3.2.44 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx\)

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

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Rubi [A]  time = 0.14, 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} \begin {gather*} -\frac {15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac {5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac {42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac {21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac {10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac {15 a^2 b^7 (8 a B+3 A b)}{2 x^2}-\frac {a^9 (a B+10 A b)}{9 x^9}-\frac {5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac {a^{10} A}{10 x^{10}}-\frac {5 a b^8 (9 a B+2 A b)}{x}+b^9 \log (x) (10 a B+A b)+b^{10} B x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

-(a^10*A)/(10*x^10) - (a^9*(10*A*b + a*B))/(9*x^9) - (5*a^8*b*(9*A*b + 2*a*B))/(8*x^8) - (15*a^7*b^2*(8*A*b +
3*a*B))/(7*x^7) - (5*a^6*b^3*(7*A*b + 4*a*B))/x^6 - (42*a^5*b^4*(6*A*b + 5*a*B))/(5*x^5) - (21*a^4*b^5*(5*A*b
+ 6*a*B))/(2*x^4) - (10*a^3*b^6*(4*A*b + 7*a*B))/x^3 - (15*a^2*b^7*(3*A*b + 8*a*B))/(2*x^2) - (5*a*b^8*(2*A*b
+ 9*a*B))/x + b^10*B*x + b^9*(A*b + 10*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^{11}} \, dx &=\int \left (b^{10} B+\frac {a^{10} A}{x^{11}}+\frac {a^9 (10 A b+a B)}{x^{10}}+\frac {5 a^8 b (9 A b+2 a B)}{x^9}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^8}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^7}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^6}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^5}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x^4}+\frac {15 a^2 b^7 (3 A b+8 a B)}{x^3}+\frac {5 a b^8 (2 A b+9 a B)}{x^2}+\frac {b^9 (A b+10 a B)}{x}\right ) \, dx\\ &=-\frac {a^{10} A}{10 x^{10}}-\frac {a^9 (10 A b+a B)}{9 x^9}-\frac {5 a^8 b (9 A b+2 a B)}{8 x^8}-\frac {15 a^7 b^2 (8 A b+3 a B)}{7 x^7}-\frac {5 a^6 b^3 (7 A b+4 a B)}{x^6}-\frac {42 a^5 b^4 (6 A b+5 a B)}{5 x^5}-\frac {21 a^4 b^5 (5 A b+6 a B)}{2 x^4}-\frac {10 a^3 b^6 (4 A b+7 a B)}{x^3}-\frac {15 a^2 b^7 (3 A b+8 a B)}{2 x^2}-\frac {5 a b^8 (2 A b+9 a B)}{x}+b^{10} B x+b^9 (A b+10 a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 209, normalized size = 0.97 \begin {gather*} -\frac {a^{10} (9 A+10 B x)}{90 x^{10}}-\frac {5 a^9 b (8 A+9 B x)}{36 x^9}-\frac {45 a^8 b^2 (7 A+8 B x)}{56 x^8}-\frac {20 a^7 b^3 (6 A+7 B x)}{7 x^7}-\frac {7 a^6 b^4 (5 A+6 B x)}{x^6}-\frac {63 a^5 b^5 (4 A+5 B x)}{5 x^5}-\frac {35 a^4 b^6 (3 A+4 B x)}{2 x^4}-\frac {20 a^3 b^7 (2 A+3 B x)}{x^3}-\frac {45 a^2 b^8 (A+2 B x)}{2 x^2}+b^9 \log (x) (10 a B+A b)-\frac {10 a A b^9}{x}+b^{10} B x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

(-10*a*A*b^9)/x + b^10*B*x - (45*a^2*b^8*(A + 2*B*x))/(2*x^2) - (20*a^3*b^7*(2*A + 3*B*x))/x^3 - (35*a^4*b^6*(
3*A + 4*B*x))/(2*x^4) - (63*a^5*b^5*(4*A + 5*B*x))/(5*x^5) - (7*a^6*b^4*(5*A + 6*B*x))/x^6 - (20*a^7*b^3*(6*A
+ 7*B*x))/(7*x^7) - (45*a^8*b^2*(7*A + 8*B*x))/(56*x^8) - (5*a^9*b*(8*A + 9*B*x))/(36*x^9) - (a^10*(9*A + 10*B
*x))/(90*x^10) + b^9*(A*b + 10*a*B)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/x^11, x]

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fricas [A]  time = 1.08, size = 245, normalized size = 1.13 \begin {gather*} \frac {2520 \, B b^{10} x^{11} + 2520 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} \log \relax (x) - 252 \, A a^{10} - 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*B*b^10*x^11 + 2520*(10*B*a*b^9 + A*b^10)*x^10*log(x) - 252*A*a^10 - 12600*(9*B*a^2*b^8 + 2*A*a*b^
9)*x^9 - 18900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 26460*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 - 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 5400*(3*B*a
^8*b^2 + 8*A*a^7*b^3)*x^3 - 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

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giac [A]  time = 1.01, size = 240, normalized size = 1.11 \begin {gather*} B b^{10} x + {\left (10 \, B a b^{9} + A b^{10}\right )} \log \left ({\left | x \right |}\right ) - \frac {252 \, A a^{10} + 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="giac")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(abs(x)) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1890
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
x^6 + 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

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maple [A]  time = 0.01, size = 240, normalized size = 1.11 \begin {gather*} A \,b^{10} \ln \relax (x )+10 B a \,b^{9} \ln \relax (x )+B \,b^{10} x -\frac {10 A a \,b^{9}}{x}-\frac {45 B \,a^{2} b^{8}}{x}-\frac {45 A \,a^{2} b^{8}}{2 x^{2}}-\frac {60 B \,a^{3} b^{7}}{x^{2}}-\frac {40 A \,a^{3} b^{7}}{x^{3}}-\frac {70 B \,a^{4} b^{6}}{x^{3}}-\frac {105 A \,a^{4} b^{6}}{2 x^{4}}-\frac {63 B \,a^{5} b^{5}}{x^{4}}-\frac {252 A \,a^{5} b^{5}}{5 x^{5}}-\frac {42 B \,a^{6} b^{4}}{x^{5}}-\frac {35 A \,a^{6} b^{4}}{x^{6}}-\frac {20 B \,a^{7} b^{3}}{x^{6}}-\frac {120 A \,a^{7} b^{3}}{7 x^{7}}-\frac {45 B \,a^{8} b^{2}}{7 x^{7}}-\frac {45 A \,a^{8} b^{2}}{8 x^{8}}-\frac {5 B \,a^{9} b}{4 x^{8}}-\frac {10 A \,a^{9} b}{9 x^{9}}-\frac {B \,a^{10}}{9 x^{9}}-\frac {A \,a^{10}}{10 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^11,x)

[Out]

b^10*B*x-120/7*a^7*b^3/x^7*A-45/7*a^8*b^2/x^7*B-35*a^6*b^4/x^6*A-20*a^7*b^3/x^6*B-252/5*a^5*b^5/x^5*A-42*a^6*b
^4/x^5*B-10*a*b^9/x*A-45*a^2*b^8/x*B+A*ln(x)*b^10+10*B*ln(x)*a*b^9-45/2*a^2*b^8/x^2*A-60*a^3*b^7/x^2*B-45/8*a^
8*b^2/x^8*A-5/4*a^9*b/x^8*B-10/9*a^9/x^9*A*b-1/9*a^10/x^9*B-1/10*a^10*A/x^10-40*a^3*b^7/x^3*A-70*a^4*b^6/x^3*B
-105/2*a^4*b^6/x^4*A-63*a^5*b^5/x^4*B

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maxima [A]  time = 1.04, size = 239, normalized size = 1.11 \begin {gather*} B b^{10} x + {\left (10 \, B a b^{9} + A b^{10}\right )} \log \relax (x) - \frac {252 \, A a^{10} + 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="maxima")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(x) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 18900*(8*
B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

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mupad [B]  time = 0.37, size = 231, normalized size = 1.07 \begin {gather*} \ln \relax (x)\,\left (A\,b^{10}+10\,B\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{9}+\frac {10\,A\,b\,a^9}{9}\right )+\frac {A\,a^{10}}{10}+x^2\,\left (\frac {5\,B\,a^9\,b}{4}+\frac {45\,A\,a^8\,b^2}{8}\right )+x^9\,\left (45\,B\,a^2\,b^8+10\,A\,a\,b^9\right )+x^4\,\left (20\,B\,a^7\,b^3+35\,A\,a^6\,b^4\right )+x^8\,\left (60\,B\,a^3\,b^7+\frac {45\,A\,a^2\,b^8}{2}\right )+x^7\,\left (70\,B\,a^4\,b^6+40\,A\,a^3\,b^7\right )+x^6\,\left (63\,B\,a^5\,b^5+\frac {105\,A\,a^4\,b^6}{2}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{7}+\frac {120\,A\,a^7\,b^3}{7}\right )+x^5\,\left (42\,B\,a^6\,b^4+\frac {252\,A\,a^5\,b^5}{5}\right )}{x^{10}}+B\,b^{10}\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^11,x)

[Out]

log(x)*(A*b^10 + 10*B*a*b^9) - (x*((B*a^10)/9 + (10*A*a^9*b)/9) + (A*a^10)/10 + x^2*((45*A*a^8*b^2)/8 + (5*B*a
^9*b)/4) + x^9*(45*B*a^2*b^8 + 10*A*a*b^9) + x^4*(35*A*a^6*b^4 + 20*B*a^7*b^3) + x^8*((45*A*a^2*b^8)/2 + 60*B*
a^3*b^7) + x^7*(40*A*a^3*b^7 + 70*B*a^4*b^6) + x^6*((105*A*a^4*b^6)/2 + 63*B*a^5*b^5) + x^3*((120*A*a^7*b^3)/7
+ (45*B*a^8*b^2)/7) + x^5*((252*A*a^5*b^5)/5 + 42*B*a^6*b^4))/x^10 + B*b^10*x

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sympy [A]  time = 16.82, size = 252, normalized size = 1.17 \begin {gather*} B b^{10} x + b^{9} \left (A b + 10 B a\right ) \log {\relax (x )} + \frac {- 252 A a^{10} + x^{9} \left (- 25200 A a b^{9} - 113400 B a^{2} b^{8}\right ) + x^{8} \left (- 56700 A a^{2} b^{8} - 151200 B a^{3} b^{7}\right ) + x^{7} \left (- 100800 A a^{3} b^{7} - 176400 B a^{4} b^{6}\right ) + x^{6} \left (- 132300 A a^{4} b^{6} - 158760 B a^{5} b^{5}\right ) + x^{5} \left (- 127008 A a^{5} b^{5} - 105840 B a^{6} b^{4}\right ) + x^{4} \left (- 88200 A a^{6} b^{4} - 50400 B a^{7} b^{3}\right ) + x^{3} \left (- 43200 A a^{7} b^{3} - 16200 B a^{8} b^{2}\right ) + x^{2} \left (- 14175 A a^{8} b^{2} - 3150 B a^{9} b\right ) + x \left (- 2800 A a^{9} b - 280 B a^{10}\right )}{2520 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**11,x)

[Out]

B*b**10*x + b**9*(A*b + 10*B*a)*log(x) + (-252*A*a**10 + x**9*(-25200*A*a*b**9 - 113400*B*a**2*b**8) + x**8*(-
56700*A*a**2*b**8 - 151200*B*a**3*b**7) + x**7*(-100800*A*a**3*b**7 - 176400*B*a**4*b**6) + x**6*(-132300*A*a*
*4*b**6 - 158760*B*a**5*b**5) + x**5*(-127008*A*a**5*b**5 - 105840*B*a**6*b**4) + x**4*(-88200*A*a**6*b**4 - 5
0400*B*a**7*b**3) + x**3*(-43200*A*a**7*b**3 - 16200*B*a**8*b**2) + x**2*(-14175*A*a**8*b**2 - 3150*B*a**9*b)
+ x*(-2800*A*a**9*b - 280*B*a**10))/(2520*x**10)

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