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

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

[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]

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Rubi [A]  time = 0.473494, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^{10} A}{10 x^{10}}-\frac{a^9 (a B+10 A b)}{9 x^9}-\frac{5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac{15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac{5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac{42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac{21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac{10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac{15 a^2 b^7 (8 a B+3 A b)}{2 x^2}+b^9 \log (x) (10 a B+A b)-\frac{5 a b^8 (9 a B+2 A b)}{x}+b^{10} B x \]

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]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{10}}{10 x^{10}} - \frac{a^{9} \left (10 A b + B a\right )}{9 x^{9}} - \frac{5 a^{8} b \left (9 A b + 2 B a\right )}{8 x^{8}} - \frac{15 a^{7} b^{2} \left (8 A b + 3 B a\right )}{7 x^{7}} - \frac{5 a^{6} b^{3} \left (7 A b + 4 B a\right )}{x^{6}} - \frac{42 a^{5} b^{4} \left (6 A b + 5 B a\right )}{5 x^{5}} - \frac{21 a^{4} b^{5} \left (5 A b + 6 B a\right )}{2 x^{4}} - \frac{10 a^{3} b^{6} \left (4 A b + 7 B a\right )}{x^{3}} - \frac{15 a^{2} b^{7} \left (3 A b + 8 B a\right )}{2 x^{2}} - \frac{5 a b^{8} \left (2 A b + 9 B a\right )}{x} + b^{10} \int B\, dx + b^{9} \left (A b + 10 B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**11,x)

[Out]

-A*a**10/(10*x**10) - a**9*(10*A*b + B*a)/(9*x**9) - 5*a**8*b*(9*A*b + 2*B*a)/(8
*x**8) - 15*a**7*b**2*(8*A*b + 3*B*a)/(7*x**7) - 5*a**6*b**3*(7*A*b + 4*B*a)/x**
6 - 42*a**5*b**4*(6*A*b + 5*B*a)/(5*x**5) - 21*a**4*b**5*(5*A*b + 6*B*a)/(2*x**4
) - 10*a**3*b**6*(4*A*b + 7*B*a)/x**3 - 15*a**2*b**7*(3*A*b + 8*B*a)/(2*x**2) -
5*a*b**8*(2*A*b + 9*B*a)/x + b**10*Integral(B, x) + b**9*(A*b + 10*B*a)*log(x)

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Mathematica [A]  time = 0.156, size = 209, normalized size = 0.97 \[ -\frac{a^{10} (9 A+10 B x)}{90 x^{10}}-\frac{5 a^9 b (8 A+9 B x)}{36 x^9}-\frac{45 a^8 b^2 (7 A+8 B x)}{56 x^8}-\frac{20 a^7 b^3 (6 A+7 B x)}{7 x^7}-\frac{7 a^6 b^4 (5 A+6 B x)}{x^6}-\frac{63 a^5 b^5 (4 A+5 B x)}{5 x^5}-\frac{35 a^4 b^6 (3 A+4 B x)}{2 x^4}-\frac{20 a^3 b^7 (2 A+3 B x)}{x^3}-\frac{45 a^2 b^8 (A+2 B x)}{2 x^2}+b^9 \log (x) (10 a B+A b)-\frac{10 a A b^9}{x}+b^{10} B x \]

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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Maple [A]  time = 0.014, size = 240, normalized size = 1.1 \[{b}^{10}Bx-{\frac{45\,{a}^{8}{b}^{2}A}{8\,{x}^{8}}}-{\frac{5\,{a}^{9}bB}{4\,{x}^{8}}}-{\frac{120\,{a}^{7}{b}^{3}A}{7\,{x}^{7}}}-{\frac{45\,{a}^{8}{b}^{2}B}{7\,{x}^{7}}}-{\frac{10\,{a}^{9}bA}{9\,{x}^{9}}}-{\frac{{a}^{10}B}{9\,{x}^{9}}}+A\ln \left ( x \right ){b}^{10}+10\,B\ln \left ( x \right ) a{b}^{9}-{\frac{45\,A{a}^{2}{b}^{8}}{2\,{x}^{2}}}-60\,{\frac{B{a}^{3}{b}^{7}}{{x}^{2}}}-{\frac{252\,{a}^{5}{b}^{5}A}{5\,{x}^{5}}}-42\,{\frac{{a}^{6}{b}^{4}B}{{x}^{5}}}-10\,{\frac{a{b}^{9}A}{x}}-45\,{\frac{{a}^{2}{b}^{8}B}{x}}-40\,{\frac{{a}^{3}{b}^{7}A}{{x}^{3}}}-70\,{\frac{{a}^{4}{b}^{6}B}{{x}^{3}}}-{\frac{105\,A{a}^{4}{b}^{6}}{2\,{x}^{4}}}-63\,{\frac{{a}^{5}{b}^{5}B}{{x}^{4}}}-{\frac{A{a}^{10}}{10\,{x}^{10}}}-35\,{\frac{{a}^{6}{b}^{4}A}{{x}^{6}}}-20\,{\frac{{a}^{7}{b}^{3}B}{{x}^{6}}} \]

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-45/8*a^8*b^2/x^8*A-5/4*a^9*b/x^8*B-120/7*a^7*b^3/x^7*A-45/7*a^8*b^2/x^7
*B-10/9*a^9/x^9*A*b-1/9*a^10/x^9*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-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-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-1/1
0*a^10*A/x^10-35*a^6*b^4/x^6*A-20*a^7*b^3/x^6*B

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Maxima [A]  time = 1.37552, size = 323, normalized size = 1.5 \[ B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )} \log \left (x\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/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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Fricas [A]  time = 0.209803, size = 331, normalized size = 1.53 \[ \frac{2520 \, B b^{10} x^{11} + 2520 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} \log \left (x\right ) - 252 \, A a^{10} - 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/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 - 2
5200*(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 - 2
1168*(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 - 5
400*(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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Sympy [A]  time = 63.4233, size = 236, normalized size = 1.09 \[ B b^{10} x + b^{9} \left (A b + 10 B a\right ) \log{\left (x \right )} - \frac{252 A a^{10} + x^{9} \left (25200 A a b^{9} + 113400 B a^{2} b^{8}\right ) + x^{8} \left (56700 A a^{2} b^{8} + 151200 B a^{3} b^{7}\right ) + x^{7} \left (100800 A a^{3} b^{7} + 176400 B a^{4} b^{6}\right ) + x^{6} \left (132300 A a^{4} b^{6} + 158760 B a^{5} b^{5}\right ) + x^{5} \left (127008 A a^{5} b^{5} + 105840 B a^{6} b^{4}\right ) + x^{4} \left (88200 A a^{6} b^{4} + 50400 B a^{7} b^{3}\right ) + x^{3} \left (43200 A a^{7} b^{3} + 16200 B a^{8} b^{2}\right ) + x^{2} \left (14175 A a^{8} b^{2} + 3150 B a^{9} b\right ) + x \left (2800 A a^{9} b + 280 B a^{10}\right )}{2520 x^{10}} \]

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 + 1
13400*B*a**2*b**8) + x**8*(56700*A*a**2*b**8 + 151200*B*a**3*b**7) + x**7*(10080
0*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
+ 50400*B*a**7*b**3) + x**3*(43200*A*a**7*b**3 + 16200*B*a**8*b**2) + x**2*(1417
5*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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GIAC/XCAS [A]  time = 0.319432, size = 324, normalized size = 1.5 \[ B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*ln(abs(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