Optimal. Leaf size=181 \[ \frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac {12155 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{19/2} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 211}
\begin {gather*} \frac {12155 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{19/2} \sqrt {b}}+\frac {12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {x}{18 a \left (a+b x^2\right )^9} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 \int \frac {1}{\left (a+b x^2\right )^9} \, dx}{18 a}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 \int \frac {1}{\left (a+b x^2\right )^8} \, dx}{96 a^2}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 \int \frac {1}{\left (a+b x^2\right )^7} \, dx}{1344 a^3}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {12155 \int \frac {1}{\left (a+b x^2\right )^6} \, dx}{16128 a^4}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 \int \frac {1}{\left (a+b x^2\right )^5} \, dx}{3584 a^5}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 \int \frac {1}{\left (a+b x^2\right )^4} \, dx}{4096 a^6}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 \int \frac {1}{\left (a+b x^2\right )^3} \, dx}{24576 a^7}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {12155 \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{32768 a^8}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac {12155 \int \frac {1}{a+b x^2} \, dx}{65536 a^9}\\ &=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac {12155 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{19/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 131, normalized size = 0.72 \begin {gather*} \frac {\frac {3363003 a^8 x+16759722 a^7 b x^3+44765658 a^6 b^2 x^5+73947042 a^5 b^3 x^7+79659008 a^4 b^4 x^9+56404062 a^3 b^5 x^{11}+25423398 a^2 b^6 x^{13}+6636630 a b^7 x^{15}+765765 b^8 x^{17}}{a^9 \left (a+b x^2\right )^9}+\frac {765765 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{19/2} \sqrt {b}}}{4128768} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 204, normalized size = 1.13
method | result | size |
risch | \(\frac {\frac {53381 x}{65536 a}+\frac {399041 b \,x^{3}}{98304 a^{2}}+\frac {355283 b^{2} x^{5}}{32768 a^{3}}+\frac {4108169 b^{3} x^{7}}{229376 a^{4}}+\frac {2431 b^{4} x^{9}}{126 a^{5}}+\frac {3133559 b^{5} x^{11}}{229376 a^{6}}+\frac {201773 b^{6} x^{13}}{32768 a^{7}}+\frac {158015 b^{7} x^{15}}{98304 a^{8}}+\frac {12155 b^{8} x^{17}}{65536 a^{9}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {12155 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, a^{9}}+\frac {12155 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, a^{9}}\) | \(150\) |
default | \(\frac {x}{18 a \left (b \,x^{2}+a \right )^{9}}+\frac {\frac {17 x}{288 a \left (b \,x^{2}+a \right )^{8}}+\frac {17 \left (\frac {15 x}{224 a \left (b \,x^{2}+a \right )^{7}}+\frac {15 \left (\frac {13 x}{168 a \left (b \,x^{2}+a \right )^{6}}+\frac {13 \left (\frac {11 x}{120 a \left (b \,x^{2}+a \right )^{5}}+\frac {11 \left (\frac {9 x}{80 a \left (b \,x^{2}+a \right )^{4}}+\frac {9 \left (\frac {7 x}{48 a \left (b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (b \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )}{14 a}\right )}{16 a}\right )}{18 a}}{a}\) | \(204\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 212, normalized size = 1.17 \begin {gather*} \frac {765765 \, b^{8} x^{17} + 6636630 \, a b^{7} x^{15} + 25423398 \, a^{2} b^{6} x^{13} + 56404062 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 73947042 \, a^{5} b^{3} x^{7} + 44765658 \, a^{6} b^{2} x^{5} + 16759722 \, a^{7} b x^{3} + 3363003 \, a^{8} x}{4128768 \, {\left (a^{9} b^{9} x^{18} + 9 \, a^{10} b^{8} x^{16} + 36 \, a^{11} b^{7} x^{14} + 84 \, a^{12} b^{6} x^{12} + 126 \, a^{13} b^{5} x^{10} + 126 \, a^{14} b^{4} x^{8} + 84 \, a^{15} b^{3} x^{6} + 36 \, a^{16} b^{2} x^{4} + 9 \, a^{17} b x^{2} + a^{18}\right )}} + \frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 315 vs.
\(2 (153) = 306\).
time = 0.68, size = 650, normalized size = 3.59 \begin {gather*} \left [\frac {1531530 \, a b^{9} x^{17} + 13273260 \, a^{2} b^{8} x^{15} + 50846796 \, a^{3} b^{7} x^{13} + 112808124 \, a^{4} b^{6} x^{11} + 159318016 \, a^{5} b^{5} x^{9} + 147894084 \, a^{6} b^{4} x^{7} + 89531316 \, a^{7} b^{3} x^{5} + 33519444 \, a^{8} b^{2} x^{3} + 6726006 \, a^{9} b x - 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}, \frac {765765 \, a b^{9} x^{17} + 6636630 \, a^{2} b^{8} x^{15} + 25423398 \, a^{3} b^{7} x^{13} + 56404062 \, a^{4} b^{6} x^{11} + 79659008 \, a^{5} b^{5} x^{9} + 73947042 \, a^{6} b^{4} x^{7} + 44765658 \, a^{7} b^{3} x^{5} + 16759722 \, a^{8} b^{2} x^{3} + 3363003 \, a^{9} b x + 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.48, size = 272, normalized size = 1.50 \begin {gather*} - \frac {12155 \sqrt {- \frac {1}{a^{19} b}} \log {\left (- a^{10} \sqrt {- \frac {1}{a^{19} b}} + x \right )}}{131072} + \frac {12155 \sqrt {- \frac {1}{a^{19} b}} \log {\left (a^{10} \sqrt {- \frac {1}{a^{19} b}} + x \right )}}{131072} + \frac {3363003 a^{8} x + 16759722 a^{7} b x^{3} + 44765658 a^{6} b^{2} x^{5} + 73947042 a^{5} b^{3} x^{7} + 79659008 a^{4} b^{4} x^{9} + 56404062 a^{3} b^{5} x^{11} + 25423398 a^{2} b^{6} x^{13} + 6636630 a b^{7} x^{15} + 765765 b^{8} x^{17}}{4128768 a^{18} + 37158912 a^{17} b x^{2} + 148635648 a^{16} b^{2} x^{4} + 346816512 a^{15} b^{3} x^{6} + 520224768 a^{14} b^{4} x^{8} + 520224768 a^{13} b^{5} x^{10} + 346816512 a^{12} b^{6} x^{12} + 148635648 a^{11} b^{7} x^{14} + 37158912 a^{10} b^{8} x^{16} + 4128768 a^{9} b^{9} x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.39, size = 122, normalized size = 0.67 \begin {gather*} \frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{9}} + \frac {765765 \, b^{8} x^{17} + 6636630 \, a b^{7} x^{15} + 25423398 \, a^{2} b^{6} x^{13} + 56404062 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 73947042 \, a^{5} b^{3} x^{7} + 44765658 \, a^{6} b^{2} x^{5} + 16759722 \, a^{7} b x^{3} + 3363003 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.74, size = 209, normalized size = 1.15 \begin {gather*} \frac {\frac {53381\,x}{65536\,a}+\frac {399041\,b\,x^3}{98304\,a^2}+\frac {355283\,b^2\,x^5}{32768\,a^3}+\frac {4108169\,b^3\,x^7}{229376\,a^4}+\frac {2431\,b^4\,x^9}{126\,a^5}+\frac {3133559\,b^5\,x^{11}}{229376\,a^6}+\frac {201773\,b^6\,x^{13}}{32768\,a^7}+\frac {158015\,b^7\,x^{15}}{98304\,a^8}+\frac {12155\,b^8\,x^{17}}{65536\,a^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {12155\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{19/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________