3.229 \(\int \frac{x^5}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=69 \[ \frac{x^6}{504 a^4 (a+b x)^6}+\frac{x^6}{84 a^3 (a+b x)^7}+\frac{x^6}{24 a^2 (a+b x)^8}+\frac{x^6}{9 a (a+b x)^9} \]

[Out]

x^6/(9*a*(a + b*x)^9) + x^6/(24*a^2*(a + b*x)^8) + x^6/(84*a^3*(a + b*x)^7) + x^
6/(504*a^4*(a + b*x)^6)

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Rubi [A]  time = 0.0525418, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^6}{504 a^4 (a+b x)^6}+\frac{x^6}{84 a^3 (a+b x)^7}+\frac{x^6}{24 a^2 (a+b x)^8}+\frac{x^6}{9 a (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x)^10,x]

[Out]

x^6/(9*a*(a + b*x)^9) + x^6/(24*a^2*(a + b*x)^8) + x^6/(84*a^3*(a + b*x)^7) + x^
6/(504*a^4*(a + b*x)^6)

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Rubi in Sympy [A]  time = 9.42943, size = 58, normalized size = 0.84 \[ \frac{x^{6}}{9 a \left (a + b x\right )^{9}} + \frac{x^{6}}{24 a^{2} \left (a + b x\right )^{8}} + \frac{x^{6}}{84 a^{3} \left (a + b x\right )^{7}} + \frac{x^{6}}{504 a^{4} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)**10,x)

[Out]

x**6/(9*a*(a + b*x)**9) + x**6/(24*a**2*(a + b*x)**8) + x**6/(84*a**3*(a + b*x)*
*7) + x**6/(504*a**4*(a + b*x)**6)

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Mathematica [A]  time = 0.0218206, size = 64, normalized size = 0.93 \[ -\frac{a^5+9 a^4 b x+36 a^3 b^2 x^2+84 a^2 b^3 x^3+126 a b^4 x^4+126 b^5 x^5}{504 b^6 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x)^10,x]

[Out]

-(a^5 + 9*a^4*b*x + 36*a^3*b^2*x^2 + 84*a^2*b^3*x^3 + 126*a*b^4*x^4 + 126*b^5*x^
5)/(504*b^6*(a + b*x)^9)

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Maple [A]  time = 0.01, size = 86, normalized size = 1.3 \[ -{\frac{5\,{a}^{4}}{8\,{b}^{6} \left ( bx+a \right ) ^{8}}}-{\frac{5\,{a}^{2}}{3\,{b}^{6} \left ( bx+a \right ) ^{6}}}+{\frac{{a}^{5}}{9\,{b}^{6} \left ( bx+a \right ) ^{9}}}+{\frac{a}{{b}^{6} \left ( bx+a \right ) ^{5}}}+{\frac{10\,{a}^{3}}{7\,{b}^{6} \left ( bx+a \right ) ^{7}}}-{\frac{1}{4\, \left ( bx+a \right ) ^{4}{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)^10,x)

[Out]

-5/8*a^4/b^6/(b*x+a)^8-5/3*a^2/b^6/(b*x+a)^6+1/9*a^5/b^6/(b*x+a)^9+a/b^6/(b*x+a)
^5+10/7*a^3/b^6/(b*x+a)^7-1/4/(b*x+a)^4/b^6

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Maxima [A]  time = 1.36016, size = 207, normalized size = 3. \[ -\frac{126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \,{\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*
x + a^5)/(b^15*x^9 + 9*a*b^14*x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*
b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^2 + 9*a^8*b^7*x + a^
9*b^6)

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Fricas [A]  time = 0.206536, size = 207, normalized size = 3. \[ -\frac{126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \,{\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*
x + a^5)/(b^15*x^9 + 9*a*b^14*x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*
b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^2 + 9*a^8*b^7*x + a^
9*b^6)

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Sympy [A]  time = 3.57153, size = 163, normalized size = 2.36 \[ - \frac{a^{5} + 9 a^{4} b x + 36 a^{3} b^{2} x^{2} + 84 a^{2} b^{3} x^{3} + 126 a b^{4} x^{4} + 126 b^{5} x^{5}}{504 a^{9} b^{6} + 4536 a^{8} b^{7} x + 18144 a^{7} b^{8} x^{2} + 42336 a^{6} b^{9} x^{3} + 63504 a^{5} b^{10} x^{4} + 63504 a^{4} b^{11} x^{5} + 42336 a^{3} b^{12} x^{6} + 18144 a^{2} b^{13} x^{7} + 4536 a b^{14} x^{8} + 504 b^{15} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)**10,x)

[Out]

-(a**5 + 9*a**4*b*x + 36*a**3*b**2*x**2 + 84*a**2*b**3*x**3 + 126*a*b**4*x**4 +
126*b**5*x**5)/(504*a**9*b**6 + 4536*a**8*b**7*x + 18144*a**7*b**8*x**2 + 42336*
a**6*b**9*x**3 + 63504*a**5*b**10*x**4 + 63504*a**4*b**11*x**5 + 42336*a**3*b**1
2*x**6 + 18144*a**2*b**13*x**7 + 4536*a*b**14*x**8 + 504*b**15*x**9)

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GIAC/XCAS [A]  time = 0.206471, size = 84, normalized size = 1.22 \[ -\frac{126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \,{\left (b x + a\right )}^{9} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*
x + a^5)/((b*x + a)^9*b^6)