3.231 \(\int \frac{x^3}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=64 \[ \frac{a^3}{9 b^4 (a+b x)^9}-\frac{3 a^2}{8 b^4 (a+b x)^8}+\frac{3 a}{7 b^4 (a+b x)^7}-\frac{1}{6 b^4 (a+b x)^6} \]

[Out]

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

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Rubi [A]  time = 0.0714916, antiderivative size = 64, 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 \[ \frac{a^3}{9 b^4 (a+b x)^9}-\frac{3 a^2}{8 b^4 (a+b x)^8}+\frac{3 a}{7 b^4 (a+b x)^7}-\frac{1}{6 b^4 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.3559, size = 60, normalized size = 0.94 \[ \frac{a^{3}}{9 b^{4} \left (a + b x\right )^{9}} - \frac{3 a^{2}}{8 b^{4} \left (a + b x\right )^{8}} + \frac{3 a}{7 b^{4} \left (a + b x\right )^{7}} - \frac{1}{6 b^{4} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**10,x)

[Out]

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

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Mathematica [A]  time = 0.0141554, size = 42, normalized size = 0.66 \[ -\frac{a^3+9 a^2 b x+36 a b^2 x^2+84 b^3 x^3}{504 b^4 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x)^10,x]

[Out]

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

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Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \[{\frac{{a}^{3}}{9\,{b}^{4} \left ( bx+a \right ) ^{9}}}-{\frac{3\,{a}^{2}}{8\,{b}^{4} \left ( bx+a \right ) ^{8}}}+{\frac{3\,a}{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{1}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^10,x)

[Out]

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

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Maxima [A]  time = 1.33597, size = 177, normalized size = 2.77 \[ -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211935, size = 177, normalized size = 2.77 \[ -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.27566, size = 139, normalized size = 2.17 \[ - \frac{a^{3} + 9 a^{2} b x + 36 a b^{2} x^{2} + 84 b^{3} x^{3}}{504 a^{9} b^{4} + 4536 a^{8} b^{5} x + 18144 a^{7} b^{6} x^{2} + 42336 a^{6} b^{7} x^{3} + 63504 a^{5} b^{8} x^{4} + 63504 a^{4} b^{9} x^{5} + 42336 a^{3} b^{10} x^{6} + 18144 a^{2} b^{11} x^{7} + 4536 a b^{12} x^{8} + 504 b^{13} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**10,x)

[Out]

-(a**3 + 9*a**2*b*x + 36*a*b**2*x**2 + 84*b**3*x**3)/(504*a**9*b**4 + 4536*a**8*
b**5*x + 18144*a**7*b**6*x**2 + 42336*a**6*b**7*x**3 + 63504*a**5*b**8*x**4 + 63
504*a**4*b**9*x**5 + 42336*a**3*b**10*x**6 + 18144*a**2*b**11*x**7 + 4536*a*b**1
2*x**8 + 504*b**13*x**9)

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GIAC/XCAS [A]  time = 0.203175, size = 54, normalized size = 0.84 \[ -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b x + a\right )}^{9} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^10,x, algorithm="giac")

[Out]

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