Optimal. Leaf size=211 \[ -\frac {a^{15}}{15 x^{15}}-\frac {30 a^{14} b}{29 x^{29/2}}-\frac {15 a^{13} b^2}{2 x^{14}}-\frac {910 a^{12} b^3}{27 x^{27/2}}-\frac {105 a^{11} b^4}{x^{13}}-\frac {6006 a^{10} b^5}{25 x^{25/2}}-\frac {5005 a^9 b^6}{12 x^{12}}-\frac {12870 a^8 b^7}{23 x^{23/2}}-\frac {585 a^7 b^8}{x^{11}}-\frac {1430 a^6 b^9}{3 x^{21/2}}-\frac {3003 a^5 b^{10}}{10 x^{10}}-\frac {2730 a^4 b^{11}}{19 x^{19/2}}-\frac {455 a^3 b^{12}}{9 x^9}-\frac {210 a^2 b^{13}}{17 x^{17/2}}-\frac {15 a b^{14}}{8 x^8}-\frac {2 b^{15}}{15 x^{15/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {15 a^{13} b^2}{2 x^{14}}-\frac {910 a^{12} b^3}{27 x^{27/2}}-\frac {105 a^{11} b^4}{x^{13}}-\frac {6006 a^{10} b^5}{25 x^{25/2}}-\frac {5005 a^9 b^6}{12 x^{12}}-\frac {12870 a^8 b^7}{23 x^{23/2}}-\frac {585 a^7 b^8}{x^{11}}-\frac {1430 a^6 b^9}{3 x^{21/2}}-\frac {3003 a^5 b^{10}}{10 x^{10}}-\frac {2730 a^4 b^{11}}{19 x^{19/2}}-\frac {455 a^3 b^{12}}{9 x^9}-\frac {210 a^2 b^{13}}{17 x^{17/2}}-\frac {30 a^{14} b}{29 x^{29/2}}-\frac {a^{15}}{15 x^{15}}-\frac {15 a b^{14}}{8 x^8}-\frac {2 b^{15}}{15 x^{15/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{16}} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{31}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^{15}}{x^{31}}+\frac {15 a^{14} b}{x^{30}}+\frac {105 a^{13} b^2}{x^{29}}+\frac {455 a^{12} b^3}{x^{28}}+\frac {1365 a^{11} b^4}{x^{27}}+\frac {3003 a^{10} b^5}{x^{26}}+\frac {5005 a^9 b^6}{x^{25}}+\frac {6435 a^8 b^7}{x^{24}}+\frac {6435 a^7 b^8}{x^{23}}+\frac {5005 a^6 b^9}{x^{22}}+\frac {3003 a^5 b^{10}}{x^{21}}+\frac {1365 a^4 b^{11}}{x^{20}}+\frac {455 a^3 b^{12}}{x^{19}}+\frac {105 a^2 b^{13}}{x^{18}}+\frac {15 a b^{14}}{x^{17}}+\frac {b^{15}}{x^{16}}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{15 x^{15}}-\frac {30 a^{14} b}{29 x^{29/2}}-\frac {15 a^{13} b^2}{2 x^{14}}-\frac {910 a^{12} b^3}{27 x^{27/2}}-\frac {105 a^{11} b^4}{x^{13}}-\frac {6006 a^{10} b^5}{25 x^{25/2}}-\frac {5005 a^9 b^6}{12 x^{12}}-\frac {12870 a^8 b^7}{23 x^{23/2}}-\frac {585 a^7 b^8}{x^{11}}-\frac {1430 a^6 b^9}{3 x^{21/2}}-\frac {3003 a^5 b^{10}}{10 x^{10}}-\frac {2730 a^4 b^{11}}{19 x^{19/2}}-\frac {455 a^3 b^{12}}{9 x^9}-\frac {210 a^2 b^{13}}{17 x^{17/2}}-\frac {15 a b^{14}}{8 x^8}-\frac {2 b^{15}}{15 x^{15/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 211, normalized size = 1.00 \[ -\frac {a^{15}}{15 x^{15}}-\frac {30 a^{14} b}{29 x^{29/2}}-\frac {15 a^{13} b^2}{2 x^{14}}-\frac {910 a^{12} b^3}{27 x^{27/2}}-\frac {105 a^{11} b^4}{x^{13}}-\frac {6006 a^{10} b^5}{25 x^{25/2}}-\frac {5005 a^9 b^6}{12 x^{12}}-\frac {12870 a^8 b^7}{23 x^{23/2}}-\frac {585 a^7 b^8}{x^{11}}-\frac {1430 a^6 b^9}{3 x^{21/2}}-\frac {3003 a^5 b^{10}}{10 x^{10}}-\frac {2730 a^4 b^{11}}{19 x^{19/2}}-\frac {455 a^3 b^{12}}{9 x^9}-\frac {210 a^2 b^{13}}{17 x^{17/2}}-\frac {15 a b^{14}}{8 x^8}-\frac {2 b^{15}}{15 x^{15/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 168, normalized size = 0.80 \[ -\frac {2181340125 \, a b^{14} x^{7} + 58815393000 \, a^{3} b^{12} x^{6} + 349363434420 \, a^{5} b^{10} x^{5} + 680578119000 \, a^{7} b^{8} x^{4} + 485226992250 \, a^{9} b^{6} x^{3} + 122155047000 \, a^{11} b^{4} x^{2} + 8725360500 \, a^{13} b^{2} x + 77558760 \, a^{15} + 16 \, {\left (9694845 \, b^{15} x^{7} + 898198875 \, a^{2} b^{13} x^{6} + 10447471125 \, a^{4} b^{11} x^{5} + 34659070875 \, a^{6} b^{9} x^{4} + 40686735375 \, a^{8} b^{7} x^{3} + 17468171721 \, a^{10} b^{5} x^{2} + 2450641375 \, a^{12} b^{3} x + 75218625 \, a^{14} b\right )} \sqrt {x}}{1163381400 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 167, normalized size = 0.79 \[ -\frac {155117520 \, b^{15} x^{\frac {15}{2}} + 2181340125 \, a b^{14} x^{7} + 14371182000 \, a^{2} b^{13} x^{\frac {13}{2}} + 58815393000 \, a^{3} b^{12} x^{6} + 167159538000 \, a^{4} b^{11} x^{\frac {11}{2}} + 349363434420 \, a^{5} b^{10} x^{5} + 554545134000 \, a^{6} b^{9} x^{\frac {9}{2}} + 680578119000 \, a^{7} b^{8} x^{4} + 650987766000 \, a^{8} b^{7} x^{\frac {7}{2}} + 485226992250 \, a^{9} b^{6} x^{3} + 279490747536 \, a^{10} b^{5} x^{\frac {5}{2}} + 122155047000 \, a^{11} b^{4} x^{2} + 39210262000 \, a^{12} b^{3} x^{\frac {3}{2}} + 8725360500 \, a^{13} b^{2} x + 1203498000 \, a^{14} b \sqrt {x} + 77558760 \, a^{15}}{1163381400 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 168, normalized size = 0.80 \[ -\frac {2 b^{15}}{15 x^{\frac {15}{2}}}-\frac {15 a \,b^{14}}{8 x^{8}}-\frac {210 a^{2} b^{13}}{17 x^{\frac {17}{2}}}-\frac {455 a^{3} b^{12}}{9 x^{9}}-\frac {2730 a^{4} b^{11}}{19 x^{\frac {19}{2}}}-\frac {3003 a^{5} b^{10}}{10 x^{10}}-\frac {1430 a^{6} b^{9}}{3 x^{\frac {21}{2}}}-\frac {585 a^{7} b^{8}}{x^{11}}-\frac {12870 a^{8} b^{7}}{23 x^{\frac {23}{2}}}-\frac {5005 a^{9} b^{6}}{12 x^{12}}-\frac {6006 a^{10} b^{5}}{25 x^{\frac {25}{2}}}-\frac {105 a^{11} b^{4}}{x^{13}}-\frac {910 a^{12} b^{3}}{27 x^{\frac {27}{2}}}-\frac {15 a^{13} b^{2}}{2 x^{14}}-\frac {30 a^{14} b}{29 x^{\frac {29}{2}}}-\frac {a^{15}}{15 x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 167, normalized size = 0.79 \[ -\frac {155117520 \, b^{15} x^{\frac {15}{2}} + 2181340125 \, a b^{14} x^{7} + 14371182000 \, a^{2} b^{13} x^{\frac {13}{2}} + 58815393000 \, a^{3} b^{12} x^{6} + 167159538000 \, a^{4} b^{11} x^{\frac {11}{2}} + 349363434420 \, a^{5} b^{10} x^{5} + 554545134000 \, a^{6} b^{9} x^{\frac {9}{2}} + 680578119000 \, a^{7} b^{8} x^{4} + 650987766000 \, a^{8} b^{7} x^{\frac {7}{2}} + 485226992250 \, a^{9} b^{6} x^{3} + 279490747536 \, a^{10} b^{5} x^{\frac {5}{2}} + 122155047000 \, a^{11} b^{4} x^{2} + 39210262000 \, a^{12} b^{3} x^{\frac {3}{2}} + 8725360500 \, a^{13} b^{2} x + 1203498000 \, a^{14} b \sqrt {x} + 77558760 \, a^{15}}{1163381400 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 167, normalized size = 0.79 \[ -\frac {\frac {a^{15}}{15}+\frac {2\,b^{15}\,x^{15/2}}{15}+\frac {15\,a^{13}\,b^2\,x}{2}+\frac {30\,a^{14}\,b\,\sqrt {x}}{29}+\frac {15\,a\,b^{14}\,x^7}{8}+105\,a^{11}\,b^4\,x^2+\frac {5005\,a^9\,b^6\,x^3}{12}+585\,a^7\,b^8\,x^4+\frac {3003\,a^5\,b^{10}\,x^5}{10}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{27}+\frac {455\,a^3\,b^{12}\,x^6}{9}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{25}+\frac {12870\,a^8\,b^7\,x^{7/2}}{23}+\frac {1430\,a^6\,b^9\,x^{9/2}}{3}+\frac {2730\,a^4\,b^{11}\,x^{11/2}}{19}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{17}}{x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 48.09, size = 216, normalized size = 1.02 \[ - \frac {a^{15}}{15 x^{15}} - \frac {30 a^{14} b}{29 x^{\frac {29}{2}}} - \frac {15 a^{13} b^{2}}{2 x^{14}} - \frac {910 a^{12} b^{3}}{27 x^{\frac {27}{2}}} - \frac {105 a^{11} b^{4}}{x^{13}} - \frac {6006 a^{10} b^{5}}{25 x^{\frac {25}{2}}} - \frac {5005 a^{9} b^{6}}{12 x^{12}} - \frac {12870 a^{8} b^{7}}{23 x^{\frac {23}{2}}} - \frac {585 a^{7} b^{8}}{x^{11}} - \frac {1430 a^{6} b^{9}}{3 x^{\frac {21}{2}}} - \frac {3003 a^{5} b^{10}}{10 x^{10}} - \frac {2730 a^{4} b^{11}}{19 x^{\frac {19}{2}}} - \frac {455 a^{3} b^{12}}{9 x^{9}} - \frac {210 a^{2} b^{13}}{17 x^{\frac {17}{2}}} - \frac {15 a b^{14}}{8 x^{8}} - \frac {2 b^{15}}{15 x^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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