3.128 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{12}} \, dx\)

Optimal. Leaf size=153 \[ -\frac{a^{10} B}{10 x^{10}}-\frac{10 a^9 b B}{9 x^9}-\frac{45 a^8 b^2 B}{8 x^8}-\frac{120 a^7 b^3 B}{7 x^7}-\frac{35 a^6 b^4 B}{x^6}-\frac{252 a^5 b^5 B}{5 x^5}-\frac{105 a^4 b^6 B}{2 x^4}-\frac{40 a^3 b^7 B}{x^3}-\frac{45 a^2 b^8 B}{2 x^2}-\frac{A (a+b x)^{11}}{11 a x^{11}}-\frac{10 a b^9 B}{x}+b^{10} B \log (x) \]

[Out]

-(a^10*B)/(10*x^10) - (10*a^9*b*B)/(9*x^9) - (45*a^8*b^2*B)/(8*x^8) - (120*a^7*b
^3*B)/(7*x^7) - (35*a^6*b^4*B)/x^6 - (252*a^5*b^5*B)/(5*x^5) - (105*a^4*b^6*B)/(
2*x^4) - (40*a^3*b^7*B)/x^3 - (45*a^2*b^8*B)/(2*x^2) - (10*a*b^9*B)/x - (A*(a +
b*x)^11)/(11*a*x^11) + b^10*B*Log[x]

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Rubi [A]  time = 0.187065, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^{10} B}{10 x^{10}}-\frac{10 a^9 b B}{9 x^9}-\frac{45 a^8 b^2 B}{8 x^8}-\frac{120 a^7 b^3 B}{7 x^7}-\frac{35 a^6 b^4 B}{x^6}-\frac{252 a^5 b^5 B}{5 x^5}-\frac{105 a^4 b^6 B}{2 x^4}-\frac{40 a^3 b^7 B}{x^3}-\frac{45 a^2 b^8 B}{2 x^2}-\frac{A (a+b x)^{11}}{11 a x^{11}}-\frac{10 a b^9 B}{x}+b^{10} B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^12,x]

[Out]

-(a^10*B)/(10*x^10) - (10*a^9*b*B)/(9*x^9) - (45*a^8*b^2*B)/(8*x^8) - (120*a^7*b
^3*B)/(7*x^7) - (35*a^6*b^4*B)/x^6 - (252*a^5*b^5*B)/(5*x^5) - (105*a^4*b^6*B)/(
2*x^4) - (40*a^3*b^7*B)/x^3 - (45*a^2*b^8*B)/(2*x^2) - (10*a*b^9*B)/x - (A*(a +
b*x)^11)/(11*a*x^11) + b^10*B*Log[x]

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Rubi in Sympy [A]  time = 48.5731, size = 160, normalized size = 1.05 \[ - \frac{A \left (a + b x\right )^{11}}{11 a x^{11}} - \frac{B a^{10}}{10 x^{10}} - \frac{10 B a^{9} b}{9 x^{9}} - \frac{45 B a^{8} b^{2}}{8 x^{8}} - \frac{120 B a^{7} b^{3}}{7 x^{7}} - \frac{35 B a^{6} b^{4}}{x^{6}} - \frac{252 B a^{5} b^{5}}{5 x^{5}} - \frac{105 B a^{4} b^{6}}{2 x^{4}} - \frac{40 B a^{3} b^{7}}{x^{3}} - \frac{45 B a^{2} b^{8}}{2 x^{2}} - \frac{10 B a b^{9}}{x} + B b^{10} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**12,x)

[Out]

-A*(a + b*x)**11/(11*a*x**11) - B*a**10/(10*x**10) - 10*B*a**9*b/(9*x**9) - 45*B
*a**8*b**2/(8*x**8) - 120*B*a**7*b**3/(7*x**7) - 35*B*a**6*b**4/x**6 - 252*B*a**
5*b**5/(5*x**5) - 105*B*a**4*b**6/(2*x**4) - 40*B*a**3*b**7/x**3 - 45*B*a**2*b**
8/(2*x**2) - 10*B*a*b**9/x + B*b**10*log(x)

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Mathematica [A]  time = 0.220915, size = 212, normalized size = 1.39 \[ -\frac{a^{10} (10 A+11 B x)}{110 x^{11}}-\frac{a^9 b (9 A+10 B x)}{9 x^{10}}-\frac{5 a^8 b^2 (8 A+9 B x)}{8 x^9}-\frac{15 a^7 b^3 (7 A+8 B x)}{7 x^8}-\frac{5 a^6 b^4 (6 A+7 B x)}{x^7}-\frac{42 a^5 b^5 (5 A+6 B x)}{5 x^6}-\frac{21 a^4 b^6 (4 A+5 B x)}{2 x^5}-\frac{10 a^3 b^7 (3 A+4 B x)}{x^4}-\frac{15 a^2 b^8 (2 A+3 B x)}{2 x^3}-\frac{5 a b^9 (A+2 B x)}{x^2}-\frac{A b^{10}}{x}+b^{10} B \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^12,x]

[Out]

-((A*b^10)/x) - (5*a*b^9*(A + 2*B*x))/x^2 - (15*a^2*b^8*(2*A + 3*B*x))/(2*x^3) -
(10*a^3*b^7*(3*A + 4*B*x))/x^4 - (21*a^4*b^6*(4*A + 5*B*x))/(2*x^5) - (42*a^5*b
^5*(5*A + 6*B*x))/(5*x^6) - (5*a^6*b^4*(6*A + 7*B*x))/x^7 - (15*a^7*b^3*(7*A + 8
*B*x))/(7*x^8) - (5*a^8*b^2*(8*A + 9*B*x))/(8*x^9) - (a^9*b*(9*A + 10*B*x))/(9*x
^10) - (a^10*(10*A + 11*B*x))/(110*x^11) + b^10*B*Log[x]

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Maple [A]  time = 0.013, size = 244, normalized size = 1.6 \[ -15\,{\frac{{a}^{7}{b}^{3}A}{{x}^{8}}}-{\frac{45\,{a}^{8}{b}^{2}B}{8\,{x}^{8}}}-{\frac{A{a}^{10}}{11\,{x}^{11}}}-30\,{\frac{{a}^{6}{b}^{4}A}{{x}^{7}}}-{\frac{120\,{a}^{7}{b}^{3}B}{7\,{x}^{7}}}-5\,{\frac{{a}^{8}{b}^{2}A}{{x}^{9}}}-{\frac{10\,{a}^{9}bB}{9\,{x}^{9}}}+{b}^{10}B\ln \left ( x \right ) -5\,{\frac{a{b}^{9}A}{{x}^{2}}}-{\frac{45\,{a}^{2}{b}^{8}B}{2\,{x}^{2}}}-42\,{\frac{A{a}^{4}{b}^{6}}{{x}^{5}}}-{\frac{252\,{a}^{5}{b}^{5}B}{5\,{x}^{5}}}-{\frac{A{b}^{10}}{x}}-10\,{\frac{a{b}^{9}B}{x}}-15\,{\frac{A{a}^{2}{b}^{8}}{{x}^{3}}}-40\,{\frac{B{a}^{3}{b}^{7}}{{x}^{3}}}-30\,{\frac{{a}^{3}{b}^{7}A}{{x}^{4}}}-{\frac{105\,{a}^{4}{b}^{6}B}{2\,{x}^{4}}}-{\frac{{a}^{9}bA}{{x}^{10}}}-{\frac{{a}^{10}B}{10\,{x}^{10}}}-42\,{\frac{{a}^{5}{b}^{5}A}{{x}^{6}}}-35\,{\frac{{a}^{6}{b}^{4}B}{{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^12,x)

[Out]

-15*a^7*b^3/x^8*A-45/8*a^8*b^2*B/x^8-1/11*A*a^10/x^11-30*a^6*b^4/x^7*A-120/7*a^7
*b^3*B/x^7-5*a^8*b^2/x^9*A-10/9*a^9*b*B/x^9+b^10*B*ln(x)-5*a*b^9/x^2*A-45/2*a^2*
b^8*B/x^2-42*a^4*b^6/x^5*A-252/5*a^5*b^5*B/x^5-b^10/x*A-10*a*b^9*B/x-15*a^2*b^8/
x^3*A-40*a^3*b^7*B/x^3-30*a^3*b^7/x^4*A-105/2*a^4*b^6*B/x^4-a^9/x^10*A*b-1/10*a^
10*B/x^10-42*a^5*b^5/x^6*A-35*a^6*b^4*B/x^6

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Maxima [A]  time = 1.44525, size = 327, normalized size = 2.14 \[ B b^{10} \log \left (x\right ) - \frac{2520 \, A a^{10} + 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^12,x, algorithm="maxima")

[Out]

B*b^10*log(x) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^10 + 69300*
(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 207900*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19404
0*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 519
75*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 2772*
(B*a^10 + 10*A*a^9*b)*x)/x^11

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Fricas [A]  time = 0.201775, size = 331, normalized size = 2.16 \[ \frac{27720 \, B b^{10} x^{11} \log \left (x\right ) - 2520 \, A a^{10} - 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} - 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^12,x, algorithm="fricas")

[Out]

1/27720*(27720*B*b^10*x^11*log(x) - 2520*A*a^10 - 27720*(10*B*a*b^9 + A*b^10)*x^
10 - 69300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 - 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^
8 - 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 232848*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
x^6 - 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*x^4 - 51975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 15400*(2*B*a^9*b + 9*A*a^8*b^2)*
x^2 - 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11

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Sympy [A]  time = 89.541, size = 241, normalized size = 1.58 \[ B b^{10} \log{\left (x \right )} - \frac{2520 A a^{10} + x^{10} \left (27720 A b^{10} + 277200 B a b^{9}\right ) + x^{9} \left (138600 A a b^{9} + 623700 B a^{2} b^{8}\right ) + x^{8} \left (415800 A a^{2} b^{8} + 1108800 B a^{3} b^{7}\right ) + x^{7} \left (831600 A a^{3} b^{7} + 1455300 B a^{4} b^{6}\right ) + x^{6} \left (1164240 A a^{4} b^{6} + 1397088 B a^{5} b^{5}\right ) + x^{5} \left (1164240 A a^{5} b^{5} + 970200 B a^{6} b^{4}\right ) + x^{4} \left (831600 A a^{6} b^{4} + 475200 B a^{7} b^{3}\right ) + x^{3} \left (415800 A a^{7} b^{3} + 155925 B a^{8} b^{2}\right ) + x^{2} \left (138600 A a^{8} b^{2} + 30800 B a^{9} b\right ) + x \left (27720 A a^{9} b + 2772 B a^{10}\right )}{27720 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**12,x)

[Out]

B*b**10*log(x) - (2520*A*a**10 + x**10*(27720*A*b**10 + 277200*B*a*b**9) + x**9*
(138600*A*a*b**9 + 623700*B*a**2*b**8) + x**8*(415800*A*a**2*b**8 + 1108800*B*a*
*3*b**7) + x**7*(831600*A*a**3*b**7 + 1455300*B*a**4*b**6) + x**6*(1164240*A*a**
4*b**6 + 1397088*B*a**5*b**5) + x**5*(1164240*A*a**5*b**5 + 970200*B*a**6*b**4)
+ x**4*(831600*A*a**6*b**4 + 475200*B*a**7*b**3) + x**3*(415800*A*a**7*b**3 + 15
5925*B*a**8*b**2) + x**2*(138600*A*a**8*b**2 + 30800*B*a**9*b) + x*(27720*A*a**9
*b + 2772*B*a**10))/(27720*x**11)

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GIAC/XCAS [A]  time = 0.363109, size = 328, normalized size = 2.14 \[ B b^{10}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2520 \, A a^{10} + 27720 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^12,x, algorithm="giac")

[Out]

B*b^10*ln(abs(x)) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^10 + 69
300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 207
900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 1
94040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 +
51975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 2
772*(B*a^10 + 10*A*a^9*b)*x)/x^11