3.118 \(\int \frac{(a+b x)^{10} (A+B x)}{x^2} \, dx\)

Optimal. Leaf size=217 \[ -\frac{a^{10} A}{x}+a^9 \log (x) (a B+10 A b)+5 a^8 b x (2 a B+9 A b)+\frac{15}{2} a^7 b^2 x^2 (3 a B+8 A b)+10 a^6 b^3 x^3 (4 a B+7 A b)+\frac{21}{2} a^5 b^4 x^4 (5 a B+6 A b)+\frac{42}{5} a^4 b^5 x^5 (6 a B+5 A b)+5 a^3 b^6 x^6 (7 a B+4 A b)+\frac{15}{7} a^2 b^7 x^7 (8 a B+3 A b)+\frac{1}{9} b^9 x^9 (10 a B+A b)+\frac{5}{8} a b^8 x^8 (9 a B+2 A b)+\frac{1}{10} b^{10} B x^{10} \]

[Out]

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

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Rubi [A]  time = 0.472862, antiderivative size = 217, 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 \[ -\frac{a^{10} A}{x}+a^9 \log (x) (a B+10 A b)+5 a^8 b x (2 a B+9 A b)+\frac{15}{2} a^7 b^2 x^2 (3 a B+8 A b)+10 a^6 b^3 x^3 (4 a B+7 A b)+\frac{21}{2} a^5 b^4 x^4 (5 a B+6 A b)+\frac{42}{5} a^4 b^5 x^5 (6 a B+5 A b)+5 a^3 b^6 x^6 (7 a B+4 A b)+\frac{15}{7} a^2 b^7 x^7 (8 a B+3 A b)+\frac{1}{9} b^9 x^9 (10 a B+A b)+\frac{5}{8} a b^8 x^8 (9 a B+2 A b)+\frac{1}{10} b^{10} B x^{10} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.267617, size = 209, normalized size = 0.96 \[ -\frac{a^{10} A}{x}+a^9 \log (x) (a B+10 A b)+10 a^9 b B x+\frac{45}{2} a^8 b^2 x (2 A+B x)+20 a^7 b^3 x^2 (3 A+2 B x)+\frac{35}{2} a^6 b^4 x^3 (4 A+3 B x)+\frac{63}{5} a^5 b^5 x^4 (5 A+4 B x)+7 a^4 b^6 x^5 (6 A+5 B x)+\frac{20}{7} a^3 b^7 x^6 (7 A+6 B x)+\frac{45}{56} a^2 b^8 x^7 (8 A+7 B x)+\frac{5}{36} a b^9 x^8 (9 A+8 B x)+\frac{1}{90} b^{10} x^9 (10 A+9 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 239, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{10}}{10}}+{\frac{A{x}^{9}{b}^{10}}{9}}+{\frac{10\,B{x}^{9}a{b}^{9}}{9}}+{\frac{5\,A{x}^{8}a{b}^{9}}{4}}+{\frac{45\,B{x}^{8}{a}^{2}{b}^{8}}{8}}+{\frac{45\,A{x}^{7}{a}^{2}{b}^{8}}{7}}+{\frac{120\,B{x}^{7}{a}^{3}{b}^{7}}{7}}+20\,A{x}^{6}{a}^{3}{b}^{7}+35\,B{x}^{6}{a}^{4}{b}^{6}+42\,A{x}^{5}{a}^{4}{b}^{6}+{\frac{252\,B{x}^{5}{a}^{5}{b}^{5}}{5}}+63\,A{x}^{4}{a}^{5}{b}^{5}+{\frac{105\,B{x}^{4}{a}^{6}{b}^{4}}{2}}+70\,A{x}^{3}{a}^{6}{b}^{4}+40\,B{x}^{3}{a}^{7}{b}^{3}+60\,A{x}^{2}{a}^{7}{b}^{3}+{\frac{45\,B{x}^{2}{a}^{8}{b}^{2}}{2}}+45\,Ax{a}^{8}{b}^{2}+10\,Bx{a}^{9}b+10\,A\ln \left ( x \right ){a}^{9}b+B\ln \left ( x \right ){a}^{10}-{\frac{A{a}^{10}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.37758, size = 323, normalized size = 1.49 \[ \frac{1}{10} \, B b^{10} x^{10} - \frac{A a^{10}}{x} + \frac{1}{9} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{9} + \frac{5}{8} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{8} + \frac{15}{7} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{7} + 5 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{6} + \frac{42}{5} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{5} + \frac{21}{2} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{4} + 10 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{3} + \frac{15}{2} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{2} + 5 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x +{\left (B a^{10} + 10 \, A a^{9} b\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201489, size = 331, normalized size = 1.53 \[ \frac{252 \, B b^{10} x^{11} - 2520 \, A a^{10} + 280 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1575 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5400 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 12600 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21168 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 26460 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 25200 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 18900 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 12600 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2520 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x \log \left (x\right )}{2520 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.78683, size = 248, normalized size = 1.14 \[ - \frac{A a^{10}}{x} + \frac{B b^{10} x^{10}}{10} + a^{9} \left (10 A b + B a\right ) \log{\left (x \right )} + x^{9} \left (\frac{A b^{10}}{9} + \frac{10 B a b^{9}}{9}\right ) + x^{8} \left (\frac{5 A a b^{9}}{4} + \frac{45 B a^{2} b^{8}}{8}\right ) + x^{7} \left (\frac{45 A a^{2} b^{8}}{7} + \frac{120 B a^{3} b^{7}}{7}\right ) + x^{6} \left (20 A a^{3} b^{7} + 35 B a^{4} b^{6}\right ) + x^{5} \left (42 A a^{4} b^{6} + \frac{252 B a^{5} b^{5}}{5}\right ) + x^{4} \left (63 A a^{5} b^{5} + \frac{105 B a^{6} b^{4}}{2}\right ) + x^{3} \left (70 A a^{6} b^{4} + 40 B a^{7} b^{3}\right ) + x^{2} \left (60 A a^{7} b^{3} + \frac{45 B a^{8} b^{2}}{2}\right ) + x \left (45 A a^{8} b^{2} + 10 B a^{9} b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.293738, size = 323, normalized size = 1.49 \[ \frac{1}{10} \, B b^{10} x^{10} + \frac{10}{9} \, B a b^{9} x^{9} + \frac{1}{9} \, A b^{10} x^{9} + \frac{45}{8} \, B a^{2} b^{8} x^{8} + \frac{5}{4} \, A a b^{9} x^{8} + \frac{120}{7} \, B a^{3} b^{7} x^{7} + \frac{45}{7} \, A a^{2} b^{8} x^{7} + 35 \, B a^{4} b^{6} x^{6} + 20 \, A a^{3} b^{7} x^{6} + \frac{252}{5} \, B a^{5} b^{5} x^{5} + 42 \, A a^{4} b^{6} x^{5} + \frac{105}{2} \, B a^{6} b^{4} x^{4} + 63 \, A a^{5} b^{5} x^{4} + 40 \, B a^{7} b^{3} x^{3} + 70 \, A a^{6} b^{4} x^{3} + \frac{45}{2} \, B a^{8} b^{2} x^{2} + 60 \, A a^{7} b^{3} x^{2} + 10 \, B a^{9} b x + 45 \, A a^{8} b^{2} x - \frac{A a^{10}}{x} +{\left (B a^{10} + 10 \, A a^{9} b\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^2,x, algorithm="giac")

[Out]

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