3.155 \(\int \frac {(a+b x)^{10} (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=216 \[ -\frac {a^{10} A}{7 x^7}-\frac {a^9 (a B+10 A b)}{6 x^6}-\frac {a^8 b (2 a B+9 A b)}{x^5}-\frac {15 a^7 b^2 (3 a B+8 A b)}{4 x^4}-\frac {10 a^6 b^3 (4 a B+7 A b)}{x^3}-\frac {21 a^5 b^4 (5 a B+6 A b)}{x^2}-\frac {42 a^4 b^5 (6 a B+5 A b)}{x}+30 a^3 b^6 \log (x) (7 a B+4 A b)+15 a^2 b^7 x (8 a B+3 A b)+\frac {1}{3} b^9 x^3 (10 a B+A b)+\frac {5}{2} a b^8 x^2 (9 a B+2 A b)+\frac {1}{4} b^{10} B x^4 \]

[Out]

-1/7*a^10*A/x^7-1/6*a^9*(10*A*b+B*a)/x^6-a^8*b*(9*A*b+2*B*a)/x^5-15/4*a^7*b^2*(8*A*b+3*B*a)/x^4-10*a^6*b^3*(7*
A*b+4*B*a)/x^3-21*a^5*b^4*(6*A*b+5*B*a)/x^2-42*a^4*b^5*(5*A*b+6*B*a)/x+15*a^2*b^7*(3*A*b+8*B*a)*x+5/2*a*b^8*(2
*A*b+9*B*a)*x^2+1/3*b^9*(A*b+10*B*a)*x^3+1/4*b^10*B*x^4+30*a^3*b^6*(4*A*b+7*B*a)*ln(x)

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Rubi [A]  time = 0.15, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \[ -\frac {15 a^7 b^2 (3 a B+8 A b)}{4 x^4}-\frac {10 a^6 b^3 (4 a B+7 A b)}{x^3}-\frac {21 a^5 b^4 (5 a B+6 A b)}{x^2}-\frac {42 a^4 b^5 (6 a B+5 A b)}{x}+15 a^2 b^7 x (8 a B+3 A b)+30 a^3 b^6 \log (x) (7 a B+4 A b)-\frac {a^9 (a B+10 A b)}{6 x^6}-\frac {a^8 b (2 a B+9 A b)}{x^5}-\frac {a^{10} A}{7 x^7}+\frac {5}{2} a b^8 x^2 (9 a B+2 A b)+\frac {1}{3} b^9 x^3 (10 a B+A b)+\frac {1}{4} b^{10} B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^8,x]

[Out]

-(a^10*A)/(7*x^7) - (a^9*(10*A*b + a*B))/(6*x^6) - (a^8*b*(9*A*b + 2*a*B))/x^5 - (15*a^7*b^2*(8*A*b + 3*a*B))/
(4*x^4) - (10*a^6*b^3*(7*A*b + 4*a*B))/x^3 - (21*a^5*b^4*(6*A*b + 5*a*B))/x^2 - (42*a^4*b^5*(5*A*b + 6*a*B))/x
+ 15*a^2*b^7*(3*A*b + 8*a*B)*x + (5*a*b^8*(2*A*b + 9*a*B)*x^2)/2 + (b^9*(A*b + 10*a*B)*x^3)/3 + (b^10*B*x^4)/
4 + 30*a^3*b^6*(4*A*b + 7*a*B)*Log[x]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^8} \, dx &=\int \left (15 a^2 b^7 (3 A b+8 a B)+\frac {a^{10} A}{x^8}+\frac {a^9 (10 A b+a B)}{x^7}+\frac {5 a^8 b (9 A b+2 a B)}{x^6}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^5}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^4}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^3}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^2}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x}+5 a b^8 (2 A b+9 a B) x+b^9 (A b+10 a B) x^2+b^{10} B x^3\right ) \, dx\\ &=-\frac {a^{10} A}{7 x^7}-\frac {a^9 (10 A b+a B)}{6 x^6}-\frac {a^8 b (9 A b+2 a B)}{x^5}-\frac {15 a^7 b^2 (8 A b+3 a B)}{4 x^4}-\frac {10 a^6 b^3 (7 A b+4 a B)}{x^3}-\frac {21 a^5 b^4 (6 A b+5 a B)}{x^2}-\frac {42 a^4 b^5 (5 A b+6 a B)}{x}+15 a^2 b^7 (3 A b+8 a B) x+\frac {5}{2} a b^8 (2 A b+9 a B) x^2+\frac {1}{3} b^9 (A b+10 a B) x^3+\frac {1}{4} b^{10} B x^4+30 a^3 b^6 (4 A b+7 a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 210, normalized size = 0.97 \[ -\frac {a^{10} (6 A+7 B x)}{42 x^7}-\frac {a^9 b (5 A+6 B x)}{3 x^6}-\frac {9 a^8 b^2 (4 A+5 B x)}{4 x^5}-\frac {10 a^7 b^3 (3 A+4 B x)}{x^4}-\frac {35 a^6 b^4 (2 A+3 B x)}{x^3}-\frac {126 a^5 b^5 (A+2 B x)}{x^2}-\frac {210 a^4 A b^6}{x}+30 a^3 b^6 \log (x) (7 a B+4 A b)+120 a^3 b^7 B x+\frac {45}{2} a^2 b^8 x (2 A+B x)+\frac {5}{3} a b^9 x^2 (3 A+2 B x)+\frac {1}{12} b^{10} x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^8,x]

[Out]

(-210*a^4*A*b^6)/x + 120*a^3*b^7*B*x + (45*a^2*b^8*x*(2*A + B*x))/2 - (126*a^5*b^5*(A + 2*B*x))/x^2 + (5*a*b^9
*x^2*(3*A + 2*B*x))/3 - (35*a^6*b^4*(2*A + 3*B*x))/x^3 + (b^10*x^3*(4*A + 3*B*x))/12 - (10*a^7*b^3*(3*A + 4*B*
x))/x^4 - (9*a^8*b^2*(4*A + 5*B*x))/(4*x^5) - (a^9*b*(5*A + 6*B*x))/(3*x^6) - (a^10*(6*A + 7*B*x))/(42*x^7) +
30*a^3*b^6*(4*A*b + 7*a*B)*Log[x]

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fricas [A]  time = 0.82, size = 245, normalized size = 1.13 \[ \frac {21 \, B b^{10} x^{11} - 12 \, A a^{10} + 28 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 210 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1260 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 2520 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} \log \relax (x) - 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 1764 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 840 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 315 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 84 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 14 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

1/84*(21*B*b^10*x^11 - 12*A*a^10 + 28*(10*B*a*b^9 + A*b^10)*x^10 + 210*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1260*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 2520*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7*log(x) - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*x^6 - 1764*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 315*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^3 - 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 14*(B*a^10 + 10*A*a^9*b)*x)/x^7

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giac [A]  time = 1.00, size = 241, normalized size = 1.12 \[ \frac {1}{4} \, B b^{10} x^{4} + \frac {10}{3} \, B a b^{9} x^{3} + \frac {1}{3} \, A b^{10} x^{3} + \frac {45}{2} \, B a^{2} b^{8} x^{2} + 5 \, A a b^{9} x^{2} + 120 \, B a^{3} b^{7} x + 45 \, A a^{2} b^{8} x + 30 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, A a^{10} + 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 1764 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 840 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 315 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 84 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 14 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="giac")

[Out]

1/4*B*b^10*x^4 + 10/3*B*a*b^9*x^3 + 1/3*A*b^10*x^3 + 45/2*B*a^2*b^8*x^2 + 5*A*a*b^9*x^2 + 120*B*a^3*b^7*x + 45
*A*a^2*b^8*x + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*log(abs(x)) - 1/84*(12*A*a^10 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*x^6 + 1764*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 315*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^3 + 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 14*(B*a^10 + 10*A*a^9*b)*x)/x^7

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maple [A]  time = 0.01, size = 240, normalized size = 1.11 \[ \frac {B \,b^{10} x^{4}}{4}+\frac {A \,b^{10} x^{3}}{3}+\frac {10 B a \,b^{9} x^{3}}{3}+5 A a \,b^{9} x^{2}+\frac {45 B \,a^{2} b^{8} x^{2}}{2}+120 A \,a^{3} b^{7} \ln \relax (x )+45 A \,a^{2} b^{8} x +210 B \,a^{4} b^{6} \ln \relax (x )+120 B \,a^{3} b^{7} x -\frac {210 A \,a^{4} b^{6}}{x}-\frac {252 B \,a^{5} b^{5}}{x}-\frac {126 A \,a^{5} b^{5}}{x^{2}}-\frac {105 B \,a^{6} b^{4}}{x^{2}}-\frac {70 A \,a^{6} b^{4}}{x^{3}}-\frac {40 B \,a^{7} b^{3}}{x^{3}}-\frac {30 A \,a^{7} b^{3}}{x^{4}}-\frac {45 B \,a^{8} b^{2}}{4 x^{4}}-\frac {9 A \,a^{8} b^{2}}{x^{5}}-\frac {2 B \,a^{9} b}{x^{5}}-\frac {5 A \,a^{9} b}{3 x^{6}}-\frac {B \,a^{10}}{6 x^{6}}-\frac {A \,a^{10}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^8,x)

[Out]

1/4*b^10*B*x^4+1/3*A*x^3*b^10+10/3*B*x^3*a*b^9+5*A*x^2*a*b^9+45/2*B*x^2*a^2*b^8+45*a^2*b^8*A*x+120*a^3*b^7*B*x
-5/3*a^9/x^6*A*b-1/6*a^10/x^6*B-1/7*a^10*A/x^7-9*a^8*b^2/x^5*A-2*a^9*b/x^5*B-210*a^4*b^6/x*A-252*a^5*b^5/x*B+1
20*A*ln(x)*a^3*b^7+210*B*ln(x)*a^4*b^6-126*a^5*b^5/x^2*A-105*a^6*b^4/x^2*B-70*a^6*b^4/x^3*A-40*a^7*b^3/x^3*B-3
0*a^7*b^3/x^4*A-45/4*a^8*b^2/x^4*B

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maxima [A]  time = 1.07, size = 241, normalized size = 1.12 \[ \frac {1}{4} \, B b^{10} x^{4} + \frac {1}{3} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{3} + \frac {5}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{2} + 15 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x + 30 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} \log \relax (x) - \frac {12 \, A a^{10} + 3528 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 1764 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 840 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 315 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 84 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 14 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

1/4*B*b^10*x^4 + 1/3*(10*B*a*b^9 + A*b^10)*x^3 + 5/2*(9*B*a^2*b^8 + 2*A*a*b^9)*x^2 + 15*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*x + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*log(x) - 1/84*(12*A*a^10 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 176
4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3
+ 84*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 14*(B*a^10 + 10*A*a^9*b)*x)/x^7

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mupad [B]  time = 0.08, size = 227, normalized size = 1.05 \[ x^3\,\left (\frac {A\,b^{10}}{3}+\frac {10\,B\,a\,b^9}{3}\right )-\frac {x\,\left (\frac {B\,a^{10}}{6}+\frac {5\,A\,b\,a^9}{3}\right )+\frac {A\,a^{10}}{7}+x^2\,\left (2\,B\,a^9\,b+9\,A\,a^8\,b^2\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{4}+30\,A\,a^7\,b^3\right )+x^4\,\left (40\,B\,a^7\,b^3+70\,A\,a^6\,b^4\right )+x^5\,\left (105\,B\,a^6\,b^4+126\,A\,a^5\,b^5\right )+x^6\,\left (252\,B\,a^5\,b^5+210\,A\,a^4\,b^6\right )}{x^7}+\ln \relax (x)\,\left (210\,B\,a^4\,b^6+120\,A\,a^3\,b^7\right )+\frac {B\,b^{10}\,x^4}{4}+15\,a^2\,b^7\,x\,\left (3\,A\,b+8\,B\,a\right )+\frac {5\,a\,b^8\,x^2\,\left (2\,A\,b+9\,B\,a\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^8,x)

[Out]

x^3*((A*b^10)/3 + (10*B*a*b^9)/3) - (x*((B*a^10)/6 + (5*A*a^9*b)/3) + (A*a^10)/7 + x^2*(9*A*a^8*b^2 + 2*B*a^9*
b) + x^3*(30*A*a^7*b^3 + (45*B*a^8*b^2)/4) + x^4*(70*A*a^6*b^4 + 40*B*a^7*b^3) + x^5*(126*A*a^5*b^5 + 105*B*a^
6*b^4) + x^6*(210*A*a^4*b^6 + 252*B*a^5*b^5))/x^7 + log(x)*(120*A*a^3*b^7 + 210*B*a^4*b^6) + (B*b^10*x^4)/4 +
15*a^2*b^7*x*(3*A*b + 8*B*a) + (5*a*b^8*x^2*(2*A*b + 9*B*a))/2

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sympy [A]  time = 6.07, size = 253, normalized size = 1.17 \[ \frac {B b^{10} x^{4}}{4} + 30 a^{3} b^{6} \left (4 A b + 7 B a\right ) \log {\relax (x )} + x^{3} \left (\frac {A b^{10}}{3} + \frac {10 B a b^{9}}{3}\right ) + x^{2} \left (5 A a b^{9} + \frac {45 B a^{2} b^{8}}{2}\right ) + x \left (45 A a^{2} b^{8} + 120 B a^{3} b^{7}\right ) + \frac {- 12 A a^{10} + x^{6} \left (- 17640 A a^{4} b^{6} - 21168 B a^{5} b^{5}\right ) + x^{5} \left (- 10584 A a^{5} b^{5} - 8820 B a^{6} b^{4}\right ) + x^{4} \left (- 5880 A a^{6} b^{4} - 3360 B a^{7} b^{3}\right ) + x^{3} \left (- 2520 A a^{7} b^{3} - 945 B a^{8} b^{2}\right ) + x^{2} \left (- 756 A a^{8} b^{2} - 168 B a^{9} b\right ) + x \left (- 140 A a^{9} b - 14 B a^{10}\right )}{84 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**8,x)

[Out]

B*b**10*x**4/4 + 30*a**3*b**6*(4*A*b + 7*B*a)*log(x) + x**3*(A*b**10/3 + 10*B*a*b**9/3) + x**2*(5*A*a*b**9 + 4
5*B*a**2*b**8/2) + x*(45*A*a**2*b**8 + 120*B*a**3*b**7) + (-12*A*a**10 + x**6*(-17640*A*a**4*b**6 - 21168*B*a*
*5*b**5) + x**5*(-10584*A*a**5*b**5 - 8820*B*a**6*b**4) + x**4*(-5880*A*a**6*b**4 - 3360*B*a**7*b**3) + x**3*(
-2520*A*a**7*b**3 - 945*B*a**8*b**2) + x**2*(-756*A*a**8*b**2 - 168*B*a**9*b) + x*(-140*A*a**9*b - 14*B*a**10)
)/(84*x**7)

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