Optimal. Leaf size=207 \[ -\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{a^{15}}{16 x^{16}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]
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Rubi [A] time = 0.11063, antiderivative size = 207, 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} \[ -\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{a^{15}}{16 x^{16}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]
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^{17}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{33}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^{15}}{x^{33}}+\frac{15 a^{14} b}{x^{32}}+\frac{105 a^{13} b^2}{x^{31}}+\frac{455 a^{12} b^3}{x^{30}}+\frac{1365 a^{11} b^4}{x^{29}}+\frac{3003 a^{10} b^5}{x^{28}}+\frac{5005 a^9 b^6}{x^{27}}+\frac{6435 a^8 b^7}{x^{26}}+\frac{6435 a^7 b^8}{x^{25}}+\frac{5005 a^6 b^9}{x^{24}}+\frac{3003 a^5 b^{10}}{x^{23}}+\frac{1365 a^4 b^{11}}{x^{22}}+\frac{455 a^3 b^{12}}{x^{21}}+\frac{105 a^2 b^{13}}{x^{20}}+\frac{15 a b^{14}}{x^{19}}+\frac{b^{15}}{x^{18}}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{16 x^{16}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}}\\ \end{align*}
Mathematica [A] time = 0.104368, size = 207, normalized size = 1. \[ -\frac{7 a^{13} b^2}{x^{15}}-\frac{910 a^{12} b^3}{29 x^{29/2}}-\frac{195 a^{11} b^4}{2 x^{14}}-\frac{2002 a^{10} b^5}{9 x^{27/2}}-\frac{385 a^9 b^6}{x^{13}}-\frac{2574 a^8 b^7}{5 x^{25/2}}-\frac{2145 a^7 b^8}{4 x^{12}}-\frac{10010 a^6 b^9}{23 x^{23/2}}-\frac{273 a^5 b^{10}}{x^{11}}-\frac{130 a^4 b^{11}}{x^{21/2}}-\frac{91 a^3 b^{12}}{2 x^{10}}-\frac{210 a^2 b^{13}}{19 x^{19/2}}-\frac{30 a^{14} b}{31 x^{31/2}}-\frac{a^{15}}{16 x^{16}}-\frac{5 a b^{14}}{3 x^9}-\frac{2 b^{15}}{17 x^{17/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 168, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{16\,{x}^{16}}}-{\frac{30\,{a}^{14}b}{31}{x}^{-{\frac{31}{2}}}}-7\,{\frac{{a}^{13}{b}^{2}}{{x}^{15}}}-{\frac{910\,{a}^{12}{b}^{3}}{29}{x}^{-{\frac{29}{2}}}}-{\frac{195\,{a}^{11}{b}^{4}}{2\,{x}^{14}}}-{\frac{2002\,{a}^{10}{b}^{5}}{9}{x}^{-{\frac{27}{2}}}}-385\,{\frac{{a}^{9}{b}^{6}}{{x}^{13}}}-{\frac{2574\,{a}^{8}{b}^{7}}{5}{x}^{-{\frac{25}{2}}}}-{\frac{2145\,{a}^{7}{b}^{8}}{4\,{x}^{12}}}-{\frac{10010\,{a}^{6}{b}^{9}}{23}{x}^{-{\frac{23}{2}}}}-273\,{\frac{{a}^{5}{b}^{10}}{{x}^{11}}}-130\,{\frac{{a}^{4}{b}^{11}}{{x}^{21/2}}}-{\frac{91\,{a}^{3}{b}^{12}}{2\,{x}^{10}}}-{\frac{210\,{a}^{2}{b}^{13}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{5\,a{b}^{14}}{3\,{x}^{9}}}-{\frac{2\,{b}^{15}}{17}{x}^{-{\frac{17}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02853, size = 225, normalized size = 1.09 \begin{align*} -\frac{565722720 \, b^{15} x^{\frac{15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac{13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac{11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac{9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac{7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac{5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac{3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt{x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25447, size = 587, normalized size = 2.84 \begin{align*} -\frac{8014405200 \, a b^{14} x^{7} + 218793261960 \, a^{3} b^{12} x^{6} + 1312759571760 \, a^{5} b^{10} x^{5} + 2578634873100 \, a^{7} b^{8} x^{4} + 1851327601200 \, a^{9} b^{6} x^{3} + 468842704200 \, a^{11} b^{4} x^{2} + 33660501840 \, a^{13} b^{2} x + 300540195 \, a^{15} + 32 \,{\left (17678835 \, b^{15} x^{7} + 1660880025 \, a^{2} b^{13} x^{6} + 19535112675 \, a^{4} b^{11} x^{5} + 65400159825 \, a^{6} b^{9} x^{4} + 77359046193 \, a^{8} b^{7} x^{3} + 33426748355 \, a^{10} b^{5} x^{2} + 4715372025 \, a^{12} b^{3} x + 145422675 \, a^{14} b\right )} \sqrt{x}}{4808643120 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 58.8739, size = 212, normalized size = 1.02 \begin{align*} - \frac{a^{15}}{16 x^{16}} - \frac{30 a^{14} b}{31 x^{\frac{31}{2}}} - \frac{7 a^{13} b^{2}}{x^{15}} - \frac{910 a^{12} b^{3}}{29 x^{\frac{29}{2}}} - \frac{195 a^{11} b^{4}}{2 x^{14}} - \frac{2002 a^{10} b^{5}}{9 x^{\frac{27}{2}}} - \frac{385 a^{9} b^{6}}{x^{13}} - \frac{2574 a^{8} b^{7}}{5 x^{\frac{25}{2}}} - \frac{2145 a^{7} b^{8}}{4 x^{12}} - \frac{10010 a^{6} b^{9}}{23 x^{\frac{23}{2}}} - \frac{273 a^{5} b^{10}}{x^{11}} - \frac{130 a^{4} b^{11}}{x^{\frac{21}{2}}} - \frac{91 a^{3} b^{12}}{2 x^{10}} - \frac{210 a^{2} b^{13}}{19 x^{\frac{19}{2}}} - \frac{5 a b^{14}}{3 x^{9}} - \frac{2 b^{15}}{17 x^{\frac{17}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13418, size = 225, normalized size = 1.09 \begin{align*} -\frac{565722720 \, b^{15} x^{\frac{15}{2}} + 8014405200 \, a b^{14} x^{7} + 53148160800 \, a^{2} b^{13} x^{\frac{13}{2}} + 218793261960 \, a^{3} b^{12} x^{6} + 625123605600 \, a^{4} b^{11} x^{\frac{11}{2}} + 1312759571760 \, a^{5} b^{10} x^{5} + 2092805114400 \, a^{6} b^{9} x^{\frac{9}{2}} + 2578634873100 \, a^{7} b^{8} x^{4} + 2475489478176 \, a^{8} b^{7} x^{\frac{7}{2}} + 1851327601200 \, a^{9} b^{6} x^{3} + 1069655947360 \, a^{10} b^{5} x^{\frac{5}{2}} + 468842704200 \, a^{11} b^{4} x^{2} + 150891904800 \, a^{12} b^{3} x^{\frac{3}{2}} + 33660501840 \, a^{13} b^{2} x + 4653525600 \, a^{14} b \sqrt{x} + 300540195 \, a^{15}}{4808643120 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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