Optimal. Leaf size=198 \[ -\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac {715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac {715 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{3/2} b^{17/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {294, 205, 211}
\begin {gather*} \frac {715 \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{3/2} b^{17/2}}+\frac {715 x}{65536 a b^8 \left (a+b x^2\right )}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {x^{15}}{18 b \left (a+b x^2\right )^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 294
Rubi steps
\begin {align*} \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}+\frac {5 \int \frac {x^{14}}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}+\frac {65 \int \frac {x^{12}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}+\frac {715 \int \frac {x^{10}}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}+\frac {715 \int \frac {x^8}{\left (a+b x^2\right )^6} \, dx}{1792 b^4}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}+\frac {143 \int \frac {x^6}{\left (a+b x^2\right )^5} \, dx}{512 b^5}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}+\frac {715 \int \frac {x^4}{\left (a+b x^2\right )^4} \, dx}{4096 b^6}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}+\frac {715 \int \frac {x^2}{\left (a+b x^2\right )^3} \, dx}{8192 b^7}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac {715 \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac {715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac {715 \int \frac {1}{a+b x^2} \, dx}{65536 a b^8}\\ &=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac {715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac {715 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{3/2} b^{17/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 138, normalized size = 0.70 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}-985866 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 a^{3/2} b^{17/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 124, normalized size = 0.63
method | result | size |
default | \(\frac {-\frac {715 a^{7} x}{65536 b^{8}}-\frac {9295 a^{6} x^{3}}{98304 b^{7}}-\frac {11869 a^{5} x^{5}}{32768 b^{6}}-\frac {184327 a^{4} x^{7}}{229376 b^{5}}-\frac {143 a^{3} x^{9}}{126 b^{4}}-\frac {241657 a^{2} x^{11}}{229376 b^{3}}-\frac {20899 a \,x^{13}}{32768 b^{2}}-\frac {23473 x^{15}}{98304 b}+\frac {715 x^{17}}{65536 a}}{\left (b \,x^{2}+a \right )^{9}}+\frac {715 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a \,b^{8} \sqrt {a b}}\) | \(124\) |
risch | \(\frac {-\frac {715 a^{7} x}{65536 b^{8}}-\frac {9295 a^{6} x^{3}}{98304 b^{7}}-\frac {11869 a^{5} x^{5}}{32768 b^{6}}-\frac {184327 a^{4} x^{7}}{229376 b^{5}}-\frac {143 a^{3} x^{9}}{126 b^{4}}-\frac {241657 a^{2} x^{11}}{229376 b^{3}}-\frac {20899 a \,x^{13}}{32768 b^{2}}-\frac {23473 x^{15}}{98304 b}+\frac {715 x^{17}}{65536 a}}{\left (b \,x^{2}+a \right )^{9}}-\frac {715 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{8} a}+\frac {715 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{8} a}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 219, normalized size = 1.11 \begin {gather*} \frac {45045 \, b^{8} x^{17} - 985866 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (a b^{17} x^{18} + 9 \, a^{2} b^{16} x^{16} + 36 \, a^{3} b^{15} x^{14} + 84 \, a^{4} b^{14} x^{12} + 126 \, a^{5} b^{13} x^{10} + 126 \, a^{6} b^{12} x^{8} + 84 \, a^{7} b^{11} x^{6} + 36 \, a^{8} b^{10} x^{4} + 9 \, a^{9} b^{9} x^{2} + a^{10} b^{8}\right )}} + \frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.85, size = 654, normalized size = 3.30 \begin {gather*} \left [\frac {90090 \, a b^{9} x^{17} - 1971732 \, a^{2} b^{8} x^{15} - 5266548 \, a^{3} b^{7} x^{13} - 8699652 \, a^{4} b^{6} x^{11} - 9371648 \, a^{5} b^{5} x^{9} - 6635772 \, a^{6} b^{4} x^{7} - 2990988 \, a^{7} b^{3} x^{5} - 780780 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\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^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}, \frac {45045 \, a b^{9} x^{17} - 985866 \, a^{2} b^{8} x^{15} - 2633274 \, a^{3} b^{7} x^{13} - 4349826 \, a^{4} b^{6} x^{11} - 4685824 \, a^{5} b^{5} x^{9} - 3317886 \, a^{6} b^{4} x^{7} - 1495494 \, a^{7} b^{3} x^{5} - 390390 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\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^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.64, size = 289, normalized size = 1.46 \begin {gather*} - \frac {715 \sqrt {- \frac {1}{a^{3} b^{17}}} \log {\left (- a^{2} b^{8} \sqrt {- \frac {1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac {715 \sqrt {- \frac {1}{a^{3} b^{17}}} \log {\left (a^{2} b^{8} \sqrt {- \frac {1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x - 390390 a^{7} b x^{3} - 1495494 a^{6} b^{2} x^{5} - 3317886 a^{5} b^{3} x^{7} - 4685824 a^{4} b^{4} x^{9} - 4349826 a^{3} b^{5} x^{11} - 2633274 a^{2} b^{6} x^{13} - 985866 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{10} b^{8} + 37158912 a^{9} b^{9} x^{2} + 148635648 a^{8} b^{10} x^{4} + 346816512 a^{7} b^{11} x^{6} + 520224768 a^{6} b^{12} x^{8} + 520224768 a^{5} b^{13} x^{10} + 346816512 a^{4} b^{14} x^{12} + 148635648 a^{3} b^{15} x^{14} + 37158912 a^{2} b^{16} x^{16} + 4128768 a b^{17} x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 128, normalized size = 0.65 \begin {gather*} \frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a b^{8}} + \frac {45045 \, b^{8} x^{17} - 985866 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.76, size = 207, normalized size = 1.05 \begin {gather*} \frac {715\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{3/2}\,b^{17/2}}-\frac {\frac {23473\,x^{15}}{98304\,b}-\frac {715\,x^{17}}{65536\,a}+\frac {20899\,a\,x^{13}}{32768\,b^2}+\frac {715\,a^7\,x}{65536\,b^8}+\frac {241657\,a^2\,x^{11}}{229376\,b^3}+\frac {143\,a^3\,x^9}{126\,b^4}+\frac {184327\,a^4\,x^7}{229376\,b^5}+\frac {11869\,a^5\,x^5}{32768\,b^6}+\frac {9295\,a^6\,x^3}{98304\,b^7}}{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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