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

Optimal. Leaf size=215 \[ -\frac{a^{10} A}{9 x^9}-\frac{a^9 (a B+10 A b)}{8 x^8}-\frac{5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac{5 a^7 b^2 (3 a B+8 A b)}{2 x^6}-\frac{6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac{21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac{14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac{15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac{15 a^2 b^7 (8 a B+3 A b)}{x}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac{1}{2} b^{10} B x^2 \]

[Out]

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

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Rubi [A]  time = 0.493365, antiderivative size = 215, 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 \[ -\frac{a^{10} A}{9 x^9}-\frac{a^9 (a B+10 A b)}{8 x^8}-\frac{5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac{5 a^7 b^2 (3 a B+8 A b)}{2 x^6}-\frac{6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac{21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac{14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac{15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac{15 a^2 b^7 (8 a B+3 A b)}{x}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac{1}{2} b^{10} B x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.171917, size = 206, normalized size = 0.96 \[ -\frac{a^{10} (8 A+9 B x)}{72 x^9}-\frac{5 a^9 b (7 A+8 B x)}{28 x^8}-\frac{15 a^8 b^2 (6 A+7 B x)}{14 x^7}-\frac{4 a^7 b^3 (5 A+6 B x)}{x^6}-\frac{21 a^6 b^4 (4 A+5 B x)}{2 x^5}-\frac{21 a^5 b^5 (3 A+4 B x)}{x^4}-\frac{35 a^4 b^6 (2 A+3 B x)}{x^3}-\frac{60 a^3 b^7 (A+2 B x)}{x^2}-\frac{45 a^2 A b^8}{x}+5 a b^8 \log (x) (9 a B+2 A b)+10 a b^9 B x+\frac{1}{2} b^{10} x (2 A+B x) \]

Antiderivative was successfully verified.

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

[Out]

(-45*a^2*A*b^8)/x + 10*a*b^9*B*x + (b^10*x*(2*A + B*x))/2 - (60*a^3*b^7*(A + 2*B
*x))/x^2 - (35*a^4*b^6*(2*A + 3*B*x))/x^3 - (21*a^5*b^5*(3*A + 4*B*x))/x^4 - (21
*a^6*b^4*(4*A + 5*B*x))/(2*x^5) - (4*a^7*b^3*(5*A + 6*B*x))/x^6 - (15*a^8*b^2*(6
*A + 7*B*x))/(14*x^7) - (5*a^9*b*(7*A + 8*B*x))/(28*x^8) - (a^10*(8*A + 9*B*x))/
(72*x^9) + 5*a*b^8*(2*A*b + 9*a*B)*Log[x]

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Maple [A]  time = 0.013, size = 239, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{2}}{2}}+Ax{b}^{10}+10\,Bxa{b}^{9}-{\frac{5\,{a}^{9}bA}{4\,{x}^{8}}}-{\frac{{a}^{10}B}{8\,{x}^{8}}}-{\frac{45\,{a}^{8}{b}^{2}A}{7\,{x}^{7}}}-{\frac{10\,{a}^{9}bB}{7\,{x}^{7}}}-{\frac{A{a}^{10}}{9\,{x}^{9}}}+10\,A\ln \left ( x \right ) a{b}^{9}+45\,B\ln \left ( x \right ){a}^{2}{b}^{8}-60\,{\frac{{a}^{3}{b}^{7}A}{{x}^{2}}}-105\,{\frac{{a}^{4}{b}^{6}B}{{x}^{2}}}-42\,{\frac{{a}^{6}{b}^{4}A}{{x}^{5}}}-24\,{\frac{{a}^{7}{b}^{3}B}{{x}^{5}}}-45\,{\frac{A{a}^{2}{b}^{8}}{x}}-120\,{\frac{B{a}^{3}{b}^{7}}{x}}-70\,{\frac{A{a}^{4}{b}^{6}}{{x}^{3}}}-84\,{\frac{{a}^{5}{b}^{5}B}{{x}^{3}}}-63\,{\frac{{a}^{5}{b}^{5}A}{{x}^{4}}}-{\frac{105\,{a}^{6}{b}^{4}B}{2\,{x}^{4}}}-20\,{\frac{{a}^{7}{b}^{3}A}{{x}^{6}}}-{\frac{15\,{a}^{8}{b}^{2}B}{2\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^10,x)

[Out]

1/2*b^10*B*x^2+A*x*b^10+10*B*x*a*b^9-5/4*a^9/x^8*A*b-1/8*a^10/x^8*B-45/7*a^8*b^2
/x^7*A-10/7*a^9*b/x^7*B-1/9*a^10*A/x^9+10*A*ln(x)*a*b^9+45*B*ln(x)*a^2*b^8-60*a^
3*b^7/x^2*A-105*a^4*b^6/x^2*B-42*a^6*b^4/x^5*A-24*a^7*b^3/x^5*B-45*a^2*b^8/x*A-1
20*a^3*b^7/x*B-70*a^4*b^6/x^3*A-84*a^5*b^5/x^3*B-63*a^5*b^5/x^4*A-105/2*a^6*b^4/
x^4*B-20*a^7*b^3/x^6*A-15/2*a^8*b^2/x^6*B

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Maxima [A]  time = 1.3682, size = 324, normalized size = 1.51 \[ \frac{1}{2} \, B b^{10} x^{2} +{\left (10 \, B a b^{9} + A b^{10}\right )} x + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} \log \left (x\right ) - \frac{56 \, A a^{10} + 7560 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*b^10*x^2 + (10*B*a*b^9 + A*b^10)*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*log(x) -
1/504*(56*A*a^10 + 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560*(7*B*a^4*b^6 + 4*
A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 5292*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 1260*(3*B*a^8*b^2 + 8*A*a^
7*b^3)*x^3 + 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 63*(B*a^10 + 10*A*a^9*b)*x)/x^9

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Fricas [A]  time = 0.209761, size = 331, normalized size = 1.54 \[ \frac{252 \, B b^{10} x^{11} - 56 \, A a^{10} + 504 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2520 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} \log \left (x\right ) - 7560 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 7560 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 7056 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 5292 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 3024 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 1260 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 360 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 63 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/504*(252*B*b^10*x^11 - 56*A*a^10 + 504*(10*B*a*b^9 + A*b^10)*x^10 + 2520*(9*B*
a^2*b^8 + 2*A*a*b^9)*x^9*log(x) - 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 7560*(7
*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 - 5292*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 1260*(3*B*a
^8*b^2 + 8*A*a^7*b^3)*x^3 - 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 63*(B*a^10 + 10*
A*a^9*b)*x)/x^9

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Sympy [A]  time = 43.0658, size = 238, normalized size = 1.11 \[ \frac{B b^{10} x^{2}}{2} + 5 a b^{8} \left (2 A b + 9 B a\right ) \log{\left (x \right )} + x \left (A b^{10} + 10 B a b^{9}\right ) - \frac{56 A a^{10} + x^{8} \left (22680 A a^{2} b^{8} + 60480 B a^{3} b^{7}\right ) + x^{7} \left (30240 A a^{3} b^{7} + 52920 B a^{4} b^{6}\right ) + x^{6} \left (35280 A a^{4} b^{6} + 42336 B a^{5} b^{5}\right ) + x^{5} \left (31752 A a^{5} b^{5} + 26460 B a^{6} b^{4}\right ) + x^{4} \left (21168 A a^{6} b^{4} + 12096 B a^{7} b^{3}\right ) + x^{3} \left (10080 A a^{7} b^{3} + 3780 B a^{8} b^{2}\right ) + x^{2} \left (3240 A a^{8} b^{2} + 720 B a^{9} b\right ) + x \left (630 A a^{9} b + 63 B a^{10}\right )}{504 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**10*x**2/2 + 5*a*b**8*(2*A*b + 9*B*a)*log(x) + x*(A*b**10 + 10*B*a*b**9) - (
56*A*a**10 + x**8*(22680*A*a**2*b**8 + 60480*B*a**3*b**7) + x**7*(30240*A*a**3*b
**7 + 52920*B*a**4*b**6) + x**6*(35280*A*a**4*b**6 + 42336*B*a**5*b**5) + x**5*(
31752*A*a**5*b**5 + 26460*B*a**6*b**4) + x**4*(21168*A*a**6*b**4 + 12096*B*a**7*
b**3) + x**3*(10080*A*a**7*b**3 + 3780*B*a**8*b**2) + x**2*(3240*A*a**8*b**2 + 7
20*B*a**9*b) + x*(630*A*a**9*b + 63*B*a**10))/(504*x**9)

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GIAC/XCAS [A]  time = 0.284611, size = 324, normalized size = 1.51 \[ \frac{1}{2} \, B b^{10} x^{2} + 10 \, B a b^{9} x + A b^{10} x + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{56 \, A a^{10} + 7560 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*b^10*x^2 + 10*B*a*b^9*x + A*b^10*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*ln(abs(x)
) - 1/504*(56*A*a^10 + 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 5292*(5*B*a^6*b^4 +
6*A*a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 1260*(3*B*a^8*b^2 + 8*
A*a^7*b^3)*x^3 + 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 63*(B*a^10 + 10*A*a^9*b)*x)
/x^9