 60*(B*a^6 + 6*A*a^5*b)*d*e^6 - 2*(7*B*b^6*d*e^6 - 6*(6*B*a*b^5 + A*b^6)*e^7)*x^6
+ 3*(7*B*b^6*d^2*e^5 - 6*(6*B*a*b^5 + A*b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 5*(7*B*b^6*d^3*e
^4 - 6*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*
x^4 + 10*(7*B*b^6*d^4*e^3 - 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 20*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 30*(7*B*b^6*d^5*e^2 - 6*(6*B*a*b^5 + A*b
^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^6 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 - 60*(6*B*b^6*d^6*e - 5*(6*B*a*b^5 + A*b^6)*d^5*e^
2 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 10*(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)*x + 60*(7*B*b^6*d^7 - 6*(6*B*a*b^5 + A*b^6)*d^6*e + 15*(5*B*
a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4
- 6*(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 + (7*B*b^6*d^6*e - 6*(6*B*a*b^5 + A*b^6)*d^
5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*
a^3*b^3)*d^2*e^5 - 6*(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)*log(e*x + d))/(e^9*x + d*e^
8)

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giac [B]  time = 1.25, size = 936, normalized size = 3.38 \begin {gather*} \frac {1}{60} \, {\left (10 \, B b^{6} - \frac {12 \, {\left (7 \, B b^{6} d e - 6 \, B a b^{5} e^{2} - A b^{6} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {45 \, {\left (7 \, B b^{6} d^{2} e^{2} - 12 \, B a b^{5} d e^{3} - 2 \, A b^{6} d e^{3} + 5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {100 \, {\left (7 \, B b^{6} d^{3} e^{3} - 18 \, B a b^{5} d^{2} e^{4} - 3 \, A b^{6} d^{2} e^{4} + 15 \, B a^{2} b^{4} d e^{5} + 6 \, A a b^{5} d e^{5} - 4 \, B a^{3} b^{3} e^{6} - 3 \, A a^{2} b^{4} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {150 \, {\left (7 \, B b^{6} d^{4} e^{4} - 24 \, B a b^{5} d^{3} e^{5} - 4 \, A b^{6} d^{3} e^{5} + 30 \, B a^{2} b^{4} d^{2} e^{6} + 12 \, A a b^{5} d^{2} e^{6} - 16 \, B a^{3} b^{3} d e^{7} - 12 \, A a^{2} b^{4} d e^{7} + 3 \, B a^{4} b^{2} e^{8} + 4 \, A a^{3} b^{3} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {180 \, {\left (7 \, B b^{6} d^{5} e^{5} - 30 \, B a b^{5} d^{4} e^{6} - 5 \, A b^{6} d^{4} e^{6} + 50 \, B a^{2} b^{4} d^{3} e^{7} + 20 \, A a b^{5} d^{3} e^{7} - 40 \, B a^{3} b^{3} d^{2} e^{8} - 30 \, A a^{2} b^{4} d^{2} e^{8} + 15 \, B a^{4} b^{2} d e^{9} + 20 \, A a^{3} b^{3} d e^{9} - 2 \, B a^{5} b e^{10} - 5 \, A a^{4} b^{2} e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )} {\left (x e + d\right )}^{6} e^{\left (-8\right )} - {\left (7 \, B b^{6} d^{6} - 36 \, B a b^{5} d^{5} e - 6 \, A b^{6} d^{5} e + 75 \, B a^{2} b^{4} d^{4} e^{2} + 30 \, A a b^{5} d^{4} e^{2} - 80 \, B a^{3} b^{3} d^{3} e^{3} - 60 \, A a^{2} b^{4} d^{3} e^{3} + 45 \, B a^{4} b^{2} d^{2} e^{4} + 60 \, A a^{3} b^{3} d^{2} e^{4} - 12 \, B a^{5} b d e^{5} - 30 \, A a^{4} b^{2} d e^{5} + B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} e^{\left (-8\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B b^{6} d^{7} e^{6}}{x e + d} - \frac {6 \, B a b^{5} d^{6} e^{7}}{x e + d} - \frac {A b^{6} d^{6} e^{7}}{x e + d} + \frac {15 \, B a^{2} b^{4} d^{5} e^{8}}{x e + d} + \frac {6 \, A a b^{5} d^{5} e^{8}}{x e + d} - \frac {20 \, B a^{3} b^{3} d^{4} e^{9}}{x e + d} - \frac {15 \, A a^{2} b^{4} d^{4} e^{9}}{x e + d} + \frac {15 \, B a^{4} b^{2} d^{3} e^{10}}{x e + d} + \frac {20 \, A a^{3} b^{3} d^{3} e^{10}}{x e + d} - \frac {6 \, B a^{5} b d^{2} e^{11}}{x e + d} - \frac {15 \, A a^{4} b^{2} d^{2} e^{11}}{x e + d} + \frac {B a^{6} d e^{12}}{x e + d} + \frac {6 \, A a^{5} b d e^{12}}{x e + d} - \frac {A a^{6} e^{13}}{x e + d}\right )} e^{\left (-14\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)^2,x, algorithm="giac")

[Out]

1/60*(10*B*b^6 - 12*(7*B*b^6*d*e - 6*B*a*b^5*e^2 - A*b^6*e^2)*e^(-1)/(x*e + d) + 45*(7*B*b^6*d^2*e^2 - 12*B*a*
b^5*d*e^3 - 2*A*b^6*d*e^3 + 5*B*a^2*b^4*e^4 + 2*A*a*b^5*e^4)*e^(-2)/(x*e + d)^2 - 100*(7*B*b^6*d^3*e^3 - 18*B*
a*b^5*d^2*e^4 - 3*A*b^6*d^2*e^4 + 15*B*a^2*b^4*d*e^5 + 6*A*a*b^5*d*e^5 - 4*B*a^3*b^3*e^6 - 3*A*a^2*b^4*e^6)*e^
(-3)/(x*e + d)^3 + 150*(7*B*b^6*d^4*e^4 - 24*B*a*b^5*d^3*e^5 - 4*A*b^6*d^3*e^5 + 30*B*a^2*b^4*d^2*e^6 + 12*A*a
*b^5*d^2*e^6 - 16*B*a^3*b^3*d*e^7 - 12*A*a^2*b^4*d*e^7 + 3*B*a^4*b^2*e^8 + 4*A*a^3*b^3*e^8)*e^(-4)/(x*e + d)^4
- 180*(7*B*b^6*d^5*e^5 - 30*B*a*b^5*d^4*e^6 - 5*A*b^6*d^4*e^6 + 50*B*a^2*b^4*d^3*e^7 + 20*A*a*b^5*d^3*e^7 - 4
0*B*a^3*b^3*d^2*e^8 - 30*A*a^2*b^4*d^2*e^8 + 15*B*a^4*b^2*d*e^9 + 20*A*a^3*b^3*d*e^9 - 2*B*a^5*b*e^10 - 5*A*a^
4*b^2*e^10)*e^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - (7*B*b^6*d^6 - 36*B*a*b^5*d^5*e - 6*A*b^6*d^5*e + 75*B*a^
2*b^4*d^4*e^2 + 30*A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 - 60*A*a^2*b^4*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 + 60*A
*a^3*b^3*d^2*e^4 - 12*B*a^5*b*d*e^5 - 30*A*a^4*b^2*d*e^5 + B*a^6*e^6 + 6*A*a^5*b*e^6)*e^(-8)*log(abs(x*e + d)*
e^(-1)/(x*e + d)^2) + (B*b^6*d^7*e^6/(x*e + d) - 6*B*a*b^5*d^6*e^7/(x*e + d) - A*b^6*d^6*e^7/(x*e + d) + 15*B*
a^2*b^4*d^5*e^8/(x*e + d) + 6*A*a*b^5*d^5*e^8/(x*e + d) - 20*B*a^3*b^3*d^4*e^9/(x*e + d) - 15*A*a^2*b^4*d^4*e^
9/(x*e + d) + 15*B*a^4*b^2*d^3*e^10/(x*e + d) + 20*A*a^3*b^3*d^3*e^10/(x*e + d) - 6*B*a^5*b*d^2*e^11/(x*e + d)
- 15*A*a^4*b^2*d^2*e^11/(x*e + d) + B*a^6*d*e^12/(x*e + d) + 6*A*a^5*b*d*e^12/(x*e + d) - A*a^6*e^13/(x*e + d
))*e^(-14)

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maple [B]  time = 0.02, size = 1047, normalized size = 3.78 \begin {gather*} \frac {B \,b^{6} x^{6}}{6 e^{2}}+\frac {A \,b^{6} x^{5}}{5 e^{2}}+\frac {6 B a \,b^{5} x^{5}}{5 e^{2}}-\frac {2 B \,b^{6} d \,x^{5}}{5 e^{3}}+\frac {3 A a \,b^{5} x^{4}}{2 e^{2}}-\frac {A \,b^{6} d \,x^{4}}{2 e^{3}}+\frac {15 B \,a^{2} b^{4} x^{4}}{4 e^{2}}-\frac {3 B a \,b^{5} d \,x^{4}}{e^{3}}+\frac {3 B \,b^{6} d^{2} x^{4}}{4 e^{4}}+\frac {5 A \,a^{2} b^{4} x^{3}}{e^{2}}-\frac {4 A a \,b^{5} d \,x^{3}}{e^{3}}+\frac {A \,b^{6} d^{2} x^{3}}{e^{4}}+\frac {20 B \,a^{3} b^{3} x^{3}}{3 e^{2}}-\frac {10 B \,a^{2} b^{4} d \,x^{3}}{e^{3}}+\frac {6 B a \,b^{5} d^{2} x^{3}}{e^{4}}-\frac {4 B \,b^{6} d^{3} x^{3}}{3 e^{5}}+\frac {10 A \,a^{3} b^{3} x^{2}}{e^{2}}-\frac {15 A \,a^{2} b^{4} d \,x^{2}}{e^{3}}+\frac {9 A a \,b^{5} d^{2} x^{2}}{e^{4}}-\frac {2 A \,b^{6} d^{3} x^{2}}{e^{5}}+\frac {15 B \,a^{4} b^{2} x^{2}}{2 e^{2}}-\frac {20 B \,a^{3} b^{3} d \,x^{2}}{e^{3}}+\frac {45 B \,a^{2} b^{4} d^{2} x^{2}}{2 e^{4}}-\frac {12 B a \,b^{5} d^{3} x^{2}}{e^{5}}+\frac {5 B \,b^{6} d^{4} x^{2}}{2 e^{6}}-\frac {A \,a^{6}}{\left (e x +d \right ) e}+\frac {6 A \,a^{5} b d}{\left (e x +d \right ) e^{2}}+\frac {6 A \,a^{5} b \ln \left (e x +d \right )}{e^{2}}-\frac {15 A \,a^{4} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {30 A \,a^{4} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {15 A \,a^{4} b^{2} x}{e^{2}}+\frac {20 A \,a^{3} b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {60 A \,a^{3} b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {40 A \,a^{3} b^{3} d x}{e^{3}}-\frac {15 A \,a^{2} b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {60 A \,a^{2} b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {45 A \,a^{2} b^{4} d^{2} x}{e^{4}}+\frac {6 A a \,b^{5} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {30 A a \,b^{5} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {24 A a \,b^{5} d^{3} x}{e^{5}}-\frac {A \,b^{6} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 A \,b^{6} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 A \,b^{6} d^{4} x}{e^{6}}+\frac {B \,a^{6} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{6} \ln \left (e x +d \right )}{e^{2}}-\frac {6 B \,a^{5} b \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {12 B \,a^{5} b d \ln \left (e x +d \right )}{e^{3}}+\frac {6 B \,a^{5} b x}{e^{2}}+\frac {15 B \,a^{4} b^{2} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {45 B \,a^{4} b^{2} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {30 B \,a^{4} b^{2} d x}{e^{3}}-\frac {20 B \,a^{3} b^{3} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {80 B \,a^{3} b^{3} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {60 B \,a^{3} b^{3} d^{2} x}{e^{4}}+\frac {15 B \,a^{2} b^{4} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {75 B \,a^{2} b^{4} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {60 B \,a^{2} b^{4} d^{3} x}{e^{5}}-\frac {6 B a \,b^{5} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {36 B a \,b^{5} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {30 B a \,b^{5} d^{4} x}{e^{6}}+\frac {B \,b^{6} d^{7}}{\left (e x +d \right ) e^{8}}+\frac {7 B \,b^{6} d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {6 B \,b^{6} d^{5} 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)^2,x)

[Out]

1/e^2*ln(e*x+d)*B*a^6+1/6*b^6/e^2*B*x^6+1/5*b^6/e^2*A*x^5-1/e/(e*x+d)*A*a^6-36/e^7*ln(e*x+d)*B*a*b^5*d^5+15/e^
6/(e*x+d)*B*a^2*b^4*d^5-6/e^7/(e*x+d)*B*a*b^5*d^6-3*b^5/e^3*B*x^4*a*d+6/e^2/(e*x+d)*A*d*a^5*b-15/e^3/(e*x+d)*A
*a^4*b^2*d^2+20/e^4/(e*x+d)*A*a^3*b^3*d^3-15/e^5/(e*x+d)*A*a^2*b^4*d^4+6/e^6/(e*x+d)*A*a*b^5*d^5-60*b^4/e^5*B*
a^2*d^3*x+30*b^5/e^6*B*a*d^4*x+15/4*b^4/e^2*B*x^4*a^2+3/4*b^6/e^4*B*x^4*d^2-10*b^4/e^3*B*x^3*a^2*d-40*b^3/e^3*
A*a^3*d*x+45/2*b^4/e^4*B*x^2*a^2*d^2-12*b^5/e^5*B*x^2*a*d^3+45*b^4/e^4*A*a^2*d^2*x-24*b^5/e^5*A*a*d^3*x-30*b^2
/e^3*B*a^4*d*x+60*b^3/e^4*B*a^3*d^2*x-4*b^5/e^3*A*x^3*a*d-2/5*b^6/e^3*B*x^5*d+3/2*b^5/e^2*A*x^4*a+6/5*b^5/e^2*
B*x^5*a-2*b^6/e^5*A*x^2*d^3+15/2*b^2/e^2*B*x^2*a^4+5/2*b^6/e^6*B*x^2*d^4+15*b^2/e^2*A*a^4*x+5*b^6/e^6*A*d^4*x-
1/2*b^6/e^3*A*x^4*d-6*b^6/e^7*B*d^5*x+b^6/e^4*A*x^3*d^2+5*b^4/e^2*A*x^3*a^2+7/e^8*ln(e*x+d)*B*b^6*d^6-6/e^7*ln
(e*x+d)*A*b^6*d^5-1/e^7/(e*x+d)*A*b^6*d^6+1/e^8/(e*x+d)*B*b^6*d^7+1/e^2/(e*x+d)*B*d*a^6+6/e^2*ln(e*x+d)*A*a^5*
b+6*b/e^2*B*a^5*x+20/3*b^3/e^2*B*x^3*a^3-4/3*b^6/e^5*B*x^3*d^3+10*b^3/e^2*A*x^2*a^3-20*b^3/e^3*B*x^2*a^3*d+6*b
^5/e^4*B*x^3*a*d^2-15*b^4/e^3*A*x^2*a^2*d+9*b^5/e^4*A*x^2*a*d^2-60/e^5*ln(e*x+d)*A*a^2*b^4*d^3+30/e^6*ln(e*x+d
)*A*a*b^5*d^4-12/e^3*ln(e*x+d)*B*a^5*b*d+45/e^4*ln(e*x+d)*B*a^4*b^2*d^2-80/e^5*ln(e*x+d)*B*a^3*b^3*d^3+75/e^6*
ln(e*x+d)*B*a^2*b^4*d^4-6/e^3/(e*x+d)*B*a^5*b*d^2+15/e^4/(e*x+d)*B*a^4*b^2*d^3-20/e^5/(e*x+d)*B*a^3*b^3*d^4-30
/e^3*ln(e*x+d)*A*a^4*b^2*d+60/e^4*ln(e*x+d)*A*a^3*b^3*d^2

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maxima [B]  time = 0.61, size = 771, normalized size = 2.78 \begin {gather*} \frac {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}}{e^{9} x + d e^{8}} + \frac {10 \, B b^{6} e^{5} x^{6} - 12 \, {\left (2 \, B b^{6} d e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{5}\right )} x^{5} + 15 \, {\left (3 \, B b^{6} d^{2} e^{3} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{5}\right )} x^{4} - 20 \, {\left (4 \, B b^{6} d^{3} e^{2} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{6} d^{4} e - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{2} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 60 \, {\left (6 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{60 \, e^{7}} + \frac {{\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\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 )} \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)^2,x, algorithm="maxima")

[Out]

(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)/(e^9*x + d*e^8) + 1/60*(10*B*b^6*e^5*x^6 - 12*(2*B*b^6*d*e^4 - (6*B*a*b^5 + A*b^6)*e^5)*x^5 +
15*(3*B*b^6*d^2*e^3 - 2*(6*B*a*b^5 + A*b^6)*d*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^5)*x^4 - 20*(4*B*b^6*d^3*e^2
- 3*(6*B*a*b^5 + A*b^6)*d^2*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^4 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3
+ 30*(5*B*b^6*d^4*e - 4*(6*B*a*b^5 + A*b^6)*d^3*e^2 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^3 - 10*(4*B*a^3*b^3 +
3*A*a^2*b^4)*d*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^5)*x^2 - 60*(6*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e +
12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*d*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^5)*x)/e^7 + (7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(
2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*log(e*x + d)/e^8

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mupad [B]  time = 1.17, size = 1228, normalized size = 4.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^2,x)

[Out]

x^2*((d^2*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e
^4))/(2*e^2) - (d*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 +
(B*b^6*d^2)/e^4))/e - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e^2 + (5*a^2*b^3*(3*A*b + 4*B*a))/e^2
))/e + (5*a^3*b^2*(4*A*b + 3*B*a))/(2*e^2)) - x^4*((d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/(2*e) - (3*
a*b^4*(2*A*b + 5*B*a))/(4*e^2) + (B*b^6*d^2)/(4*e^4)) - x*((d^2*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^
6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/e - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*
d)/e^3))/e^2 + (5*a^2*b^3*(3*A*b + 4*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)
/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 -
(2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/e - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (
2*B*b^6*d)/e^3))/e^2 + (5*a^2*b^3*(3*A*b + 4*B*a))/e^2))/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))/e^2) + x^3*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a
))/e^2 + (B*b^6*d^2)/e^4))/(3*e) - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/(3*e^2) + (5*a^2*b^3*(3*A
*b + 4*B*a))/(3*e^2)) + x^5*((A*b^6 + 6*B*a*b^5)/(5*e^2) - (2*B*b^6*d)/(5*e^3)) + (log(d + e*x)*(B*a^6*e^6 + 7
*B*b^6*d^6 + 6*A*a^5*b*e^6 - 6*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 - 30*A*a^4*b^2*d*e^5 - 60*A*a^2*b^4*d^3*e^3 +
60*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 - 36*B*a*b^5*d^5*e -
12*B*a^5*b*d*e^5))/e^8 - (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*(d*e^7 + e^8*x)) + (B*b^6*x^6)/(6*e
^2)

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sympy [B]  time = 4.06, size = 782, normalized size = 2.82 \begin {gather*} \frac {B b^{6} x^{6}}{6 e^{2}} + x^{5} \left (\frac {A b^{6}}{5 e^{2}} + \frac {6 B a b^{5}}{5 e^{2}} - \frac {2 B b^{6} d}{5 e^{3}}\right ) + x^{4} \left (\frac {3 A a b^{5}}{2 e^{2}} - \frac {A b^{6} d}{2 e^{3}} + \frac {15 B a^{2} b^{4}}{4 e^{2}} - \frac {3 B a b^{5} d}{e^{3}} + \frac {3 B b^{6} d^{2}}{4 e^{4}}\right ) + x^{3} \left (\frac {5 A a^{2} b^{4}}{e^{2}} - \frac {4 A a b^{5} d}{e^{3}} + \frac {A b^{6} d^{2}}{e^{4}} + \frac {20 B a^{3} b^{3}}{3 e^{2}} - \frac {10 B a^{2} b^{4} d}{e^{3}} + \frac {6 B a b^{5} d^{2}}{e^{4}} - \frac {4 B b^{6} d^{3}}{3 e^{5}}\right ) + x^{2} \left (\frac {10 A a^{3} b^{3}}{e^{2}} - \frac {15 A a^{2} b^{4} d}{e^{3}} + \frac {9 A a b^{5} d^{2}}{e^{4}} - \frac {2 A b^{6} d^{3}}{e^{5}} + \frac {15 B a^{4} b^{2}}{2 e^{2}} - \frac {20 B a^{3} b^{3} d}{e^{3}} + \frac {45 B a^{2} b^{4} d^{2}}{2 e^{4}} - \frac {12 B a b^{5} d^{3}}{e^{5}} + \frac {5 B b^{6} d^{4}}{2 e^{6}}\right ) + x \left (\frac {15 A a^{4} b^{2}}{e^{2}} - \frac {40 A a^{3} b^{3} d}{e^{3}} + \frac {45 A a^{2} b^{4} d^{2}}{e^{4}} - \frac {24 A a b^{5} d^{3}}{e^{5}} + \frac {5 A b^{6} d^{4}}{e^{6}} + \frac {6 B a^{5} b}{e^{2}} - \frac {30 B a^{4} b^{2} d}{e^{3}} + \frac {60 B a^{3} b^{3} d^{2}}{e^{4}} - \frac {60 B a^{2} b^{4} d^{3}}{e^{5}} + \frac {30 B a b^{5} d^{4}}{e^{6}} - \frac {6 B b^{6} d^{5}}{e^{7}}\right ) + \frac {- A a^{6} e^{7} + 6 A a^{5} b d e^{6} - 15 A a^{4} b^{2} d^{2} e^{5} + 20 A a^{3} b^{3} d^{3} e^{4} - 15 A a^{2} b^{4} d^{4} e^{3} + 6 A a b^{5} d^{5} e^{2} - A b^{6} d^{6} e + B a^{6} d e^{6} - 6 B a^{5} b d^{2} e^{5} + 15 B a^{4} b^{2} d^{3} e^{4} - 20 B a^{3} b^{3} d^{4} e^{3} + 15 B a^{2} b^{4} d^{5} e^{2} - 6 B a b^{5} d^{6} e + B b^{6} d^{7}}{d e^{8} + e^{9} x} + \frac {\left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right ) \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)**2,x)

[Out]

B*b**6*x**6/(6*e**2) + x**5*(A*b**6/(5*e**2) + 6*B*a*b**5/(5*e**2) - 2*B*b**6*d/(5*e**3)) + x**4*(3*A*a*b**5/(
2*e**2) - A*b**6*d/(2*e**3) + 15*B*a**2*b**4/(4*e**2) - 3*B*a*b**5*d/e**3 + 3*B*b**6*d**2/(4*e**4)) + x**3*(5*
A*a**2*b**4/e**2 - 4*A*a*b**5*d/e**3 + A*b**6*d**2/e**4 + 20*B*a**3*b**3/(3*e**2) - 10*B*a**2*b**4*d/e**3 + 6*
B*a*b**5*d**2/e**4 - 4*B*b**6*d**3/(3*e**5)) + x**2*(10*A*a**3*b**3/e**2 - 15*A*a**2*b**4*d/e**3 + 9*A*a*b**5*
d**2/e**4 - 2*A*b**6*d**3/e**5 + 15*B*a**4*b**2/(2*e**2) - 20*B*a**3*b**3*d/e**3 + 45*B*a**2*b**4*d**2/(2*e**4
) - 12*B*a*b**5*d**3/e**5 + 5*B*b**6*d**4/(2*e**6)) + x*(15*A*a**4*b**2/e**2 - 40*A*a**3*b**3*d/e**3 + 45*A*a*
*2*b**4*d**2/e**4 - 24*A*a*b**5*d**3/e**5 + 5*A*b**6*d**4/e**6 + 6*B*a**5*b/e**2 - 30*B*a**4*b**2*d/e**3 + 60*
B*a**3*b**3*d**2/e**4 - 60*B*a**2*b**4*d**3/e**5 + 30*B*a*b**5*d**4/e**6 - 6*B*b**6*d**5/e**7) + (-A*a**6*e**7
+ 6*A*a**5*b*d*e**6 - 15*A*a**4*b**2*d**2*e**5 + 20*A*a**3*b**3*d**3*e**4 - 15*A*a**2*b**4*d**4*e**3 + 6*A*a*
b**5*d**5*e**2 - A*b**6*d**6*e + B*a**6*d*e**6 - 6*B*a**5*b*d**2*e**5 + 15*B*a**4*b**2*d**3*e**4 - 20*B*a**3*b
**3*d**4*e**3 + 15*B*a**2*b**4*d**5*e**2 - 6*B*a*b**5*d**6*e + B*b**6*d**7)/(d*e**8 + e**9*x) + (a*e - b*d)**5
*(6*A*b*e + B*a*e - 7*B*b*d)*log(d + e*x)/e**8

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