Optimal. Leaf size=184 \[ -\frac {1}{2 a^{10} x^2}-\frac {b}{18 a^2 \left (a+b x^2\right )^9}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {10 b \log (x)}{a^{11}}+\frac {5 b \log \left (a+b x^2\right )}{a^{11}} \]
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Rubi [A]
time = 0.13, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46}
\begin {gather*} \frac {5 b \log \left (a+b x^2\right )}{a^{11}}-\frac {10 b \log (x)}{a^{11}}-\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {1}{2 a^{10} x^2}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {b}{18 a^2 \left (a+b x^2\right )^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^{10} x^2}-\frac {10 b}{a^{11} x}+\frac {b^2}{a^2 (a+b x)^{10}}+\frac {2 b^2}{a^3 (a+b x)^9}+\frac {3 b^2}{a^4 (a+b x)^8}+\frac {4 b^2}{a^5 (a+b x)^7}+\frac {5 b^2}{a^6 (a+b x)^6}+\frac {6 b^2}{a^7 (a+b x)^5}+\frac {7 b^2}{a^8 (a+b x)^4}+\frac {8 b^2}{a^9 (a+b x)^3}+\frac {9 b^2}{a^{10} (a+b x)^2}+\frac {10 b^2}{a^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^{10} x^2}-\frac {b}{18 a^2 \left (a+b x^2\right )^9}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {10 b \log (x)}{a^{11}}+\frac {5 b \log \left (a+b x^2\right )}{a^{11}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 136, normalized size = 0.74 \begin {gather*} -\frac {\frac {a \left (252 a^9+7129 a^8 b x^2+41481 a^7 b^2 x^4+120564 a^6 b^3 x^6+210756 a^5 b^4 x^8+236754 a^4 b^5 x^{10}+173250 a^3 b^6 x^{12}+80220 a^2 b^7 x^{14}+21420 a b^8 x^{16}+2520 b^9 x^{18}\right )}{x^2 \left (a+b x^2\right )^9}+5040 b \log (x)-2520 b \log \left (a+b x^2\right )}{504 a^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 191, normalized size = 1.04
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {45 b^{2} x^{4}}{a^{3}}+\frac {270 b^{3} x^{6}}{a^{4}}+\frac {770 b^{4} x^{8}}{a^{5}}+\frac {2625 b^{5} x^{10}}{2 a^{6}}+\frac {2877 b^{6} x^{12}}{2 a^{7}}+\frac {1029 b^{7} x^{14}}{a^{8}}+\frac {3267 b^{8} x^{16}}{7 a^{9}}+\frac {6849 b^{9} x^{18}}{56 a^{10}}+\frac {7129 b^{10} x^{20}}{504 a^{11}}}{x^{2} \left (b \,x^{2}+a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {5 b \ln \left (b \,x^{2}+a \right )}{a^{11}}\) | \(142\) |
risch | \(\frac {-\frac {1}{2 a}-\frac {7129 b \,x^{2}}{504 a^{2}}-\frac {4609 b^{2} x^{4}}{56 a^{3}}-\frac {3349 b^{3} x^{6}}{14 a^{4}}-\frac {2509 b^{4} x^{8}}{6 a^{5}}-\frac {1879 b^{5} x^{10}}{4 a^{6}}-\frac {1375 b^{6} x^{12}}{4 a^{7}}-\frac {955 b^{7} x^{14}}{6 a^{8}}-\frac {85 b^{8} x^{16}}{2 a^{9}}-\frac {5 b^{9} x^{18}}{a^{10}}}{x^{2} \left (b \,x^{2}+a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {5 b \ln \left (-b \,x^{2}-a \right )}{a^{11}}\) | \(143\) |
default | \(\frac {b^{2} \left (-\frac {a^{5}}{b \left (b \,x^{2}+a \right )^{5}}-\frac {3 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {4 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {9 a}{b \left (b \,x^{2}+a \right )}-\frac {7 a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}-\frac {3 a^{4}}{2 b \left (b \,x^{2}+a \right )^{4}}-\frac {a^{8}}{4 b \left (b \,x^{2}+a \right )^{8}}-\frac {2 a^{6}}{3 b \left (b \,x^{2}+a \right )^{6}}\right )}{2 a^{11}}-\frac {1}{2 a^{10} x^{2}}-\frac {10 b \ln \left (x \right )}{a^{11}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 231, normalized size = 1.26 \begin {gather*} -\frac {2520 \, b^{9} x^{18} + 21420 \, a b^{8} x^{16} + 80220 \, a^{2} b^{7} x^{14} + 173250 \, a^{3} b^{6} x^{12} + 236754 \, a^{4} b^{5} x^{10} + 210756 \, a^{5} b^{4} x^{8} + 120564 \, a^{6} b^{3} x^{6} + 41481 \, a^{7} b^{2} x^{4} + 7129 \, a^{8} b x^{2} + 252 \, a^{9}}{504 \, {\left (a^{10} b^{9} x^{20} + 9 \, a^{11} b^{8} x^{18} + 36 \, a^{12} b^{7} x^{16} + 84 \, a^{13} b^{6} x^{14} + 126 \, a^{14} b^{5} x^{12} + 126 \, a^{15} b^{4} x^{10} + 84 \, a^{16} b^{3} x^{8} + 36 \, a^{17} b^{2} x^{6} + 9 \, a^{18} b x^{4} + a^{19} x^{2}\right )}} + \frac {5 \, b \log \left (b x^{2} + a\right )}{a^{11}} - \frac {5 \, b \log \left (x^{2}\right )}{a^{11}} \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. 427 vs.
\(2 (166) = 332\).
time = 1.42, size = 427, normalized size = 2.32 \begin {gather*} -\frac {2520 \, a b^{9} x^{18} + 21420 \, a^{2} b^{8} x^{16} + 80220 \, a^{3} b^{7} x^{14} + 173250 \, a^{4} b^{6} x^{12} + 236754 \, a^{5} b^{5} x^{10} + 210756 \, a^{6} b^{4} x^{8} + 120564 \, a^{7} b^{3} x^{6} + 41481 \, a^{8} b^{2} x^{4} + 7129 \, a^{9} b x^{2} + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 5040 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (x\right )}{504 \, {\left (a^{11} b^{9} x^{20} + 9 \, a^{12} b^{8} x^{18} + 36 \, a^{13} b^{7} x^{16} + 84 \, a^{14} b^{6} x^{14} + 126 \, a^{15} b^{5} x^{12} + 126 \, a^{16} b^{4} x^{10} + 84 \, a^{17} b^{3} x^{8} + 36 \, a^{18} b^{2} x^{6} + 9 \, a^{19} b x^{4} + a^{20} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.67, size = 245, normalized size = 1.33 \begin {gather*} \frac {- 252 a^{9} - 7129 a^{8} b x^{2} - 41481 a^{7} b^{2} x^{4} - 120564 a^{6} b^{3} x^{6} - 210756 a^{5} b^{4} x^{8} - 236754 a^{4} b^{5} x^{10} - 173250 a^{3} b^{6} x^{12} - 80220 a^{2} b^{7} x^{14} - 21420 a b^{8} x^{16} - 2520 b^{9} x^{18}}{504 a^{19} x^{2} + 4536 a^{18} b x^{4} + 18144 a^{17} b^{2} x^{6} + 42336 a^{16} b^{3} x^{8} + 63504 a^{15} b^{4} x^{10} + 63504 a^{14} b^{5} x^{12} + 42336 a^{13} b^{6} x^{14} + 18144 a^{12} b^{7} x^{16} + 4536 a^{11} b^{8} x^{18} + 504 a^{10} b^{9} x^{20}} - \frac {10 b \log {\left (x \right )}}{a^{11}} + \frac {5 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 159, normalized size = 0.86 \begin {gather*} -\frac {5 \, b \log \left (x^{2}\right )}{a^{11}} + \frac {5 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{11}} + \frac {10 \, b x^{2} - a}{2 \, a^{11} x^{2}} - \frac {7129 \, b^{10} x^{18} + 66429 \, a b^{9} x^{16} + 275796 \, a^{2} b^{8} x^{14} + 669984 \, a^{3} b^{7} x^{12} + 1050336 \, a^{4} b^{6} x^{10} + 1103256 \, a^{5} b^{5} x^{8} + 777840 \, a^{6} b^{4} x^{6} + 356040 \, a^{7} b^{3} x^{4} + 96570 \, a^{8} b^{2} x^{2} + 11990 \, a^{9} b}{504 \, {\left (b x^{2} + a\right )}^{9} a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 229, normalized size = 1.24 \begin {gather*} \frac {5\,b\,\ln \left (b\,x^2+a\right )}{a^{11}}-\frac {\frac {1}{2\,a}+\frac {7129\,b\,x^2}{504\,a^2}+\frac {4609\,b^2\,x^4}{56\,a^3}+\frac {3349\,b^3\,x^6}{14\,a^4}+\frac {2509\,b^4\,x^8}{6\,a^5}+\frac {1879\,b^5\,x^{10}}{4\,a^6}+\frac {1375\,b^6\,x^{12}}{4\,a^7}+\frac {955\,b^7\,x^{14}}{6\,a^8}+\frac {85\,b^8\,x^{16}}{2\,a^9}+\frac {5\,b^9\,x^{18}}{a^{10}}}{a^9\,x^2+9\,a^8\,b\,x^4+36\,a^7\,b^2\,x^6+84\,a^6\,b^3\,x^8+126\,a^5\,b^4\,x^{10}+126\,a^4\,b^5\,x^{12}+84\,a^3\,b^6\,x^{14}+36\,a^2\,b^7\,x^{16}+9\,a\,b^8\,x^{18}+b^9\,x^{20}}-\frac {10\,b\,\ln \left (x\right )}{a^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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