### 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 = 0.0114004, antiderivative size = 37, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0065565, 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 + 9*x + 36*x^2 + 84*x^3)/(504*(1 + x)^9)

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Maple [A]  time = 0.005, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{9\, \left ( 1+x \right ) ^{9}}}-{\frac{3}{8\, \left ( 1+x \right ) ^{8}}}+{\frac{3}{7\, \left ( 1+x \right ) ^{7}}}-{\frac{1}{6\, \left ( 1+x \right ) ^{6}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Maxima [B]  time = 0.944489, size = 84, normalized size = 2.27 \begin{align*} -\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 )}} \end{align*}

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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Fricas [B]  time = 2.03816, size = 157, normalized size = 4.24 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 0.139897, size = 61, normalized size = 1.65 \begin{align*} - \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} \end{align*}

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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Giac [A]  time = 1.06152, size = 30, normalized size = 0.81 \begin{align*} -\frac{84 \, x^{3} + 36 \, x^{2} + 9 \, x + 1}{504 \,{\left (x + 1\right )}^{9}} \end{align*}

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