Optimal. Leaf size=37 \[ \frac {1}{9 (1+x)^9}-\frac {3}{8 (1+x)^8}+\frac {3}{7 (1+x)^7}-\frac {1}{6 (1+x)^6} \]
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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 = {45}
\begin {gather*} -\frac {1}{6 (x+1)^6}+\frac {3}{7 (x+1)^7}-\frac {3}{8 (x+1)^8}+\frac {1}{9 (x+1)^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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 = 0.00, size = 24, normalized size = 0.65 \begin {gather*} -\frac {1+9 x+36 x^2+84 x^3}{504 (1+x)^9} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.14, size = 62, normalized size = 1.68 \begin {gather*} \frac {-1-9 x-36 x^2-84 x^3}{504+4536 x+18144 x^2+42336 x^3+63504 x^4+63504 x^5+42336 x^6+18144 x^7+4536 x^8+504 x^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.81
method | result | size |
norman | \(\frac {-\frac {1}{6} x^{3}-\frac {1}{14} x^{2}-\frac {1}{56} x -\frac {1}{504}}{\left (1+x \right )^{9}}\) | \(22\) |
risch | \(\frac {-\frac {1}{6} x^{3}-\frac {1}{14} x^{2}-\frac {1}{56} x -\frac {1}{504}}{\left (1+x \right )^{9}}\) | \(22\) |
gosper | \(-\frac {84 x^{3}+36 x^{2}+9 x +1}{504 \left (1+x \right )^{9}}\) | \(23\) |
default | \(\frac {1}{9 \left (1+x \right )^{9}}-\frac {3}{8 \left (1+x \right )^{8}}+\frac {3}{7 \left (1+x \right )^{7}}-\frac {1}{6 \left (1+x \right )^{6}}\) | \(30\) |
meijerg | \(\frac {x^{4} \left (x^{5}+9 x^{4}+36 x^{3}+84 x^{2}+126 x +126\right )}{504 \left (1+x \right )^{9}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (29) = 58\).
time = 0.26, size = 62, normalized size = 1.68 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (29) = 58\).
time = 0.34, size = 62, normalized size = 1.68 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 61, normalized size = 1.65 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 27, normalized size = 0.73 \begin {gather*} \frac {-84 x^{3}-36 x^{2}-9 x-1}{504 \left (x+1\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 29, normalized size = 0.78 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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