3.224 \(\int \frac{x^{10}}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=159 \[ -\frac{a^{10}}{9 b^{11} (a+b x)^9}+\frac{5 a^9}{4 b^{11} (a+b x)^8}-\frac{45 a^8}{7 b^{11} (a+b x)^7}+\frac{20 a^7}{b^{11} (a+b x)^6}-\frac{42 a^6}{b^{11} (a+b x)^5}+\frac{63 a^5}{b^{11} (a+b x)^4}-\frac{70 a^4}{b^{11} (a+b x)^3}+\frac{60 a^3}{b^{11} (a+b x)^2}-\frac{45 a^2}{b^{11} (a+b x)}-\frac{10 a \log (a+b x)}{b^{11}}+\frac{x}{b^{10}} \]

[Out]

x/b^10 - a^10/(9*b^11*(a + b*x)^9) + (5*a^9)/(4*b^11*(a + b*x)^8) - (45*a^8)/(7*
b^11*(a + b*x)^7) + (20*a^7)/(b^11*(a + b*x)^6) - (42*a^6)/(b^11*(a + b*x)^5) +
(63*a^5)/(b^11*(a + b*x)^4) - (70*a^4)/(b^11*(a + b*x)^3) + (60*a^3)/(b^11*(a +
b*x)^2) - (45*a^2)/(b^11*(a + b*x)) - (10*a*Log[a + b*x])/b^11

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Rubi [A]  time = 0.253597, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^{10}}{9 b^{11} (a+b x)^9}+\frac{5 a^9}{4 b^{11} (a+b x)^8}-\frac{45 a^8}{7 b^{11} (a+b x)^7}+\frac{20 a^7}{b^{11} (a+b x)^6}-\frac{42 a^6}{b^{11} (a+b x)^5}+\frac{63 a^5}{b^{11} (a+b x)^4}-\frac{70 a^4}{b^{11} (a+b x)^3}+\frac{60 a^3}{b^{11} (a+b x)^2}-\frac{45 a^2}{b^{11} (a+b x)}-\frac{10 a \log (a+b x)}{b^{11}}+\frac{x}{b^{10}} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a + b*x)^10,x]

[Out]

x/b^10 - a^10/(9*b^11*(a + b*x)^9) + (5*a^9)/(4*b^11*(a + b*x)^8) - (45*a^8)/(7*
b^11*(a + b*x)^7) + (20*a^7)/(b^11*(a + b*x)^6) - (42*a^6)/(b^11*(a + b*x)^5) +
(63*a^5)/(b^11*(a + b*x)^4) - (70*a^4)/(b^11*(a + b*x)^3) + (60*a^3)/(b^11*(a +
b*x)^2) - (45*a^2)/(b^11*(a + b*x)) - (10*a*Log[a + b*x])/b^11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{9 b^{11} \left (a + b x\right )^{9}} + \frac{5 a^{9}}{4 b^{11} \left (a + b x\right )^{8}} - \frac{45 a^{8}}{7 b^{11} \left (a + b x\right )^{7}} + \frac{20 a^{7}}{b^{11} \left (a + b x\right )^{6}} - \frac{42 a^{6}}{b^{11} \left (a + b x\right )^{5}} + \frac{63 a^{5}}{b^{11} \left (a + b x\right )^{4}} - \frac{70 a^{4}}{b^{11} \left (a + b x\right )^{3}} + \frac{60 a^{3}}{b^{11} \left (a + b x\right )^{2}} - \frac{45 a^{2}}{b^{11} \left (a + b x\right )} - \frac{10 a \log{\left (a + b x \right )}}{b^{11}} + \int \frac{1}{b^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**10/(9*b**11*(a + b*x)**9) + 5*a**9/(4*b**11*(a + b*x)**8) - 45*a**8/(7*b**11
*(a + b*x)**7) + 20*a**7/(b**11*(a + b*x)**6) - 42*a**6/(b**11*(a + b*x)**5) + 6
3*a**5/(b**11*(a + b*x)**4) - 70*a**4/(b**11*(a + b*x)**3) + 60*a**3/(b**11*(a +
b*x)**2) - 45*a**2/(b**11*(a + b*x)) - 10*a*log(a + b*x)/b**11 + Integral(b**(-
10), x)

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Mathematica [A]  time = 0.0518203, size = 137, normalized size = 0.86 \[ -\frac{4861 a^{10}+41229 a^9 b x+153576 a^8 b^2 x^2+328104 a^7 b^3 x^3+439236 a^6 b^4 x^4+375732 a^5 b^5 x^5+197568 a^4 b^6 x^6+54432 a^3 b^7 x^7+2268 a^2 b^8 x^8-2268 a b^9 x^9+2520 a (a+b x)^9 \log (a+b x)-252 b^{10} x^{10}}{252 b^{11} (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a + b*x)^10,x]

[Out]

-(4861*a^10 + 41229*a^9*b*x + 153576*a^8*b^2*x^2 + 328104*a^7*b^3*x^3 + 439236*a
^6*b^4*x^4 + 375732*a^5*b^5*x^5 + 197568*a^4*b^6*x^6 + 54432*a^3*b^7*x^7 + 2268*
a^2*b^8*x^8 - 2268*a*b^9*x^9 - 252*b^10*x^10 + 2520*a*(a + b*x)^9*Log[a + b*x])/
(252*b^11*(a + b*x)^9)

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Maple [A]  time = 0.017, size = 154, normalized size = 1. \[{\frac{x}{{b}^{10}}}-{\frac{{a}^{10}}{9\,{b}^{11} \left ( bx+a \right ) ^{9}}}+{\frac{5\,{a}^{9}}{4\,{b}^{11} \left ( bx+a \right ) ^{8}}}-{\frac{45\,{a}^{8}}{7\,{b}^{11} \left ( bx+a \right ) ^{7}}}+20\,{\frac{{a}^{7}}{{b}^{11} \left ( bx+a \right ) ^{6}}}-42\,{\frac{{a}^{6}}{{b}^{11} \left ( bx+a \right ) ^{5}}}+63\,{\frac{{a}^{5}}{{b}^{11} \left ( bx+a \right ) ^{4}}}-70\,{\frac{{a}^{4}}{{b}^{11} \left ( bx+a \right ) ^{3}}}+60\,{\frac{{a}^{3}}{{b}^{11} \left ( bx+a \right ) ^{2}}}-45\,{\frac{{a}^{2}}{{b}^{11} \left ( bx+a \right ) }}-10\,{\frac{a\ln \left ( bx+a \right ) }{{b}^{11}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(b*x+a)^10,x)

[Out]

x/b^10-1/9*a^10/b^11/(b*x+a)^9+5/4*a^9/b^11/(b*x+a)^8-45/7*a^8/b^11/(b*x+a)^7+20
*a^7/b^11/(b*x+a)^6-42*a^6/b^11/(b*x+a)^5+63*a^5/b^11/(b*x+a)^4-70*a^4/b^11/(b*x
+a)^3+60*a^3/b^11/(b*x+a)^2-45*a^2/b^11/(b*x+a)-10*a*ln(b*x+a)/b^11

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Maxima [A]  time = 1.35872, size = 285, normalized size = 1.79 \[ -\frac{11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \,{\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac{x}{b^{10}} - \frac{10 \, a \log \left (b x + a\right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/252*(11340*a^2*b^8*x^8 + 75600*a^3*b^7*x^7 + 229320*a^4*b^6*x^6 + 407484*a^5*
b^5*x^5 + 460404*a^6*b^4*x^4 + 337176*a^7*b^3*x^3 + 155844*a^8*b^2*x^2 + 41481*a
^9*b*x + 4861*a^10)/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6
+ 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 + 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a
^8*b^12*x + a^9*b^11) + x/b^10 - 10*a*log(b*x + a)/b^11

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Fricas [A]  time = 0.206964, size = 424, normalized size = 2.67 \[ \frac{252 \, b^{10} x^{10} + 2268 \, a b^{9} x^{9} - 2268 \, a^{2} b^{8} x^{8} - 54432 \, a^{3} b^{7} x^{7} - 197568 \, a^{4} b^{6} x^{6} - 375732 \, a^{5} b^{5} x^{5} - 439236 \, a^{6} b^{4} x^{4} - 328104 \, a^{7} b^{3} x^{3} - 153576 \, a^{8} b^{2} x^{2} - 41229 \, a^{9} b x - 4861 \, a^{10} - 2520 \,{\left (a b^{9} x^{9} + 9 \, a^{2} b^{8} x^{8} + 36 \, a^{3} b^{7} x^{7} + 84 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 126 \, a^{6} b^{4} x^{4} + 84 \, a^{7} b^{3} x^{3} + 36 \, a^{8} b^{2} x^{2} + 9 \, a^{9} b x + a^{10}\right )} \log \left (b x + a\right )}{252 \,{\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*(252*b^10*x^10 + 2268*a*b^9*x^9 - 2268*a^2*b^8*x^8 - 54432*a^3*b^7*x^7 - 1
97568*a^4*b^6*x^6 - 375732*a^5*b^5*x^5 - 439236*a^6*b^4*x^4 - 328104*a^7*b^3*x^3
- 153576*a^8*b^2*x^2 - 41229*a^9*b*x - 4861*a^10 - 2520*(a*b^9*x^9 + 9*a^2*b^8*
x^8 + 36*a^3*b^7*x^7 + 84*a^4*b^6*x^6 + 126*a^5*b^5*x^5 + 126*a^6*b^4*x^4 + 84*a
^7*b^3*x^3 + 36*a^8*b^2*x^2 + 9*a^9*b*x + a^10)*log(b*x + a))/(b^20*x^9 + 9*a*b^
19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4
+ 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11)

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Sympy [A]  time = 5.32416, size = 223, normalized size = 1.4 \[ - \frac{10 a \log{\left (a + b x \right )}}{b^{11}} - \frac{4861 a^{10} + 41481 a^{9} b x + 155844 a^{8} b^{2} x^{2} + 337176 a^{7} b^{3} x^{3} + 460404 a^{6} b^{4} x^{4} + 407484 a^{5} b^{5} x^{5} + 229320 a^{4} b^{6} x^{6} + 75600 a^{3} b^{7} x^{7} + 11340 a^{2} b^{8} x^{8}}{252 a^{9} b^{11} + 2268 a^{8} b^{12} x + 9072 a^{7} b^{13} x^{2} + 21168 a^{6} b^{14} x^{3} + 31752 a^{5} b^{15} x^{4} + 31752 a^{4} b^{16} x^{5} + 21168 a^{3} b^{17} x^{6} + 9072 a^{2} b^{18} x^{7} + 2268 a b^{19} x^{8} + 252 b^{20} x^{9}} + \frac{x}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(b*x+a)**10,x)

[Out]

-10*a*log(a + b*x)/b**11 - (4861*a**10 + 41481*a**9*b*x + 155844*a**8*b**2*x**2
+ 337176*a**7*b**3*x**3 + 460404*a**6*b**4*x**4 + 407484*a**5*b**5*x**5 + 229320
*a**4*b**6*x**6 + 75600*a**3*b**7*x**7 + 11340*a**2*b**8*x**8)/(252*a**9*b**11 +
2268*a**8*b**12*x + 9072*a**7*b**13*x**2 + 21168*a**6*b**14*x**3 + 31752*a**5*b
**15*x**4 + 31752*a**4*b**16*x**5 + 21168*a**3*b**17*x**6 + 9072*a**2*b**18*x**7
+ 2268*a*b**19*x**8 + 252*b**20*x**9) + x/b**10

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GIAC/XCAS [A]  time = 0.210335, size = 163, normalized size = 1.03 \[ \frac{x}{b^{10}} - \frac{10 \, a{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{11}} - \frac{11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \,{\left (b x + a\right )}^{9} b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/b^10 - 10*a*ln(abs(b*x + a))/b^11 - 1/252*(11340*a^2*b^8*x^8 + 75600*a^3*b^7*x
^7 + 229320*a^4*b^6*x^6 + 407484*a^5*b^5*x^5 + 460404*a^6*b^4*x^4 + 337176*a^7*b
^3*x^3 + 155844*a^8*b^2*x^2 + 41481*a^9*b*x + 4861*a^10)/((b*x + a)^9*b^11)