Optimal. Leaf size=199 \[ \frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{5/2} b^{15/2}}+\frac {143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac {143 x}{98304 a b^7 \left (a+b x^2\right )^2}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {x^{13}}{18 b \left (a+b x^2\right )^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 199, 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 = {288, 199, 205} \[ \frac {143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{5/2} b^{15/2}}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}+\frac {143 x}{98304 a b^7 \left (a+b x^2\right )^2}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}-\frac {x^{13}}{18 b \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 205
Rule 288
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}+\frac {13 \int \frac {x^{12}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}+\frac {143 \int \frac {x^{10}}{\left (a+b x^2\right )^8} \, dx}{288 b^2}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}+\frac {143 \int \frac {x^8}{\left (a+b x^2\right )^7} \, dx}{448 b^3}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}+\frac {143 \int \frac {x^6}{\left (a+b x^2\right )^6} \, dx}{768 b^4}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}+\frac {143 \int \frac {x^4}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}+\frac {143 \int \frac {x^2}{\left (a+b x^2\right )^4} \, dx}{4096 b^6}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^3} \, dx}{24576 b^7}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a b^7 \left (a+b x^2\right )^2}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{32768 a b^7}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a b^7 \left (a+b x^2\right )^2}+\frac {143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac {143 \int \frac {1}{a+b x^2} \, dx}{65536 a^2 b^7}\\ &=-\frac {x^{13}}{18 b \left (a+b x^2\right )^9}-\frac {13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac {143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac {143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac {143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac {143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac {143 x}{24576 b^7 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a b^7 \left (a+b x^2\right )^2}+\frac {143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac {143 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{5/2} b^{15/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 138, normalized size = 0.69 \[ \frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{20643840 a^{5/2} b^{15/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 654, normalized size = 3.29 \[ \left [\frac {90090 \, a b^{9} x^{17} + 780780 \, a^{2} b^{8} x^{15} - 5266548 \, a^{3} b^{7} x^{13} - 8699652 \, a^{4} b^{6} x^{11} - 9371648 \, a^{5} b^{5} x^{9} - 6635772 \, a^{6} b^{4} x^{7} - 2990988 \, a^{7} b^{3} x^{5} - 780780 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{41287680 \, {\left (a^{3} b^{17} x^{18} + 9 \, a^{4} b^{16} x^{16} + 36 \, a^{5} b^{15} x^{14} + 84 \, a^{6} b^{14} x^{12} + 126 \, a^{7} b^{13} x^{10} + 126 \, a^{8} b^{12} x^{8} + 84 \, a^{9} b^{11} x^{6} + 36 \, a^{10} b^{10} x^{4} + 9 \, a^{11} b^{9} x^{2} + a^{12} b^{8}\right )}}, \frac {45045 \, a b^{9} x^{17} + 390390 \, a^{2} b^{8} x^{15} - 2633274 \, a^{3} b^{7} x^{13} - 4349826 \, a^{4} b^{6} x^{11} - 4685824 \, a^{5} b^{5} x^{9} - 3317886 \, a^{6} b^{4} x^{7} - 1495494 \, a^{7} b^{3} x^{5} - 390390 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{20643840 \, {\left (a^{3} b^{17} x^{18} + 9 \, a^{4} b^{16} x^{16} + 36 \, a^{5} b^{15} x^{14} + 84 \, a^{6} b^{14} x^{12} + 126 \, a^{7} b^{13} x^{10} + 126 \, a^{8} b^{12} x^{8} + 84 \, a^{9} b^{11} x^{6} + 36 \, a^{10} b^{10} x^{4} + 9 \, a^{11} b^{9} x^{2} + a^{12} b^{8}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.64, size = 128, normalized size = 0.64 \[ \frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{2} b^{7}} + \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (b x^{2} + a\right )}^{9} a^{2} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 122, normalized size = 0.61 \[ \frac {143 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{2} b^{7}}+\frac {\frac {143 b \,x^{17}}{65536 a^{2}}+\frac {1859 x^{15}}{98304 a}-\frac {20899 x^{13}}{163840 b}-\frac {241657 a \,x^{11}}{1146880 b^{2}}-\frac {143 a^{2} x^{9}}{630 b^{3}}-\frac {184327 a^{3} x^{7}}{1146880 b^{4}}-\frac {11869 a^{4} x^{5}}{163840 b^{5}}-\frac {1859 a^{5} x^{3}}{98304 b^{6}}-\frac {143 a^{6} x}{65536 b^{7}}}{\left (b \,x^{2}+a \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.16, size = 221, normalized size = 1.11 \[ \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (a^{2} b^{16} x^{18} + 9 \, a^{3} b^{15} x^{16} + 36 \, a^{4} b^{14} x^{14} + 84 \, a^{5} b^{13} x^{12} + 126 \, a^{6} b^{12} x^{10} + 126 \, a^{7} b^{11} x^{8} + 84 \, a^{8} b^{10} x^{6} + 36 \, a^{9} b^{9} x^{4} + 9 \, a^{10} b^{8} x^{2} + a^{11} b^{7}\right )}} + \frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{2} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.75, size = 205, normalized size = 1.03 \[ \frac {143\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{5/2}\,b^{15/2}}-\frac {\frac {20899\,x^{13}}{163840\,b}-\frac {1859\,x^{15}}{98304\,a}+\frac {241657\,a\,x^{11}}{1146880\,b^2}+\frac {143\,a^6\,x}{65536\,b^7}-\frac {143\,b\,x^{17}}{65536\,a^2}+\frac {143\,a^2\,x^9}{630\,b^3}+\frac {184327\,a^3\,x^7}{1146880\,b^4}+\frac {11869\,a^4\,x^5}{163840\,b^5}+\frac {1859\,a^5\,x^3}{98304\,b^6}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.35, size = 291, normalized size = 1.46 \[ - \frac {143 \sqrt {- \frac {1}{a^{5} b^{15}}} \log {\left (- a^{3} b^{7} \sqrt {- \frac {1}{a^{5} b^{15}}} + x \right )}}{131072} + \frac {143 \sqrt {- \frac {1}{a^{5} b^{15}}} \log {\left (a^{3} b^{7} \sqrt {- \frac {1}{a^{5} b^{15}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x - 390390 a^{7} b x^{3} - 1495494 a^{6} b^{2} x^{5} - 3317886 a^{5} b^{3} x^{7} - 4685824 a^{4} b^{4} x^{9} - 4349826 a^{3} b^{5} x^{11} - 2633274 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{20643840 a^{11} b^{7} + 185794560 a^{10} b^{8} x^{2} + 743178240 a^{9} b^{9} x^{4} + 1734082560 a^{8} b^{10} x^{6} + 2601123840 a^{7} b^{11} x^{8} + 2601123840 a^{6} b^{12} x^{10} + 1734082560 a^{5} b^{13} x^{12} + 743178240 a^{4} b^{14} x^{14} + 185794560 a^{3} b^{15} x^{16} + 20643840 a^{2} b^{16} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________