3.313 \(\int \frac {x^3}{(1+x)^{10}} \, dx\)

Optimal. Leaf size=37 \[ -\frac {1}{6 (x+1)^6}+\frac {3}{7 (x+1)^7}-\frac {3}{8 (x+1)^8}+\frac {1}{9 (x+1)^9} \]

[Out]

1/9/(1+x)^9-3/8/(1+x)^8+3/7/(1+x)^7-1/6/(1+x)^6

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \[ -\frac {1}{6 (x+1)^6}+\frac {3}{7 (x+1)^7}-\frac {3}{8 (x+1)^8}+\frac {1}{9 (x+1)^9} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + x)^10,x]

[Out]

1/(9*(1 + x)^9) - 3/(8*(1 + x)^8) + 3/(7*(1 + x)^7) - 1/(6*(1 + x)^6)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {x^3}{(1+x)^{10}} \, dx &=\int \left (-\frac {1}{(1+x)^{10}}+\frac {3}{(1+x)^9}-\frac {3}{(1+x)^8}+\frac {1}{(1+x)^7}\right ) \, dx\\ &=\frac {1}{9 (1+x)^9}-\frac {3}{8 (1+x)^8}+\frac {3}{7 (1+x)^7}-\frac {1}{6 (1+x)^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.65 \[ -\frac {84 x^3+36 x^2+9 x+1}{504 (x+1)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + x)^10,x]

[Out]

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

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fricas [B]  time = 0.39, size = 62, normalized size = 1.68 \[ -\frac {84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1}{504 \, {\left (x^{9} + 9 \, x^{8} + 36 \, x^{7} + 84 \, x^{6} + 126 \, x^{5} + 126 \, x^{4} + 84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+x)^10,x, algorithm="fricas")

[Out]

-1/504*(84*x^3 + 36*x^2 + 9*x + 1)/(x^9 + 9*x^8 + 36*x^7 + 84*x^6 + 126*x^5 + 126*x^4 + 84*x^3 + 36*x^2 + 9*x
+ 1)

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giac [A]  time = 0.87, size = 22, normalized size = 0.59 \[ -\frac {84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1}{504 \, {\left (x + 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+x)^10,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 30, normalized size = 0.81 \[ \frac {1}{9 \left (x +1\right )^{9}}-\frac {3}{8 \left (x +1\right )^{8}}+\frac {3}{7 \left (x +1\right )^{7}}-\frac {1}{6 \left (x +1\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x+1)^10,x)

[Out]

1/9/(x+1)^9-3/8/(x+1)^8+3/7/(x+1)^7-1/6/(x+1)^6

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maxima [B]  time = 0.48, size = 62, normalized size = 1.68 \[ -\frac {84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1}{504 \, {\left (x^{9} + 9 \, x^{8} + 36 \, x^{7} + 84 \, x^{6} + 126 \, x^{5} + 126 \, x^{4} + 84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+x)^10,x, algorithm="maxima")

[Out]

-1/504*(84*x^3 + 36*x^2 + 9*x + 1)/(x^9 + 9*x^8 + 36*x^7 + 84*x^6 + 126*x^5 + 126*x^4 + 84*x^3 + 36*x^2 + 9*x
+ 1)

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mupad [B]  time = 0.12, size = 29, normalized size = 0.78 \[ \frac {3}{7\,{\left (x+1\right )}^7}-\frac {1}{6\,{\left (x+1\right )}^6}-\frac {3}{8\,{\left (x+1\right )}^8}+\frac {1}{9\,{\left (x+1\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x + 1)^10,x)

[Out]

3/(7*(x + 1)^7) - 1/(6*(x + 1)^6) - 3/(8*(x + 1)^8) + 1/(9*(x + 1)^9)

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sympy [A]  time = 0.15, size = 61, normalized size = 1.65 \[ \frac {- 84 x^{3} - 36 x^{2} - 9 x - 1}{504 x^{9} + 4536 x^{8} + 18144 x^{7} + 42336 x^{6} + 63504 x^{5} + 63504 x^{4} + 42336 x^{3} + 18144 x^{2} + 4536 x + 504} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(1+x)**10,x)

[Out]

(-84*x**3 - 36*x**2 - 9*x - 1)/(504*x**9 + 4536*x**8 + 18144*x**7 + 42336*x**6 + 63504*x**5 + 63504*x**4 + 423
36*x**3 + 18144*x**2 + 4536*x + 504)

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