Optimal. Leaf size=211 \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]
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Rubi [A] time = 0.116386, antiderivative size = 211, normalized size of antiderivative = 1., 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} \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{15}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{29}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{29}}+\frac{15 a^{14} b}{x^{28}}+\frac{105 a^{13} b^2}{x^{27}}+\frac{455 a^{12} b^3}{x^{26}}+\frac{1365 a^{11} b^4}{x^{25}}+\frac{3003 a^{10} b^5}{x^{24}}+\frac{5005 a^9 b^6}{x^{23}}+\frac{6435 a^8 b^7}{x^{22}}+\frac{6435 a^7 b^8}{x^{21}}+\frac{5005 a^6 b^9}{x^{20}}+\frac{3003 a^5 b^{10}}{x^{19}}+\frac{1365 a^4 b^{11}}{x^{18}}+\frac{455 a^3 b^{12}}{x^{17}}+\frac{105 a^2 b^{13}}{x^{16}}+\frac{15 a b^{14}}{x^{15}}+\frac{b^{15}}{x^{14}}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{14 x^{14}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.100854, size = 211, normalized size = 1. \[ -\frac{105 a^{13} b^2}{13 x^{13}}-\frac{182 a^{12} b^3}{5 x^{25/2}}-\frac{455 a^{11} b^4}{4 x^{12}}-\frac{6006 a^{10} b^5}{23 x^{23/2}}-\frac{455 a^9 b^6}{x^{11}}-\frac{4290 a^8 b^7}{7 x^{21/2}}-\frac{1287 a^7 b^8}{2 x^{10}}-\frac{10010 a^6 b^9}{19 x^{19/2}}-\frac{1001 a^5 b^{10}}{3 x^9}-\frac{2730 a^4 b^{11}}{17 x^{17/2}}-\frac{455 a^3 b^{12}}{8 x^8}-\frac{14 a^2 b^{13}}{x^{15/2}}-\frac{10 a^{14} b}{9 x^{27/2}}-\frac{a^{15}}{14 x^{14}}-\frac{15 a b^{14}}{7 x^7}-\frac{2 b^{15}}{13 x^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{14\,{x}^{14}}}-{\frac{10\,{a}^{14}b}{9}{x}^{-{\frac{27}{2}}}}-{\frac{105\,{a}^{13}{b}^{2}}{13\,{x}^{13}}}-{\frac{182\,{a}^{12}{b}^{3}}{5}{x}^{-{\frac{25}{2}}}}-{\frac{455\,{a}^{11}{b}^{4}}{4\,{x}^{12}}}-{\frac{6006\,{a}^{10}{b}^{5}}{23}{x}^{-{\frac{23}{2}}}}-455\,{\frac{{a}^{9}{b}^{6}}{{x}^{11}}}-{\frac{4290\,{a}^{8}{b}^{7}}{7}{x}^{-{\frac{21}{2}}}}-{\frac{1287\,{a}^{7}{b}^{8}}{2\,{x}^{10}}}-{\frac{10010\,{a}^{6}{b}^{9}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{1001\,{a}^{5}{b}^{10}}{3\,{x}^{9}}}-{\frac{2730\,{a}^{4}{b}^{11}}{17}{x}^{-{\frac{17}{2}}}}-{\frac{455\,{a}^{3}{b}^{12}}{8\,{x}^{8}}}-14\,{\frac{{a}^{2}{b}^{13}}{{x}^{15/2}}}-{\frac{15\,a{b}^{14}}{7\,{x}^{7}}}-{\frac{2\,{b}^{15}}{13}{x}^{-{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995123, size = 225, normalized size = 1.07 \begin{align*} -\frac{37442160 \, b^{15} x^{\frac{15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac{13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac{11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac{9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac{7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac{5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac{3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt{x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26863, size = 563, normalized size = 2.67 \begin{align*} -\frac{521515800 \, a b^{14} x^{7} + 13841898525 \, a^{3} b^{12} x^{6} + 81205804680 \, a^{5} b^{10} x^{5} + 156611194740 \, a^{7} b^{8} x^{4} + 110735188200 \, a^{9} b^{6} x^{3} + 27683797050 \, a^{11} b^{4} x^{2} + 1965713400 \, a^{13} b^{2} x + 17383860 \, a^{15} + 16 \,{\left (2340135 \, b^{15} x^{7} + 212952285 \, a^{2} b^{13} x^{6} + 2442687975 \, a^{4} b^{11} x^{5} + 8013730725 \, a^{6} b^{9} x^{4} + 9322094925 \, a^{8} b^{7} x^{3} + 3972023055 \, a^{10} b^{5} x^{2} + 553675941 \, a^{12} b^{3} x + 16900975 \, a^{14} b\right )} \sqrt{x}}{243374040 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 41.9276, size = 216, normalized size = 1.02 \begin{align*} - \frac{a^{15}}{14 x^{14}} - \frac{10 a^{14} b}{9 x^{\frac{27}{2}}} - \frac{105 a^{13} b^{2}}{13 x^{13}} - \frac{182 a^{12} b^{3}}{5 x^{\frac{25}{2}}} - \frac{455 a^{11} b^{4}}{4 x^{12}} - \frac{6006 a^{10} b^{5}}{23 x^{\frac{23}{2}}} - \frac{455 a^{9} b^{6}}{x^{11}} - \frac{4290 a^{8} b^{7}}{7 x^{\frac{21}{2}}} - \frac{1287 a^{7} b^{8}}{2 x^{10}} - \frac{10010 a^{6} b^{9}}{19 x^{\frac{19}{2}}} - \frac{1001 a^{5} b^{10}}{3 x^{9}} - \frac{2730 a^{4} b^{11}}{17 x^{\frac{17}{2}}} - \frac{455 a^{3} b^{12}}{8 x^{8}} - \frac{14 a^{2} b^{13}}{x^{\frac{15}{2}}} - \frac{15 a b^{14}}{7 x^{7}} - \frac{2 b^{15}}{13 x^{\frac{13}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11295, size = 225, normalized size = 1.07 \begin{align*} -\frac{37442160 \, b^{15} x^{\frac{15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac{13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac{11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac{9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac{7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac{5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac{3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt{x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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