3.125 \(\int \frac{(a+b x)^{10} (A+B x)}{x^9} \, dx\)

Optimal. Leaf size=216 \[ -\frac{a^{10} A}{8 x^8}-\frac{a^9 (a B+10 A b)}{7 x^7}-\frac{5 a^8 b (2 a B+9 A b)}{6 x^6}-\frac{3 a^7 b^2 (3 a B+8 A b)}{x^5}-\frac{15 a^6 b^3 (4 a B+7 A b)}{2 x^4}-\frac{14 a^5 b^4 (5 a B+6 A b)}{x^3}-\frac{21 a^4 b^5 (6 a B+5 A b)}{x^2}-\frac{30 a^3 b^6 (7 a B+4 A b)}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)+\frac{1}{2} b^9 x^2 (10 a B+A b)+5 a b^8 x (9 a B+2 A b)+\frac{1}{3} b^{10} B x^3 \]

[Out]

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

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Rubi [A]  time = 0.48848, 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 \[ -\frac{a^{10} A}{8 x^8}-\frac{a^9 (a B+10 A b)}{7 x^7}-\frac{5 a^8 b (2 a B+9 A b)}{6 x^6}-\frac{3 a^7 b^2 (3 a B+8 A b)}{x^5}-\frac{15 a^6 b^3 (4 a B+7 A b)}{2 x^4}-\frac{14 a^5 b^4 (5 a B+6 A b)}{x^3}-\frac{21 a^4 b^5 (6 a B+5 A b)}{x^2}-\frac{30 a^3 b^6 (7 a B+4 A b)}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)+\frac{1}{2} b^9 x^2 (10 a B+A b)+5 a b^8 x (9 a B+2 A b)+\frac{1}{3} b^{10} B x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^9,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{10}}{8 x^{8}} + \frac{B b^{10} x^{3}}{3} - \frac{a^{9} \left (10 A b + B a\right )}{7 x^{7}} - \frac{5 a^{8} b \left (9 A b + 2 B a\right )}{6 x^{6}} - \frac{3 a^{7} b^{2} \left (8 A b + 3 B a\right )}{x^{5}} - \frac{15 a^{6} b^{3} \left (7 A b + 4 B a\right )}{2 x^{4}} - \frac{14 a^{5} b^{4} \left (6 A b + 5 B a\right )}{x^{3}} - \frac{21 a^{4} b^{5} \left (5 A b + 6 B a\right )}{x^{2}} - \frac{30 a^{3} b^{6} \left (4 A b + 7 B a\right )}{x} + 15 a^{2} b^{7} \left (3 A b + 8 B a\right ) \log{\left (x \right )} + 5 a b^{8} x \left (2 A b + 9 B a\right ) + b^{9} \left (A b + 10 B a\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**9,x)

[Out]

-A*a**10/(8*x**8) + B*b**10*x**3/3 - a**9*(10*A*b + B*a)/(7*x**7) - 5*a**8*b*(9*
A*b + 2*B*a)/(6*x**6) - 3*a**7*b**2*(8*A*b + 3*B*a)/x**5 - 15*a**6*b**3*(7*A*b +
4*B*a)/(2*x**4) - 14*a**5*b**4*(6*A*b + 5*B*a)/x**3 - 21*a**4*b**5*(5*A*b + 6*B
*a)/x**2 - 30*a**3*b**6*(4*A*b + 7*B*a)/x + 15*a**2*b**7*(3*A*b + 8*B*a)*log(x)
+ 5*a*b**8*x*(2*A*b + 9*B*a) + b**9*(A*b + 10*B*a)*Integral(x, x)

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Mathematica [A]  time = 0.16498, size = 208, normalized size = 0.96 \[ -\frac{a^{10} (7 A+8 B x)}{56 x^8}-\frac{5 a^9 b (6 A+7 B x)}{21 x^7}-\frac{3 a^8 b^2 (5 A+6 B x)}{2 x^6}-\frac{6 a^7 b^3 (4 A+5 B x)}{x^5}-\frac{35 a^6 b^4 (3 A+4 B x)}{2 x^4}-\frac{42 a^5 b^5 (2 A+3 B x)}{x^3}-\frac{105 a^4 b^6 (A+2 B x)}{x^2}-\frac{120 a^3 A b^7}{x}+15 a^2 b^7 \log (x) (8 a B+3 A b)+45 a^2 b^8 B x+5 a b^9 x (2 A+B x)+\frac{1}{6} b^{10} x^2 (3 A+2 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^9,x]

[Out]

(-120*a^3*A*b^7)/x + 45*a^2*b^8*B*x + 5*a*b^9*x*(2*A + B*x) - (105*a^4*b^6*(A +
2*B*x))/x^2 + (b^10*x^2*(3*A + 2*B*x))/6 - (42*a^5*b^5*(2*A + 3*B*x))/x^3 - (35*
a^6*b^4*(3*A + 4*B*x))/(2*x^4) - (6*a^7*b^3*(4*A + 5*B*x))/x^5 - (3*a^8*b^2*(5*A
+ 6*B*x))/(2*x^6) - (5*a^9*b*(6*A + 7*B*x))/(21*x^7) - (a^10*(7*A + 8*B*x))/(56
*x^8) + 15*a^2*b^7*(3*A*b + 8*a*B)*Log[x]

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Maple [A]  time = 0.014, size = 240, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{10}}{2}}+5\,B{x}^{2}a{b}^{9}+10\,Axa{b}^{9}+45\,Bx{a}^{2}{b}^{8}-{\frac{A{a}^{10}}{8\,{x}^{8}}}-{\frac{10\,{a}^{9}bA}{7\,{x}^{7}}}-{\frac{{a}^{10}B}{7\,{x}^{7}}}+45\,A\ln \left ( x \right ){a}^{2}{b}^{8}+120\,B\ln \left ( x \right ){a}^{3}{b}^{7}-105\,{\frac{A{a}^{4}{b}^{6}}{{x}^{2}}}-126\,{\frac{{a}^{5}{b}^{5}B}{{x}^{2}}}-24\,{\frac{{a}^{7}{b}^{3}A}{{x}^{5}}}-9\,{\frac{{a}^{8}{b}^{2}B}{{x}^{5}}}-120\,{\frac{{a}^{3}{b}^{7}A}{x}}-210\,{\frac{{a}^{4}{b}^{6}B}{x}}-84\,{\frac{{a}^{5}{b}^{5}A}{{x}^{3}}}-70\,{\frac{{a}^{6}{b}^{4}B}{{x}^{3}}}-{\frac{105\,{a}^{6}{b}^{4}A}{2\,{x}^{4}}}-30\,{\frac{{a}^{7}{b}^{3}B}{{x}^{4}}}-{\frac{15\,{a}^{8}{b}^{2}A}{2\,{x}^{6}}}-{\frac{5\,{a}^{9}bB}{3\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^9,x)

[Out]

1/3*b^10*B*x^3+1/2*A*x^2*b^10+5*B*x^2*a*b^9+10*A*x*a*b^9+45*B*x*a^2*b^8-1/8*a^10
*A/x^8-10/7*a^9/x^7*A*b-1/7*a^10/x^7*B+45*A*ln(x)*a^2*b^8+120*B*ln(x)*a^3*b^7-10
5*a^4*b^6/x^2*A-126*a^5*b^5/x^2*B-24*a^7*b^3/x^5*A-9*a^8*b^2/x^5*B-120*a^3*b^7/x
*A-210*a^4*b^6/x*B-84*a^5*b^5/x^3*A-70*a^6*b^4/x^3*B-105/2*a^6*b^4/x^4*A-30*a^7*
b^3/x^4*B-15/2*a^8*b^2/x^6*A-5/3*a^9*b/x^6*B

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Maxima [A]  time = 1.37747, size = 325, normalized size = 1.5 \[ \frac{1}{3} \, B b^{10} x^{3} + \frac{1}{2} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{2} + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x + 15 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} \log \left (x\right ) - \frac{21 \, A a^{10} + 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^9,x, algorithm="maxima")

[Out]

1/3*B*b^10*x^3 + 1/2*(10*B*a*b^9 + A*b^10)*x^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*x +
15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*log(x) - 1/168*(21*A*a^10 + 5040*(7*B*a^4*b^6 +
4*A*a^3*b^7)*x^7 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 2352*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*x^5 + 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 504*(3*B*a^8*b^2 + 8*A*a
^7*b^3)*x^3 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 24*(B*a^10 + 10*A*a^9*b)*x)/x^
8

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Fricas [A]  time = 0.21877, size = 331, normalized size = 1.53 \[ \frac{56 \, B b^{10} x^{11} - 21 \, A a^{10} + 84 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 840 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 2520 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} \log \left (x\right ) - 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^9,x, algorithm="fricas")

[Out]

1/168*(56*B*b^10*x^11 - 21*A*a^10 + 84*(10*B*a*b^9 + A*b^10)*x^10 + 840*(9*B*a^2
*b^8 + 2*A*a*b^9)*x^9 + 2520*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8*log(x) - 5040*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*x^7 - 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 - 2352*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*x^5 - 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 504*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*x^3 - 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 24*(B*a^10 + 10*A*a^
9*b)*x)/x^8

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Sympy [A]  time = 30.8391, size = 240, normalized size = 1.11 \[ \frac{B b^{10} x^{3}}{3} + 15 a^{2} b^{7} \left (3 A b + 8 B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{10}}{2} + 5 B a b^{9}\right ) + x \left (10 A a b^{9} + 45 B a^{2} b^{8}\right ) - \frac{21 A a^{10} + x^{7} \left (20160 A a^{3} b^{7} + 35280 B a^{4} b^{6}\right ) + x^{6} \left (17640 A a^{4} b^{6} + 21168 B a^{5} b^{5}\right ) + x^{5} \left (14112 A a^{5} b^{5} + 11760 B a^{6} b^{4}\right ) + x^{4} \left (8820 A a^{6} b^{4} + 5040 B a^{7} b^{3}\right ) + x^{3} \left (4032 A a^{7} b^{3} + 1512 B a^{8} b^{2}\right ) + x^{2} \left (1260 A a^{8} b^{2} + 280 B a^{9} b\right ) + x \left (240 A a^{9} b + 24 B a^{10}\right )}{168 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**9,x)

[Out]

B*b**10*x**3/3 + 15*a**2*b**7*(3*A*b + 8*B*a)*log(x) + x**2*(A*b**10/2 + 5*B*a*b
**9) + x*(10*A*a*b**9 + 45*B*a**2*b**8) - (21*A*a**10 + x**7*(20160*A*a**3*b**7
+ 35280*B*a**4*b**6) + x**6*(17640*A*a**4*b**6 + 21168*B*a**5*b**5) + x**5*(1411
2*A*a**5*b**5 + 11760*B*a**6*b**4) + x**4*(8820*A*a**6*b**4 + 5040*B*a**7*b**3)
+ x**3*(4032*A*a**7*b**3 + 1512*B*a**8*b**2) + x**2*(1260*A*a**8*b**2 + 280*B*a*
*9*b) + x*(240*A*a**9*b + 24*B*a**10))/(168*x**8)

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GIAC/XCAS [A]  time = 0.320782, size = 325, normalized size = 1.5 \[ \frac{1}{3} \, B b^{10} x^{3} + 5 \, B a b^{9} x^{2} + \frac{1}{2} \, A b^{10} x^{2} + 45 \, B a^{2} b^{8} x + 10 \, A a b^{9} x + 15 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{21 \, A a^{10} + 5040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 3528 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2352 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 1260 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 504 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 140 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 24 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^9,x, algorithm="giac")

[Out]

1/3*B*b^10*x^3 + 5*B*a*b^9*x^2 + 1/2*A*b^10*x^2 + 45*B*a^2*b^8*x + 10*A*a*b^9*x
+ 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*ln(abs(x)) - 1/168*(21*A*a^10 + 5040*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*x^7 + 3528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 2352*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*x^5 + 1260*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 504*(3*B*a^8*b^2 +
8*A*a^7*b^3)*x^3 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 24*(B*a^10 + 10*A*a^9*b)*
x)/x^8