3.226 \(\int \frac{x^8}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=17 \[ \frac{x^9}{9 a (a+b x)^9} \]

[Out]

x^9/(9*a*(a + b*x)^9)

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Rubi [A]  time = 0.011927, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^9}{9 a (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x)^10,x]

[Out]

x^9/(9*a*(a + b*x)^9)

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Rubi in Sympy [A]  time = 2.32601, size = 12, normalized size = 0.71 \[ \frac{x^{9}}{9 a \left (a + b x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x+a)**10,x)

[Out]

x**9/(9*a*(a + b*x)**9)

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Mathematica [B]  time = 0.0282628, size = 97, normalized size = 5.71 \[ -\frac{a^8+9 a^7 b x+36 a^6 b^2 x^2+84 a^5 b^3 x^3+126 a^4 b^4 x^4+126 a^3 b^5 x^5+84 a^2 b^6 x^6+36 a b^7 x^7+9 b^8 x^8}{9 b^9 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x)^10,x]

[Out]

-(a^8 + 9*a^7*b*x + 36*a^6*b^2*x^2 + 84*a^5*b^3*x^3 + 126*a^4*b^4*x^4 + 126*a^3*
b^5*x^5 + 84*a^2*b^6*x^6 + 36*a*b^7*x^7 + 9*b^8*x^8)/(9*b^9*(a + b*x)^9)

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Maple [B]  time = 0.01, size = 131, normalized size = 7.7 \[{\frac{28\,{a}^{5}}{3\,{b}^{9} \left ( bx+a \right ) ^{6}}}-{\frac{{a}^{8}}{9\,{b}^{9} \left ( bx+a \right ) ^{9}}}-{\frac{28\,{a}^{2}}{3\,{b}^{9} \left ( bx+a \right ) ^{3}}}+14\,{\frac{{a}^{3}}{{b}^{9} \left ( bx+a \right ) ^{4}}}+4\,{\frac{a}{{b}^{9} \left ( bx+a \right ) ^{2}}}-14\,{\frac{{a}^{4}}{{b}^{9} \left ( bx+a \right ) ^{5}}}-4\,{\frac{{a}^{6}}{{b}^{9} \left ( bx+a \right ) ^{7}}}+{\frac{{a}^{7}}{{b}^{9} \left ( bx+a \right ) ^{8}}}-{\frac{1}{ \left ( bx+a \right ){b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x+a)^10,x)

[Out]

28/3*a^5/b^9/(b*x+a)^6-1/9*a^8/b^9/(b*x+a)^9-28/3*a^2/b^9/(b*x+a)^3+14*a^3/b^9/(
b*x+a)^4+4*a/b^9/(b*x+a)^2-14*a^4/b^9/(b*x+a)^5-4*a^6/b^9/(b*x+a)^7+a^7/b^9/(b*x
+a)^8-1/(b*x+a)/b^9

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Maxima [A]  time = 1.34649, size = 251, normalized size = 14.76 \[ -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \,{\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x + a)^10,x, algorithm="maxima")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*
x^4 + 84*a^5*b^3*x^3 + 36*a^6*b^2*x^2 + 9*a^7*b*x + a^8)/(b^18*x^9 + 9*a*b^17*x^
8 + 36*a^2*b^16*x^7 + 84*a^3*b^15*x^6 + 126*a^4*b^14*x^5 + 126*a^5*b^13*x^4 + 84
*a^6*b^12*x^3 + 36*a^7*b^11*x^2 + 9*a^8*b^10*x + a^9*b^9)

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Fricas [A]  time = 0.2011, size = 251, normalized size = 14.76 \[ -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \,{\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x + a)^10,x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*
x^4 + 84*a^5*b^3*x^3 + 36*a^6*b^2*x^2 + 9*a^7*b*x + a^8)/(b^18*x^9 + 9*a*b^17*x^
8 + 36*a^2*b^16*x^7 + 84*a^3*b^15*x^6 + 126*a^4*b^14*x^5 + 126*a^5*b^13*x^4 + 84
*a^6*b^12*x^3 + 36*a^7*b^11*x^2 + 9*a^8*b^10*x + a^9*b^9)

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Sympy [A]  time = 4.32046, size = 199, normalized size = 11.71 \[ - \frac{a^{8} + 9 a^{7} b x + 36 a^{6} b^{2} x^{2} + 84 a^{5} b^{3} x^{3} + 126 a^{4} b^{4} x^{4} + 126 a^{3} b^{5} x^{5} + 84 a^{2} b^{6} x^{6} + 36 a b^{7} x^{7} + 9 b^{8} x^{8}}{9 a^{9} b^{9} + 81 a^{8} b^{10} x + 324 a^{7} b^{11} x^{2} + 756 a^{6} b^{12} x^{3} + 1134 a^{5} b^{13} x^{4} + 1134 a^{4} b^{14} x^{5} + 756 a^{3} b^{15} x^{6} + 324 a^{2} b^{16} x^{7} + 81 a b^{17} x^{8} + 9 b^{18} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x+a)**10,x)

[Out]

-(a**8 + 9*a**7*b*x + 36*a**6*b**2*x**2 + 84*a**5*b**3*x**3 + 126*a**4*b**4*x**4
+ 126*a**3*b**5*x**5 + 84*a**2*b**6*x**6 + 36*a*b**7*x**7 + 9*b**8*x**8)/(9*a**
9*b**9 + 81*a**8*b**10*x + 324*a**7*b**11*x**2 + 756*a**6*b**12*x**3 + 1134*a**5
*b**13*x**4 + 1134*a**4*b**14*x**5 + 756*a**3*b**15*x**6 + 324*a**2*b**16*x**7 +
81*a*b**17*x**8 + 9*b**18*x**9)

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GIAC/XCAS [A]  time = 0.219259, size = 128, normalized size = 7.53 \[ -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \,{\left (b x + a\right )}^{9} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x + a)^10,x, algorithm="giac")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*
x^4 + 84*a^5*b^3*x^3 + 36*a^6*b^2*x^2 + 9*a^7*b*x + a^8)/((b*x + a)^9*b^9)