3.11.73 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{14}} \, dx\) [1073]

Optimal. Leaf size=292 \[ \frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{12 e^8 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{11 e^8 (d+e x)^{11}}-\frac {b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^{10}}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{9 e^8 (d+e x)^9}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{8 e^8 (d+e x)^8}+\frac {b^5 (7 b B d-A b e-6 a B e)}{7 e^8 (d+e x)^7}-\frac {b^6 B}{6 e^8 (d+e x)^6} \]

[Out]

1/13*(-a*e+b*d)^6*(-A*e+B*d)/e^8/(e*x+d)^13-1/12*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)/e^8/(e*x+d)^12+3/11*b*(
-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)/e^8/(e*x+d)^11-1/2*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)/e^8/(e*x
+d)^10+5/9*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)/e^8/(e*x+d)^9-3/8*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*
b*d)/e^8/(e*x+d)^8+1/7*b^5*(-A*b*e-6*B*a*e+7*B*b*d)/e^8/(e*x+d)^7-1/6*b^6*B/e^8/(e*x+d)^6

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Rubi [A]
time = 0.18, antiderivative size = 292, 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^5 (-6 a B e-A b e+7 b B d)}{7 e^8 (d+e x)^7}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{8 e^8 (d+e x)^8}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8 (d+e x)^9}-\frac {b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8 (d+e x)^{10}}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{12 e^8 (d+e x)^{12}}+\frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {b^6 B}{6 e^8 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]

[Out]

((b*d - a*e)^6*(B*d - A*e))/(13*e^8*(d + e*x)^13) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(12*e^8*(d + e
*x)^12) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(11*e^8*(d + e*x)^11) - (b^2*(b*d - a*e)^3*(7*b*B*
d - 4*A*b*e - 3*a*B*e))/(2*e^8*(d + e*x)^10) + (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e))/(9*e^8*(d +
e*x)^9) - (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e))/(8*e^8*(d + e*x)^8) + (b^5*(7*b*B*d - A*b*e - 6*a
*B*e))/(7*e^8*(d + e*x)^7) - (b^6*B)/(6*e^8*(d + e*x)^6)

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)^6 (A+B x)}{(d+e x)^{14}} \, dx &=\int \left (\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{14}}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{13}}+\frac {3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^{12}}-\frac {5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)^{11}}+\frac {5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7 (d+e x)^{10}}-\frac {3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e)}{e^7 (d+e x)^9}+\frac {b^5 (-7 b B d+A b e+6 a B e)}{e^7 (d+e x)^8}+\frac {b^6 B}{e^7 (d+e x)^7}\right ) \, dx\\ &=\frac {(b d-a e)^6 (B d-A e)}{13 e^8 (d+e x)^{13}}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{12 e^8 (d+e x)^{12}}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{11 e^8 (d+e x)^{11}}-\frac {b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{2 e^8 (d+e x)^{10}}+\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e)}{9 e^8 (d+e x)^9}-\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e)}{8 e^8 (d+e x)^8}+\frac {b^5 (7 b B d-A b e-6 a B e)}{7 e^8 (d+e x)^7}-\frac {b^6 B}{6 e^8 (d+e x)^6}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(292)=584\).
time = 0.20, size = 605, normalized size = 2.07 \begin {gather*} -\frac {462 a^6 e^6 (12 A e+B (d+13 e x))+252 a^5 b e^5 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+126 a^4 b^2 e^4 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+56 a^3 b^3 e^3 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+21 a^2 b^4 e^2 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a b^5 e \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+b^6 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )}{72072 e^8 (d+e x)^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^14,x]

[Out]

-1/72072*(462*a^6*e^6*(12*A*e + B*(d + 13*e*x)) + 252*a^5*b*e^5*(11*A*e*(d + 13*e*x) + 2*B*(d^2 + 13*d*e*x + 7
8*e^2*x^2)) + 126*a^4*b^2*e^4*(10*A*e*(d^2 + 13*d*e*x + 78*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 2
86*e^3*x^3)) + 56*a^3*b^3*e^3*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13*d^3*e*x +
78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 21*a^2*b^4*e^2*(8*A*e*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 2
86*d*e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*
e^5*x^5)) + 6*a*b^5*e*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x
^5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x
^6)) + b^6*(6*A*e*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 17
16*e^6*x^6) + 7*B*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d^3*e^4*x^4 + 1287*d^2*e^5*x^5 +
1716*d*e^6*x^6 + 1716*e^7*x^7)))/(e^8*(d + e*x)^13)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(813\) vs. \(2(276)=552\).
time = 0.09, size = 814, normalized size = 2.79 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^14,x,method=_RETURNVERBOSE)

[Out]

-1/13*(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)/e^8/(e*x+d)^13-3/8*b^4/e^8*(2*A*a*b*e^2-2*A*b^2*d*e+5*B*a^2*e^2-12*B*a*b*d*e+7*B*b^2
*d^2)/(e*x+d)^8-5/9*b^3/e^8*(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^3-15*B*a^2*b*d*e^2+18*B*a*b
^2*d^2*e-7*B*b^3*d^3)/(e*x+d)^9-1/7*b^5/e^8*(A*b*e+6*B*a*e-7*B*b*d)/(e*x+d)^7-1/6*b^6*B/e^8/(e*x+d)^6-1/12/e^8
*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+
B*a^6*e^6-12*B*a^5*b*d*e^5+45*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7*B
*b^6*d^6)/(e*x+d)^12-1/2*b^2/e^8*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+3*B*a^4*e^
4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a*b^3*d^3*e+7*B*b^4*d^4)/(e*x+d)^10-3/11*b/e^8*(5*A*a^4*b*e^5-20*
A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3*b^
2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/(e*x+d)^11

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (295) = 590\).
time = 0.37, size = 898, normalized size = 3.08 \begin {gather*} -\frac {12012 \, B b^{6} x^{7} e^{7} + 7 \, B b^{6} d^{7} + 5544 \, A a^{6} e^{7} + 6 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 1716 \, {\left (7 \, B b^{6} d e^{6} + 36 \, B a b^{5} e^{7} + 6 \, A b^{6} e^{7}\right )} x^{6} + 21 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 1287 \, {\left (7 \, B b^{6} d^{2} e^{5} + 105 \, B a^{2} b^{4} e^{7} + 42 \, A a b^{5} e^{7} + 6 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + 56 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 715 \, {\left (7 \, B b^{6} d^{3} e^{4} + 224 \, B a^{3} b^{3} e^{7} + 168 \, A a^{2} b^{4} e^{7} + 6 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 21 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + 126 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 286 \, {\left (7 \, B b^{6} d^{4} e^{3} + 378 \, B a^{4} b^{2} e^{7} + 504 \, A a^{3} b^{3} e^{7} + 6 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 21 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 56 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 78 \, {\left (7 \, B b^{6} d^{5} e^{2} + 504 \, B a^{5} b e^{7} + 1260 \, A a^{4} b^{2} e^{7} + 6 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 21 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 56 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 126 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + 462 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 13 \, {\left (7 \, B b^{6} d^{6} e + 462 \, B a^{6} e^{7} + 2772 \, A a^{5} b e^{7} + 6 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 21 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 56 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 126 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 252 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{72072 \, {\left (x^{13} e^{21} + 13 \, d x^{12} e^{20} + 78 \, d^{2} x^{11} e^{19} + 286 \, d^{3} x^{10} e^{18} + 715 \, d^{4} x^{9} e^{17} + 1287 \, d^{5} x^{8} e^{16} + 1716 \, d^{6} x^{7} e^{15} + 1716 \, d^{7} x^{6} e^{14} + 1287 \, d^{8} x^{5} e^{13} + 715 \, d^{9} x^{4} e^{12} + 286 \, d^{10} x^{3} e^{11} + 78 \, d^{11} x^{2} e^{10} + 13 \, d^{12} x e^{9} + d^{13} e^{8}\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)^14,x, algorithm="maxima")

[Out]

-1/72072*(12012*B*b^6*x^7*e^7 + 7*B*b^6*d^7 + 5544*A*a^6*e^7 + 6*(6*B*a*b^5*e + A*b^6*e)*d^6 + 1716*(7*B*b^6*d
*e^6 + 36*B*a*b^5*e^7 + 6*A*b^6*e^7)*x^6 + 21*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 + 1287*(7*B*b^6*d^2*e^5 +
105*B*a^2*b^4*e^7 + 42*A*a*b^5*e^7 + 6*(6*B*a*b^5*e^6 + A*b^6*e^6)*d)*x^5 + 56*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*
e^3)*d^4 + 715*(7*B*b^6*d^3*e^4 + 224*B*a^3*b^3*e^7 + 168*A*a^2*b^4*e^7 + 6*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^2 +
21*(5*B*a^2*b^4*e^6 + 2*A*a*b^5*e^6)*d)*x^4 + 126*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 + 286*(7*B*b^6*d^4*e
^3 + 378*B*a^4*b^2*e^7 + 504*A*a^3*b^3*e^7 + 6*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^3 + 21*(5*B*a^2*b^4*e^5 + 2*A*a*b
^5*e^5)*d^2 + 56*(4*B*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d)*x^3 + 252*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 78*(
7*B*b^6*d^5*e^2 + 504*B*a^5*b*e^7 + 1260*A*a^4*b^2*e^7 + 6*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 + 21*(5*B*a^2*b^4*e
^4 + 2*A*a*b^5*e^4)*d^3 + 56*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^2 + 126*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)
*d)*x^2 + 462*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d + 13*(7*B*b^6*d^6*e + 462*B*a^6*e^7 + 2772*A*a^5*b*e^7 + 6*(6*B*a*
b^5*e^2 + A*b^6*e^2)*d^5 + 21*(5*B*a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^4 + 56*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d
^3 + 126*(3*B*a^4*b^2*e^5 + 4*A*a^3*b^3*e^5)*d^2 + 252*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x)/(x^13*e^21 + 13
*d*x^12*e^20 + 78*d^2*x^11*e^19 + 286*d^3*x^10*e^18 + 715*d^4*x^9*e^17 + 1287*d^5*x^8*e^16 + 1716*d^6*x^7*e^15
+ 1716*d^7*x^6*e^14 + 1287*d^8*x^5*e^13 + 715*d^9*x^4*e^12 + 286*d^10*x^3*e^11 + 78*d^11*x^2*e^10 + 13*d^12*x
*e^9 + d^13*e^8)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (295) = 590\).
time = 1.01, size = 853, normalized size = 2.92 \begin {gather*} -\frac {7 \, B b^{6} d^{7} + {\left (12012 \, B b^{6} x^{7} + 5544 \, A a^{6} + 10296 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 27027 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 40040 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 36036 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 19656 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (12012 \, B b^{6} d x^{6} + 7722 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 15015 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 16016 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} + 9828 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 3276 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + 462 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (3003 \, B b^{6} d^{2} x^{5} + 1430 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 2002 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 1456 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 546 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + 84 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + {\left (5005 \, B b^{6} d^{3} x^{4} + 1716 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 1638 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 728 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + 126 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + {\left (2002 \, B b^{6} d^{4} x^{3} + 468 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 273 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (182 \, B b^{6} d^{5} x^{2} + 26 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 7 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + {\left (91 \, B b^{6} d^{6} x + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e}{72072 \, {\left (x^{13} e^{21} + 13 \, d x^{12} e^{20} + 78 \, d^{2} x^{11} e^{19} + 286 \, d^{3} x^{10} e^{18} + 715 \, d^{4} x^{9} e^{17} + 1287 \, d^{5} x^{8} e^{16} + 1716 \, d^{6} x^{7} e^{15} + 1716 \, d^{7} x^{6} e^{14} + 1287 \, d^{8} x^{5} e^{13} + 715 \, d^{9} x^{4} e^{12} + 286 \, d^{10} x^{3} e^{11} + 78 \, d^{11} x^{2} e^{10} + 13 \, d^{12} x e^{9} + d^{13} e^{8}\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)^14,x, algorithm="fricas")

[Out]

-1/72072*(7*B*b^6*d^7 + (12012*B*b^6*x^7 + 5544*A*a^6 + 10296*(6*B*a*b^5 + A*b^6)*x^6 + 27027*(5*B*a^2*b^4 + 2
*A*a*b^5)*x^5 + 40040*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 36036*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 19656*(2*B*a^5
*b + 5*A*a^4*b^2)*x^2 + 6006*(B*a^6 + 6*A*a^5*b)*x)*e^7 + (12012*B*b^6*d*x^6 + 7722*(6*B*a*b^5 + A*b^6)*d*x^5
+ 15015*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^4 + 16016*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^3 + 9828*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d*x^2 + 3276*(2*B*a^5*b + 5*A*a^4*b^2)*d*x + 462*(B*a^6 + 6*A*a^5*b)*d)*e^6 + 3*(3003*B*b^6*d^2*x^5 + 1
430*(6*B*a*b^5 + A*b^6)*d^2*x^4 + 2002*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 + 1456*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^
2*x^2 + 546*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x + 84*(2*B*a^5*b + 5*A*a^4*b^2)*d^2)*e^5 + (5005*B*b^6*d^3*x^4 +
1716*(6*B*a*b^5 + A*b^6)*d^3*x^3 + 1638*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^2 + 728*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^
3*x + 126*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3)*e^4 + (2002*B*b^6*d^4*x^3 + 468*(6*B*a*b^5 + A*b^6)*d^4*x^2 + 273*(
5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x + 56*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e^3 + 3*(182*B*b^6*d^5*x^2 + 26*(6*B*a*b^
5 + A*b^6)*d^5*x + 7*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5)*e^2 + (91*B*b^6*d^6*x + 6*(6*B*a*b^5 + A*b^6)*d^6)*e)/(x^1
3*e^21 + 13*d*x^12*e^20 + 78*d^2*x^11*e^19 + 286*d^3*x^10*e^18 + 715*d^4*x^9*e^17 + 1287*d^5*x^8*e^16 + 1716*d
^6*x^7*e^15 + 1716*d^7*x^6*e^14 + 1287*d^8*x^5*e^13 + 715*d^9*x^4*e^12 + 286*d^10*x^3*e^11 + 78*d^11*x^2*e^10
+ 13*d^12*x*e^9 + d^13*e^8)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**14,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (295) = 590\).
time = 1.50, size = 856, normalized size = 2.93 \begin {gather*} -\frac {{\left (12012 \, B b^{6} x^{7} e^{7} + 12012 \, B b^{6} d x^{6} e^{6} + 9009 \, B b^{6} d^{2} x^{5} e^{5} + 5005 \, B b^{6} d^{3} x^{4} e^{4} + 2002 \, B b^{6} d^{4} x^{3} e^{3} + 546 \, B b^{6} d^{5} x^{2} e^{2} + 91 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 61776 \, B a b^{5} x^{6} e^{7} + 10296 \, A b^{6} x^{6} e^{7} + 46332 \, B a b^{5} d x^{5} e^{6} + 7722 \, A b^{6} d x^{5} e^{6} + 25740 \, B a b^{5} d^{2} x^{4} e^{5} + 4290 \, A b^{6} d^{2} x^{4} e^{5} + 10296 \, B a b^{5} d^{3} x^{3} e^{4} + 1716 \, A b^{6} d^{3} x^{3} e^{4} + 2808 \, B a b^{5} d^{4} x^{2} e^{3} + 468 \, A b^{6} d^{4} x^{2} e^{3} + 468 \, B a b^{5} d^{5} x e^{2} + 78 \, A b^{6} d^{5} x e^{2} + 36 \, B a b^{5} d^{6} e + 6 \, A b^{6} d^{6} e + 135135 \, B a^{2} b^{4} x^{5} e^{7} + 54054 \, A a b^{5} x^{5} e^{7} + 75075 \, B a^{2} b^{4} d x^{4} e^{6} + 30030 \, A a b^{5} d x^{4} e^{6} + 30030 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 12012 \, A a b^{5} d^{2} x^{3} e^{5} + 8190 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 3276 \, A a b^{5} d^{3} x^{2} e^{4} + 1365 \, B a^{2} b^{4} d^{4} x e^{3} + 546 \, A a b^{5} d^{4} x e^{3} + 105 \, B a^{2} b^{4} d^{5} e^{2} + 42 \, A a b^{5} d^{5} e^{2} + 160160 \, B a^{3} b^{3} x^{4} e^{7} + 120120 \, A a^{2} b^{4} x^{4} e^{7} + 64064 \, B a^{3} b^{3} d x^{3} e^{6} + 48048 \, A a^{2} b^{4} d x^{3} e^{6} + 17472 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 13104 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 2912 \, B a^{3} b^{3} d^{3} x e^{4} + 2184 \, A a^{2} b^{4} d^{3} x e^{4} + 224 \, B a^{3} b^{3} d^{4} e^{3} + 168 \, A a^{2} b^{4} d^{4} e^{3} + 108108 \, B a^{4} b^{2} x^{3} e^{7} + 144144 \, A a^{3} b^{3} x^{3} e^{7} + 29484 \, B a^{4} b^{2} d x^{2} e^{6} + 39312 \, A a^{3} b^{3} d x^{2} e^{6} + 4914 \, B a^{4} b^{2} d^{2} x e^{5} + 6552 \, A a^{3} b^{3} d^{2} x e^{5} + 378 \, B a^{4} b^{2} d^{3} e^{4} + 504 \, A a^{3} b^{3} d^{3} e^{4} + 39312 \, B a^{5} b x^{2} e^{7} + 98280 \, A a^{4} b^{2} x^{2} e^{7} + 6552 \, B a^{5} b d x e^{6} + 16380 \, A a^{4} b^{2} d x e^{6} + 504 \, B a^{5} b d^{2} e^{5} + 1260 \, A a^{4} b^{2} d^{2} e^{5} + 6006 \, B a^{6} x e^{7} + 36036 \, A a^{5} b x e^{7} + 462 \, B a^{6} d e^{6} + 2772 \, A a^{5} b d e^{6} + 5544 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{72072 \, {\left (x e + d\right )}^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^14,x, algorithm="giac")

[Out]

-1/72072*(12012*B*b^6*x^7*e^7 + 12012*B*b^6*d*x^6*e^6 + 9009*B*b^6*d^2*x^5*e^5 + 5005*B*b^6*d^3*x^4*e^4 + 2002
*B*b^6*d^4*x^3*e^3 + 546*B*b^6*d^5*x^2*e^2 + 91*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 61776*B*a*b^5*x^6*e^7 + 10296*A*
b^6*x^6*e^7 + 46332*B*a*b^5*d*x^5*e^6 + 7722*A*b^6*d*x^5*e^6 + 25740*B*a*b^5*d^2*x^4*e^5 + 4290*A*b^6*d^2*x^4*
e^5 + 10296*B*a*b^5*d^3*x^3*e^4 + 1716*A*b^6*d^3*x^3*e^4 + 2808*B*a*b^5*d^4*x^2*e^3 + 468*A*b^6*d^4*x^2*e^3 +
468*B*a*b^5*d^5*x*e^2 + 78*A*b^6*d^5*x*e^2 + 36*B*a*b^5*d^6*e + 6*A*b^6*d^6*e + 135135*B*a^2*b^4*x^5*e^7 + 540
54*A*a*b^5*x^5*e^7 + 75075*B*a^2*b^4*d*x^4*e^6 + 30030*A*a*b^5*d*x^4*e^6 + 30030*B*a^2*b^4*d^2*x^3*e^5 + 12012
*A*a*b^5*d^2*x^3*e^5 + 8190*B*a^2*b^4*d^3*x^2*e^4 + 3276*A*a*b^5*d^3*x^2*e^4 + 1365*B*a^2*b^4*d^4*x*e^3 + 546*
A*a*b^5*d^4*x*e^3 + 105*B*a^2*b^4*d^5*e^2 + 42*A*a*b^5*d^5*e^2 + 160160*B*a^3*b^3*x^4*e^7 + 120120*A*a^2*b^4*x
^4*e^7 + 64064*B*a^3*b^3*d*x^3*e^6 + 48048*A*a^2*b^4*d*x^3*e^6 + 17472*B*a^3*b^3*d^2*x^2*e^5 + 13104*A*a^2*b^4
*d^2*x^2*e^5 + 2912*B*a^3*b^3*d^3*x*e^4 + 2184*A*a^2*b^4*d^3*x*e^4 + 224*B*a^3*b^3*d^4*e^3 + 168*A*a^2*b^4*d^4
*e^3 + 108108*B*a^4*b^2*x^3*e^7 + 144144*A*a^3*b^3*x^3*e^7 + 29484*B*a^4*b^2*d*x^2*e^6 + 39312*A*a^3*b^3*d*x^2
*e^6 + 4914*B*a^4*b^2*d^2*x*e^5 + 6552*A*a^3*b^3*d^2*x*e^5 + 378*B*a^4*b^2*d^3*e^4 + 504*A*a^3*b^3*d^3*e^4 + 3
9312*B*a^5*b*x^2*e^7 + 98280*A*a^4*b^2*x^2*e^7 + 6552*B*a^5*b*d*x*e^6 + 16380*A*a^4*b^2*d*x*e^6 + 504*B*a^5*b*
d^2*e^5 + 1260*A*a^4*b^2*d^2*e^5 + 6006*B*a^6*x*e^7 + 36036*A*a^5*b*x*e^7 + 462*B*a^6*d*e^6 + 2772*A*a^5*b*d*e
^6 + 5544*A*a^6*e^7)*e^(-8)/(x*e + d)^13

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Mupad [B]
time = 0.81, size = 921, normalized size = 3.15 \begin {gather*} -\frac {\frac {462\,B\,a^6\,d\,e^6+5544\,A\,a^6\,e^7+504\,B\,a^5\,b\,d^2\,e^5+2772\,A\,a^5\,b\,d\,e^6+378\,B\,a^4\,b^2\,d^3\,e^4+1260\,A\,a^4\,b^2\,d^2\,e^5+224\,B\,a^3\,b^3\,d^4\,e^3+504\,A\,a^3\,b^3\,d^3\,e^4+105\,B\,a^2\,b^4\,d^5\,e^2+168\,A\,a^2\,b^4\,d^4\,e^3+36\,B\,a\,b^5\,d^6\,e+42\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+6\,A\,b^6\,d^6\,e}{72072\,e^8}+\frac {x\,\left (462\,B\,a^6\,e^6+504\,B\,a^5\,b\,d\,e^5+2772\,A\,a^5\,b\,e^6+378\,B\,a^4\,b^2\,d^2\,e^4+1260\,A\,a^4\,b^2\,d\,e^5+224\,B\,a^3\,b^3\,d^3\,e^3+504\,A\,a^3\,b^3\,d^2\,e^4+105\,B\,a^2\,b^4\,d^4\,e^2+168\,A\,a^2\,b^4\,d^3\,e^3+36\,B\,a\,b^5\,d^5\,e+42\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+6\,A\,b^6\,d^5\,e\right )}{5544\,e^7}+\frac {5\,b^3\,x^4\,\left (224\,B\,a^3\,e^3+105\,B\,a^2\,b\,d\,e^2+168\,A\,a^2\,b\,e^3+36\,B\,a\,b^2\,d^2\,e+42\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )}{504\,e^4}+\frac {b^5\,x^6\,\left (6\,A\,b\,e+36\,B\,a\,e+7\,B\,b\,d\right )}{42\,e^2}+\frac {b\,x^2\,\left (504\,B\,a^5\,e^5+378\,B\,a^4\,b\,d\,e^4+1260\,A\,a^4\,b\,e^5+224\,B\,a^3\,b^2\,d^2\,e^3+504\,A\,a^3\,b^2\,d\,e^4+105\,B\,a^2\,b^3\,d^3\,e^2+168\,A\,a^2\,b^3\,d^2\,e^3+36\,B\,a\,b^4\,d^4\,e+42\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+6\,A\,b^5\,d^4\,e\right )}{924\,e^6}+\frac {b^2\,x^3\,\left (378\,B\,a^4\,e^4+224\,B\,a^3\,b\,d\,e^3+504\,A\,a^3\,b\,e^4+105\,B\,a^2\,b^2\,d^2\,e^2+168\,A\,a^2\,b^2\,d\,e^3+36\,B\,a\,b^3\,d^3\,e+42\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+6\,A\,b^4\,d^3\,e\right )}{252\,e^5}+\frac {b^4\,x^5\,\left (105\,B\,a^2\,e^2+36\,B\,a\,b\,d\,e+42\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+6\,A\,b^2\,d\,e\right )}{56\,e^3}+\frac {B\,b^6\,x^7}{6\,e}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^14,x)

[Out]

-((5544*A*a^6*e^7 + 7*B*b^6*d^7 + 6*A*b^6*d^6*e + 462*B*a^6*d*e^6 + 42*A*a*b^5*d^5*e^2 + 504*B*a^5*b*d^2*e^5 +
168*A*a^2*b^4*d^4*e^3 + 504*A*a^3*b^3*d^3*e^4 + 1260*A*a^4*b^2*d^2*e^5 + 105*B*a^2*b^4*d^5*e^2 + 224*B*a^3*b^
3*d^4*e^3 + 378*B*a^4*b^2*d^3*e^4 + 2772*A*a^5*b*d*e^6 + 36*B*a*b^5*d^6*e)/(72072*e^8) + (x*(462*B*a^6*e^6 + 7
*B*b^6*d^6 + 2772*A*a^5*b*e^6 + 6*A*b^6*d^5*e + 42*A*a*b^5*d^4*e^2 + 1260*A*a^4*b^2*d*e^5 + 168*A*a^2*b^4*d^3*
e^3 + 504*A*a^3*b^3*d^2*e^4 + 105*B*a^2*b^4*d^4*e^2 + 224*B*a^3*b^3*d^3*e^3 + 378*B*a^4*b^2*d^2*e^4 + 36*B*a*b
^5*d^5*e + 504*B*a^5*b*d*e^5))/(5544*e^7) + (5*b^3*x^4*(224*B*a^3*e^3 + 7*B*b^3*d^3 + 168*A*a^2*b*e^3 + 6*A*b^
3*d^2*e + 42*A*a*b^2*d*e^2 + 36*B*a*b^2*d^2*e + 105*B*a^2*b*d*e^2))/(504*e^4) + (b^5*x^6*(6*A*b*e + 36*B*a*e +
7*B*b*d))/(42*e^2) + (b*x^2*(504*B*a^5*e^5 + 7*B*b^5*d^5 + 1260*A*a^4*b*e^5 + 6*A*b^5*d^4*e + 42*A*a*b^4*d^3*
e^2 + 504*A*a^3*b^2*d*e^4 + 168*A*a^2*b^3*d^2*e^3 + 105*B*a^2*b^3*d^3*e^2 + 224*B*a^3*b^2*d^2*e^3 + 36*B*a*b^4
*d^4*e + 378*B*a^4*b*d*e^4))/(924*e^6) + (b^2*x^3*(378*B*a^4*e^4 + 7*B*b^4*d^4 + 504*A*a^3*b*e^4 + 6*A*b^4*d^3
*e + 42*A*a*b^3*d^2*e^2 + 168*A*a^2*b^2*d*e^3 + 105*B*a^2*b^2*d^2*e^2 + 36*B*a*b^3*d^3*e + 224*B*a^3*b*d*e^3))
/(252*e^5) + (b^4*x^5*(105*B*a^2*e^2 + 7*B*b^2*d^2 + 42*A*a*b*e^2 + 6*A*b^2*d*e + 36*B*a*b*d*e))/(56*e^3) + (B
*b^6*x^7)/(6*e))/(d^13 + e^13*x^13 + 13*d*e^12*x^12 + 78*d^11*e^2*x^2 + 286*d^10*e^3*x^3 + 715*d^9*e^4*x^4 + 1
287*d^8*e^5*x^5 + 1716*d^7*e^6*x^6 + 1716*d^6*e^7*x^7 + 1287*d^5*e^8*x^8 + 715*d^4*e^9*x^9 + 286*d^3*e^10*x^10
+ 78*d^2*e^11*x^11 + 13*d^12*e*x)

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