3.230 \(\int \frac{x^4}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{a^4}{9 b^5 (a+b x)^9}+\frac{a^3}{2 b^5 (a+b x)^8}-\frac{6 a^2}{7 b^5 (a+b x)^7}+\frac{2 a}{3 b^5 (a+b x)^6}-\frac{1}{5 b^5 (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0902403, antiderivative size = 81, 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 \[ -\frac{a^4}{9 b^5 (a+b x)^9}+\frac{a^3}{2 b^5 (a+b x)^8}-\frac{6 a^2}{7 b^5 (a+b x)^7}+\frac{2 a}{3 b^5 (a+b x)^6}-\frac{1}{5 b^5 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.1218, size = 75, normalized size = 0.93 \[ - \frac{a^{4}}{9 b^{5} \left (a + b x\right )^{9}} + \frac{a^{3}}{2 b^{5} \left (a + b x\right )^{8}} - \frac{6 a^{2}}{7 b^{5} \left (a + b x\right )^{7}} + \frac{2 a}{3 b^{5} \left (a + b x\right )^{6}} - \frac{1}{5 b^{5} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**10,x)

[Out]

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

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Mathematica [A]  time = 0.0213473, size = 53, normalized size = 0.65 \[ -\frac{a^4+9 a^3 b x+36 a^2 b^2 x^2+84 a b^3 x^3+126 b^4 x^4}{630 b^5 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x)^10,x]

[Out]

-(a^4 + 9*a^3*b*x + 36*a^2*b^2*x^2 + 84*a*b^3*x^3 + 126*b^4*x^4)/(630*b^5*(a + b
*x)^9)

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Maple [A]  time = 0.007, size = 72, normalized size = 0.9 \[ -{\frac{{a}^{4}}{9\,{b}^{5} \left ( bx+a \right ) ^{9}}}+{\frac{{a}^{3}}{2\,{b}^{5} \left ( bx+a \right ) ^{8}}}-{\frac{6\,{a}^{2}}{7\,{b}^{5} \left ( bx+a \right ) ^{7}}}+{\frac{2\,a}{3\,{b}^{5} \left ( bx+a \right ) ^{6}}}-{\frac{1}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^10,x)

[Out]

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

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Maxima [A]  time = 1.34987, size = 192, normalized size = 2.37 \[ -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9
+ 9*a*b^13*x^8 + 36*a^2*b^12*x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5
*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a^9*b^5)

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Fricas [A]  time = 0.20779, size = 192, normalized size = 2.37 \[ -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9
+ 9*a*b^13*x^8 + 36*a^2*b^12*x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5
*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a^9*b^5)

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Sympy [A]  time = 3.46004, size = 151, normalized size = 1.86 \[ - \frac{a^{4} + 9 a^{3} b x + 36 a^{2} b^{2} x^{2} + 84 a b^{3} x^{3} + 126 b^{4} x^{4}}{630 a^{9} b^{5} + 5670 a^{8} b^{6} x + 22680 a^{7} b^{7} x^{2} + 52920 a^{6} b^{8} x^{3} + 79380 a^{5} b^{9} x^{4} + 79380 a^{4} b^{10} x^{5} + 52920 a^{3} b^{11} x^{6} + 22680 a^{2} b^{12} x^{7} + 5670 a b^{13} x^{8} + 630 b^{14} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**10,x)

[Out]

-(a**4 + 9*a**3*b*x + 36*a**2*b**2*x**2 + 84*a*b**3*x**3 + 126*b**4*x**4)/(630*a
**9*b**5 + 5670*a**8*b**6*x + 22680*a**7*b**7*x**2 + 52920*a**6*b**8*x**3 + 7938
0*a**5*b**9*x**4 + 79380*a**4*b**10*x**5 + 52920*a**3*b**11*x**6 + 22680*a**2*b*
*12*x**7 + 5670*a*b**13*x**8 + 630*b**14*x**9)

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GIAC/XCAS [A]  time = 0.202393, size = 69, normalized size = 0.85 \[ -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b x + a\right )}^{9} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/((b*x + a
)^9*b^5)