Optimal. Leaf size=17 \[ \frac {x^9}{9 a (a+b x)^9} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37}
\begin {gather*} \frac {x^9}{9 a (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {x^8}{(a+b x)^{10}} \, dx &=\frac {x^9}{9 a (a+b x)^9}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(17)=34\).
time = 0.01, size = 97, normalized size = 5.71 \begin {gather*} -\frac {a^8+9 a^7 b x+36 a^6 b^2 x^2+84 a^5 b^3 x^3+126 a^4 b^4 x^4+126 a^3 b^5 x^5+84 a^2 b^6 x^6+36 a b^7 x^7+9 b^8 x^8}{9 b^9 (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(187\) vs. \(2(17)=34\).
time = 3.32, size = 185, normalized size = 10.88 \begin {gather*} \frac {-a^8-9 a^7 b x-36 a^6 b^2 x^2-84 a^5 b^3 x^3-126 a^4 b^4 x^4-126 a^3 b^5 x^5-84 a^2 b^6 x^6-36 a b^7 x^7-9 b^8 x^8}{9 b^9 \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs.
\(2(15)=30\).
time = 0.08, size = 131, normalized size = 7.71
method | result | size |
gosper | \(-\frac {9 x^{8} b^{8}+36 a \,x^{7} b^{7}+84 a^{2} x^{6} b^{6}+126 a^{3} x^{5} b^{5}+126 a^{4} x^{4} b^{4}+84 a^{5} x^{3} b^{3}+36 a^{6} x^{2} b^{2}+9 a^{7} x b +a^{8}}{9 \left (b x +a \right )^{9} b^{9}}\) | \(96\) |
norman | \(\frac {-\frac {x^{8}}{b}-\frac {4 a \,x^{7}}{b^{2}}-\frac {28 a^{2} x^{6}}{3 b^{3}}-\frac {14 a^{3} x^{5}}{b^{4}}-\frac {14 a^{4} x^{4}}{b^{5}}-\frac {28 a^{5} x^{3}}{3 b^{6}}-\frac {4 a^{6} x^{2}}{b^{7}}-\frac {a^{7} x}{b^{8}}-\frac {a^{8}}{9 b^{9}}}{\left (b x +a \right )^{9}}\) | \(99\) |
risch | \(\frac {-\frac {x^{8}}{b}-\frac {4 a \,x^{7}}{b^{2}}-\frac {28 a^{2} x^{6}}{3 b^{3}}-\frac {14 a^{3} x^{5}}{b^{4}}-\frac {14 a^{4} x^{4}}{b^{5}}-\frac {28 a^{5} x^{3}}{3 b^{6}}-\frac {4 a^{6} x^{2}}{b^{7}}-\frac {a^{7} x}{b^{8}}-\frac {a^{8}}{9 b^{9}}}{\left (b x +a \right )^{9}}\) | \(99\) |
default | \(-\frac {1}{b^{9} \left (b x +a \right )}-\frac {14 a^{4}}{b^{9} \left (b x +a \right )^{5}}+\frac {14 a^{3}}{b^{9} \left (b x +a \right )^{4}}-\frac {a^{8}}{9 b^{9} \left (b x +a \right )^{9}}+\frac {4 a}{b^{9} \left (b x +a \right )^{2}}+\frac {28 a^{5}}{3 b^{9} \left (b x +a \right )^{6}}-\frac {4 a^{6}}{b^{9} \left (b x +a \right )^{7}}+\frac {a^{7}}{b^{9} \left (b x +a \right )^{8}}-\frac {28 a^{2}}{3 b^{9} \left (b x +a \right )^{3}}\) | \(131\) |
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. 186 vs.
\(2 (15) = 30\).
time = 0.27, size = 186, normalized size = 10.94 \begin {gather*} -\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, {\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\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. 186 vs.
\(2 (15) = 30\).
time = 0.31, size = 186, normalized size = 10.94 \begin {gather*} -\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, {\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (12) = 24\).
time = 0.48, size = 199, normalized size = 11.71 \begin {gather*} \frac {- a^{8} - 9 a^{7} b x - 36 a^{6} b^{2} x^{2} - 84 a^{5} b^{3} x^{3} - 126 a^{4} b^{4} x^{4} - 126 a^{3} b^{5} x^{5} - 84 a^{2} b^{6} x^{6} - 36 a b^{7} x^{7} - 9 b^{8} x^{8}}{9 a^{9} b^{9} + 81 a^{8} b^{10} x + 324 a^{7} b^{11} x^{2} + 756 a^{6} b^{12} x^{3} + 1134 a^{5} b^{13} x^{4} + 1134 a^{4} b^{14} x^{5} + 756 a^{3} b^{15} x^{6} + 324 a^{2} b^{16} x^{7} + 81 a b^{17} x^{8} + 9 b^{18} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (15) = 30\).
time = 0.00, size = 106, normalized size = 6.24 \begin {gather*} \frac {-9 x^{8} b^{8}-36 x^{7} b^{7} a-84 x^{6} b^{6} a^{2}-126 x^{5} b^{5} a^{3}-126 x^{4} b^{4} a^{4}-84 x^{3} b^{3} a^{5}-36 x^{2} b^{2} a^{6}-9 x b a^{7}-a^{8}}{9 b^{9} \left (x b+a\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 107, normalized size = 6.29 \begin {gather*} -\frac {\frac {1}{a+b\,x}-\frac {4\,a}{{\left (a+b\,x\right )}^2}+\frac {28\,a^2}{3\,{\left (a+b\,x\right )}^3}-\frac {14\,a^3}{{\left (a+b\,x\right )}^4}+\frac {14\,a^4}{{\left (a+b\,x\right )}^5}-\frac {28\,a^5}{3\,{\left (a+b\,x\right )}^6}+\frac {4\,a^6}{{\left (a+b\,x\right )}^7}-\frac {a^7}{{\left (a+b\,x\right )}^8}+\frac {a^8}{9\,{\left (a+b\,x\right )}^9}}{b^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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