3.11.72 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{13}} \, dx\) [1072]

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

[Out]

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

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Rubi [A]
time = 0.23, 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)}{6 e^8 (d+e x)^6}-\frac {3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8 (d+e x)^7}+\frac {5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{8 e^8 (d+e x)^8}-\frac {5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8 (d+e x)^9}+\frac {3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8 (d+e x)^{10}}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{11 e^8 (d+e x)^{11}}+\frac {(b d-a e)^6 (B d-A e)}{12 e^8 (d+e x)^{12}}-\frac {b^6 B}{5 e^8 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(600\) vs. \(2(292)=584\).
time = 0.19, size = 600, normalized size = 2.05 \begin {gather*} -\frac {210 a^6 e^6 (11 A e+B (d+12 e x))+252 a^5 b e^5 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+210 a^4 b^2 e^4 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+140 a^3 b^3 e^3 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+15 a^2 b^4 e^2 \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+30 a b^5 e \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )+b^6 \left (5 A e \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )+7 B \left (d^7+12 d^6 e x+66 d^5 e^2 x^2+220 d^4 e^3 x^3+495 d^3 e^4 x^4+792 d^2 e^5 x^5+924 d e^6 x^6+792 e^7 x^7\right )\right )}{27720 e^8 (d+e x)^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^13,x]

[Out]

-1/27720*(210*a^6*e^6*(11*A*e + B*(d + 12*e*x)) + 252*a^5*b*e^5*(5*A*e*(d + 12*e*x) + B*(d^2 + 12*d*e*x + 66*e
^2*x^2)) + 210*a^4*b^2*e^4*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3
*x^3)) + 140*a^3*b^3*e^3*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x + 66*d^2
*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + 15*a^2*b^4*e^2*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^
3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)
) + 30*a*b^5*e*(A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d
^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6)) + b^6*(5*
A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*
B*(d^7 + 12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x^5 + 924*d*e^6*x^6 + 7
92*e^7*x^7)))/(e^8*(d + e*x)^12)

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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)^13,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (295) = 590\).
time = 0.39, size = 888, normalized size = 3.04 \begin {gather*} -\frac {5544 \, B b^{6} x^{7} e^{7} + 7 \, B b^{6} d^{7} + 2310 \, A a^{6} e^{7} + 5 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 924 \, {\left (7 \, B b^{6} d e^{6} + 30 \, B a b^{5} e^{7} + 5 \, A b^{6} e^{7}\right )} x^{6} + 15 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 792 \, {\left (7 \, B b^{6} d^{2} e^{5} + 75 \, B a^{2} b^{4} e^{7} + 30 \, A a b^{5} e^{7} + 5 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + 35 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 495 \, {\left (7 \, B b^{6} d^{3} e^{4} + 140 \, B a^{3} b^{3} e^{7} + 105 \, A a^{2} b^{4} e^{7} + 5 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 15 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + 70 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 220 \, {\left (7 \, B b^{6} d^{4} e^{3} + 210 \, B a^{4} b^{2} e^{7} + 280 \, A a^{3} b^{3} e^{7} + 5 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 15 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 35 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + 126 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 66 \, {\left (7 \, B b^{6} d^{5} e^{2} + 252 \, B a^{5} b e^{7} + 630 \, A a^{4} b^{2} e^{7} + 5 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 15 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 35 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 70 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + 210 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 12 \, {\left (7 \, B b^{6} d^{6} e + 210 \, B a^{6} e^{7} + 1260 \, A a^{5} b e^{7} + 5 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 15 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 35 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 70 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 126 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{27720 \, {\left (x^{12} e^{20} + 12 \, d x^{11} e^{19} + 66 \, d^{2} x^{10} e^{18} + 220 \, d^{3} x^{9} e^{17} + 495 \, d^{4} x^{8} e^{16} + 792 \, d^{5} x^{7} e^{15} + 924 \, d^{6} x^{6} e^{14} + 792 \, d^{7} x^{5} e^{13} + 495 \, d^{8} x^{4} e^{12} + 220 \, d^{9} x^{3} e^{11} + 66 \, d^{10} x^{2} e^{10} + 12 \, d^{11} x e^{9} + d^{12} 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)^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*B*b^6*x^7*e^7 + 7*B*b^6*d^7 + 2310*A*a^6*e^7 + 5*(6*B*a*b^5*e + A*b^6*e)*d^6 + 924*(7*B*b^6*d*e
^6 + 30*B*a*b^5*e^7 + 5*A*b^6*e^7)*x^6 + 15*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 + 792*(7*B*b^6*d^2*e^5 + 75*
B*a^2*b^4*e^7 + 30*A*a*b^5*e^7 + 5*(6*B*a*b^5*e^6 + A*b^6*e^6)*d)*x^5 + 35*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)
*d^4 + 495*(7*B*b^6*d^3*e^4 + 140*B*a^3*b^3*e^7 + 105*A*a^2*b^4*e^7 + 5*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^2 + 15*(
5*B*a^2*b^4*e^6 + 2*A*a*b^5*e^6)*d)*x^4 + 70*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 + 220*(7*B*b^6*d^4*e^3 +
210*B*a^4*b^2*e^7 + 280*A*a^3*b^3*e^7 + 5*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^3 + 15*(5*B*a^2*b^4*e^5 + 2*A*a*b^5*e^
5)*d^2 + 35*(4*B*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d)*x^3 + 126*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 66*(7*B*b
^6*d^5*e^2 + 252*B*a^5*b*e^7 + 630*A*a^4*b^2*e^7 + 5*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 + 15*(5*B*a^2*b^4*e^4 + 2
*A*a*b^5*e^4)*d^3 + 35*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^2 + 70*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)*d)*x^2
+ 210*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d + 12*(7*B*b^6*d^6*e + 210*B*a^6*e^7 + 1260*A*a^5*b*e^7 + 5*(6*B*a*b^5*e^2
+ A*b^6*e^2)*d^5 + 15*(5*B*a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^4 + 35*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^3 + 70
*(3*B*a^4*b^2*e^5 + 4*A*a^3*b^3*e^5)*d^2 + 126*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x)/(x^12*e^20 + 12*d*x^11*
e^19 + 66*d^2*x^10*e^18 + 220*d^3*x^9*e^17 + 495*d^4*x^8*e^16 + 792*d^5*x^7*e^15 + 924*d^6*x^6*e^14 + 792*d^7*
x^5*e^13 + 495*d^8*x^4*e^12 + 220*d^9*x^3*e^11 + 66*d^10*x^2*e^10 + 12*d^11*x*e^9 + d^12*e^8)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (295) = 590\).
time = 1.26, size = 845, normalized size = 2.89 \begin {gather*} -\frac {7 \, B b^{6} d^{7} + {\left (5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (6468 \, B b^{6} d x^{6} + 3960 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 7425 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 7700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} + 4620 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + 210 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (1848 \, B b^{6} d^{2} x^{5} + 825 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 1100 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 770 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 280 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + 42 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + 5 \, {\left (693 \, B b^{6} d^{3} x^{4} + 220 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 198 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 84 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + 14 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + 5 \, {\left (308 \, B b^{6} d^{4} x^{3} + 66 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 36 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 7 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (154 \, B b^{6} d^{5} x^{2} + 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 5 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + {\left (84 \, B b^{6} d^{6} x + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e}{27720 \, {\left (x^{12} e^{20} + 12 \, d x^{11} e^{19} + 66 \, d^{2} x^{10} e^{18} + 220 \, d^{3} x^{9} e^{17} + 495 \, d^{4} x^{8} e^{16} + 792 \, d^{5} x^{7} e^{15} + 924 \, d^{6} x^{6} e^{14} + 792 \, d^{7} x^{5} e^{13} + 495 \, d^{8} x^{4} e^{12} + 220 \, d^{9} x^{3} e^{11} + 66 \, d^{10} x^{2} e^{10} + 12 \, d^{11} x e^{9} + d^{12} 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)^13,x, algorithm="fricas")

[Out]

-1/27720*(7*B*b^6*d^7 + (5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 11880*(5*B*a^2*b^4 + 2*A
*a*b^5)*x^5 + 17325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b
+ 5*A*a^4*b^2)*x^2 + 2520*(B*a^6 + 6*A*a^5*b)*x)*e^7 + (6468*B*b^6*d*x^6 + 3960*(6*B*a*b^5 + A*b^6)*d*x^5 + 74
25*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^4 + 7700*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^3 + 4620*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*d*x^2 + 1512*(2*B*a^5*b + 5*A*a^4*b^2)*d*x + 210*(B*a^6 + 6*A*a^5*b)*d)*e^6 + 3*(1848*B*b^6*d^2*x^5 + 825*(6*
B*a*b^5 + A*b^6)*d^2*x^4 + 1100*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 + 770*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^2 +
280*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x + 42*(2*B*a^5*b + 5*A*a^4*b^2)*d^2)*e^5 + 5*(693*B*b^6*d^3*x^4 + 220*(6*
B*a*b^5 + A*b^6)*d^3*x^3 + 198*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^2 + 84*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x + 14*(
3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3)*e^4 + 5*(308*B*b^6*d^4*x^3 + 66*(6*B*a*b^5 + A*b^6)*d^4*x^2 + 36*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^4*x + 7*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e^3 + 3*(154*B*b^6*d^5*x^2 + 20*(6*B*a*b^5 + A*b^6)*d^
5*x + 5*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5)*e^2 + (84*B*b^6*d^6*x + 5*(6*B*a*b^5 + A*b^6)*d^6)*e)/(x^12*e^20 + 12*d
*x^11*e^19 + 66*d^2*x^10*e^18 + 220*d^3*x^9*e^17 + 495*d^4*x^8*e^16 + 792*d^5*x^7*e^15 + 924*d^6*x^6*e^14 + 79
2*d^7*x^5*e^13 + 495*d^8*x^4*e^12 + 220*d^9*x^3*e^11 + 66*d^10*x^2*e^10 + 12*d^11*x*e^9 + d^12*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)**13,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 = 1.78, size = 856, normalized size = 2.93 \begin {gather*} -\frac {{\left (5544 \, B b^{6} x^{7} e^{7} + 6468 \, B b^{6} d x^{6} e^{6} + 5544 \, B b^{6} d^{2} x^{5} e^{5} + 3465 \, B b^{6} d^{3} x^{4} e^{4} + 1540 \, B b^{6} d^{4} x^{3} e^{3} + 462 \, B b^{6} d^{5} x^{2} e^{2} + 84 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 27720 \, B a b^{5} x^{6} e^{7} + 4620 \, A b^{6} x^{6} e^{7} + 23760 \, B a b^{5} d x^{5} e^{6} + 3960 \, A b^{6} d x^{5} e^{6} + 14850 \, B a b^{5} d^{2} x^{4} e^{5} + 2475 \, A b^{6} d^{2} x^{4} e^{5} + 6600 \, B a b^{5} d^{3} x^{3} e^{4} + 1100 \, A b^{6} d^{3} x^{3} e^{4} + 1980 \, B a b^{5} d^{4} x^{2} e^{3} + 330 \, A b^{6} d^{4} x^{2} e^{3} + 360 \, B a b^{5} d^{5} x e^{2} + 60 \, A b^{6} d^{5} x e^{2} + 30 \, B a b^{5} d^{6} e + 5 \, A b^{6} d^{6} e + 59400 \, B a^{2} b^{4} x^{5} e^{7} + 23760 \, A a b^{5} x^{5} e^{7} + 37125 \, B a^{2} b^{4} d x^{4} e^{6} + 14850 \, A a b^{5} d x^{4} e^{6} + 16500 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 6600 \, A a b^{5} d^{2} x^{3} e^{5} + 4950 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 1980 \, A a b^{5} d^{3} x^{2} e^{4} + 900 \, B a^{2} b^{4} d^{4} x e^{3} + 360 \, A a b^{5} d^{4} x e^{3} + 75 \, B a^{2} b^{4} d^{5} e^{2} + 30 \, A a b^{5} d^{5} e^{2} + 69300 \, B a^{3} b^{3} x^{4} e^{7} + 51975 \, A a^{2} b^{4} x^{4} e^{7} + 30800 \, B a^{3} b^{3} d x^{3} e^{6} + 23100 \, A a^{2} b^{4} d x^{3} e^{6} + 9240 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 6930 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 1680 \, B a^{3} b^{3} d^{3} x e^{4} + 1260 \, A a^{2} b^{4} d^{3} x e^{4} + 140 \, B a^{3} b^{3} d^{4} e^{3} + 105 \, A a^{2} b^{4} d^{4} e^{3} + 46200 \, B a^{4} b^{2} x^{3} e^{7} + 61600 \, A a^{3} b^{3} x^{3} e^{7} + 13860 \, B a^{4} b^{2} d x^{2} e^{6} + 18480 \, A a^{3} b^{3} d x^{2} e^{6} + 2520 \, B a^{4} b^{2} d^{2} x e^{5} + 3360 \, A a^{3} b^{3} d^{2} x e^{5} + 210 \, B a^{4} b^{2} d^{3} e^{4} + 280 \, A a^{3} b^{3} d^{3} e^{4} + 16632 \, B a^{5} b x^{2} e^{7} + 41580 \, A a^{4} b^{2} x^{2} e^{7} + 3024 \, B a^{5} b d x e^{6} + 7560 \, A a^{4} b^{2} d x e^{6} + 252 \, B a^{5} b d^{2} e^{5} + 630 \, A a^{4} b^{2} d^{2} e^{5} + 2520 \, B a^{6} x e^{7} + 15120 \, A a^{5} b x e^{7} + 210 \, B a^{6} d e^{6} + 1260 \, A a^{5} b d e^{6} + 2310 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{27720 \, {\left (x e + d\right )}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^13,x, algorithm="giac")

[Out]

-1/27720*(5544*B*b^6*x^7*e^7 + 6468*B*b^6*d*x^6*e^6 + 5544*B*b^6*d^2*x^5*e^5 + 3465*B*b^6*d^3*x^4*e^4 + 1540*B
*b^6*d^4*x^3*e^3 + 462*B*b^6*d^5*x^2*e^2 + 84*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 27720*B*a*b^5*x^6*e^7 + 4620*A*b^6
*x^6*e^7 + 23760*B*a*b^5*d*x^5*e^6 + 3960*A*b^6*d*x^5*e^6 + 14850*B*a*b^5*d^2*x^4*e^5 + 2475*A*b^6*d^2*x^4*e^5
+ 6600*B*a*b^5*d^3*x^3*e^4 + 1100*A*b^6*d^3*x^3*e^4 + 1980*B*a*b^5*d^4*x^2*e^3 + 330*A*b^6*d^4*x^2*e^3 + 360*
B*a*b^5*d^5*x*e^2 + 60*A*b^6*d^5*x*e^2 + 30*B*a*b^5*d^6*e + 5*A*b^6*d^6*e + 59400*B*a^2*b^4*x^5*e^7 + 23760*A*
a*b^5*x^5*e^7 + 37125*B*a^2*b^4*d*x^4*e^6 + 14850*A*a*b^5*d*x^4*e^6 + 16500*B*a^2*b^4*d^2*x^3*e^5 + 6600*A*a*b
^5*d^2*x^3*e^5 + 4950*B*a^2*b^4*d^3*x^2*e^4 + 1980*A*a*b^5*d^3*x^2*e^4 + 900*B*a^2*b^4*d^4*x*e^3 + 360*A*a*b^5
*d^4*x*e^3 + 75*B*a^2*b^4*d^5*e^2 + 30*A*a*b^5*d^5*e^2 + 69300*B*a^3*b^3*x^4*e^7 + 51975*A*a^2*b^4*x^4*e^7 + 3
0800*B*a^3*b^3*d*x^3*e^6 + 23100*A*a^2*b^4*d*x^3*e^6 + 9240*B*a^3*b^3*d^2*x^2*e^5 + 6930*A*a^2*b^4*d^2*x^2*e^5
+ 1680*B*a^3*b^3*d^3*x*e^4 + 1260*A*a^2*b^4*d^3*x*e^4 + 140*B*a^3*b^3*d^4*e^3 + 105*A*a^2*b^4*d^4*e^3 + 46200
*B*a^4*b^2*x^3*e^7 + 61600*A*a^3*b^3*x^3*e^7 + 13860*B*a^4*b^2*d*x^2*e^6 + 18480*A*a^3*b^3*d*x^2*e^6 + 2520*B*
a^4*b^2*d^2*x*e^5 + 3360*A*a^3*b^3*d^2*x*e^5 + 210*B*a^4*b^2*d^3*e^4 + 280*A*a^3*b^3*d^3*e^4 + 16632*B*a^5*b*x
^2*e^7 + 41580*A*a^4*b^2*x^2*e^7 + 3024*B*a^5*b*d*x*e^6 + 7560*A*a^4*b^2*d*x*e^6 + 252*B*a^5*b*d^2*e^5 + 630*A
*a^4*b^2*d^2*e^5 + 2520*B*a^6*x*e^7 + 15120*A*a^5*b*x*e^7 + 210*B*a^6*d*e^6 + 1260*A*a^5*b*d*e^6 + 2310*A*a^6*
e^7)*e^(-8)/(x*e + d)^12

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Mupad [B]
time = 1.59, size = 910, normalized size = 3.12 \begin {gather*} -\frac {\frac {210\,B\,a^6\,d\,e^6+2310\,A\,a^6\,e^7+252\,B\,a^5\,b\,d^2\,e^5+1260\,A\,a^5\,b\,d\,e^6+210\,B\,a^4\,b^2\,d^3\,e^4+630\,A\,a^4\,b^2\,d^2\,e^5+140\,B\,a^3\,b^3\,d^4\,e^3+280\,A\,a^3\,b^3\,d^3\,e^4+75\,B\,a^2\,b^4\,d^5\,e^2+105\,A\,a^2\,b^4\,d^4\,e^3+30\,B\,a\,b^5\,d^6\,e+30\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+5\,A\,b^6\,d^6\,e}{27720\,e^8}+\frac {x\,\left (210\,B\,a^6\,e^6+252\,B\,a^5\,b\,d\,e^5+1260\,A\,a^5\,b\,e^6+210\,B\,a^4\,b^2\,d^2\,e^4+630\,A\,a^4\,b^2\,d\,e^5+140\,B\,a^3\,b^3\,d^3\,e^3+280\,A\,a^3\,b^3\,d^2\,e^4+75\,B\,a^2\,b^4\,d^4\,e^2+105\,A\,a^2\,b^4\,d^3\,e^3+30\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+5\,A\,b^6\,d^5\,e\right )}{2310\,e^7}+\frac {b^3\,x^4\,\left (140\,B\,a^3\,e^3+75\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+30\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{56\,e^4}+\frac {b^5\,x^6\,\left (5\,A\,b\,e+30\,B\,a\,e+7\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (252\,B\,a^5\,e^5+210\,B\,a^4\,b\,d\,e^4+630\,A\,a^4\,b\,e^5+140\,B\,a^3\,b^2\,d^2\,e^3+280\,A\,a^3\,b^2\,d\,e^4+75\,B\,a^2\,b^3\,d^3\,e^2+105\,A\,a^2\,b^3\,d^2\,e^3+30\,B\,a\,b^4\,d^4\,e+30\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )}{420\,e^6}+\frac {b^2\,x^3\,\left (210\,B\,a^4\,e^4+140\,B\,a^3\,b\,d\,e^3+280\,A\,a^3\,b\,e^4+75\,B\,a^2\,b^2\,d^2\,e^2+105\,A\,a^2\,b^2\,d\,e^3+30\,B\,a\,b^3\,d^3\,e+30\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+5\,A\,b^4\,d^3\,e\right )}{126\,e^5}+\frac {b^4\,x^5\,\left (75\,B\,a^2\,e^2+30\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{35\,e^3}+\frac {B\,b^6\,x^7}{5\,e}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^13,x)

[Out]

-((2310*A*a^6*e^7 + 7*B*b^6*d^7 + 5*A*b^6*d^6*e + 210*B*a^6*d*e^6 + 30*A*a*b^5*d^5*e^2 + 252*B*a^5*b*d^2*e^5 +
105*A*a^2*b^4*d^4*e^3 + 280*A*a^3*b^3*d^3*e^4 + 630*A*a^4*b^2*d^2*e^5 + 75*B*a^2*b^4*d^5*e^2 + 140*B*a^3*b^3*
d^4*e^3 + 210*B*a^4*b^2*d^3*e^4 + 1260*A*a^5*b*d*e^6 + 30*B*a*b^5*d^6*e)/(27720*e^8) + (x*(210*B*a^6*e^6 + 7*B
*b^6*d^6 + 1260*A*a^5*b*e^6 + 5*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 + 630*A*a^4*b^2*d*e^5 + 105*A*a^2*b^4*d^3*e^3
+ 280*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4*e^2 + 140*B*a^3*b^3*d^3*e^3 + 210*B*a^4*b^2*d^2*e^4 + 30*B*a*b^5*d
^5*e + 252*B*a^5*b*d*e^5))/(2310*e^7) + (b^3*x^4*(140*B*a^3*e^3 + 7*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^2*
e + 30*A*a*b^2*d*e^2 + 30*B*a*b^2*d^2*e + 75*B*a^2*b*d*e^2))/(56*e^4) + (b^5*x^6*(5*A*b*e + 30*B*a*e + 7*B*b*d
))/(30*e^2) + (b*x^2*(252*B*a^5*e^5 + 7*B*b^5*d^5 + 630*A*a^4*b*e^5 + 5*A*b^5*d^4*e + 30*A*a*b^4*d^3*e^2 + 280
*A*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d^2*e^3 + 75*B*a^2*b^3*d^3*e^2 + 140*B*a^3*b^2*d^2*e^3 + 30*B*a*b^4*d^4*e + 2
10*B*a^4*b*d*e^4))/(420*e^6) + (b^2*x^3*(210*B*a^4*e^4 + 7*B*b^4*d^4 + 280*A*a^3*b*e^4 + 5*A*b^4*d^3*e + 30*A*
a*b^3*d^2*e^2 + 105*A*a^2*b^2*d*e^3 + 75*B*a^2*b^2*d^2*e^2 + 30*B*a*b^3*d^3*e + 140*B*a^3*b*d*e^3))/(126*e^5)
+ (b^4*x^5*(75*B*a^2*e^2 + 7*B*b^2*d^2 + 30*A*a*b*e^2 + 5*A*b^2*d*e + 30*B*a*b*d*e))/(35*e^3) + (B*b^6*x^7)/(5
*e))/(d^12 + e^12*x^12 + 12*d*e^11*x^11 + 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8*e^4*x^4 + 792*d^7*e^5*x^
5 + 924*d^6*e^6*x^6 + 792*d^5*e^7*x^7 + 495*d^4*e^8*x^8 + 220*d^3*e^9*x^9 + 66*d^2*e^10*x^10 + 12*d^11*e*x)

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