Optimal. Leaf size=270 \[ -\frac{b^{10} \left (a+b \sqrt{x}\right )^{16}}{42493880 a^{11} x^8}+\frac{2 b^9 \left (a+b \sqrt{x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}} \]
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Rubi [A] time = 0.150993, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^{10} \left (a+b \sqrt{x}\right )^{16}}{42493880 a^{11} x^8}+\frac{2 b^9 \left (a+b \sqrt{x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^{14}} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{27}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}-\frac{(10 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{26}} \, dx,x,\sqrt{x}\right )}{13 a}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}+\frac{\left (18 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{25}} \, dx,x,\sqrt{x}\right )}{65 a^2}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{24}} \, dx,x,\sqrt{x}\right )}{65 a^3}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}+\frac{\left (42 b^4\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt{x}\right )}{1495 a^4}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}-\frac{\left (126 b^5\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt{x}\right )}{16445 a^5}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}+\frac{\left (6 b^6\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt{x}\right )}{3289 a^6}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}-\frac{\left (6 b^7\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt{x}\right )}{16445 a^7}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}+\frac{\left (18 b^8\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt{x}\right )}{312455 a^8}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}-\frac{\left (2 b^9\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt{x}\right )}{312455 a^9}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}+\frac{2 b^9 \left (a+b \sqrt{x}\right )^{16}}{5311735 a^{10} x^{17/2}}+\frac{\left (2 b^{10}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt{x}\right )}{5311735 a^{10}}\\ &=-\frac{\left (a+b \sqrt{x}\right )^{16}}{13 a x^{13}}+\frac{2 b \left (a+b \sqrt{x}\right )^{16}}{65 a^2 x^{25/2}}-\frac{3 b^2 \left (a+b \sqrt{x}\right )^{16}}{260 a^3 x^{12}}+\frac{6 b^3 \left (a+b \sqrt{x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac{21 b^4 \left (a+b \sqrt{x}\right )^{16}}{16445 a^5 x^{11}}+\frac{6 b^5 \left (a+b \sqrt{x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac{3 b^6 \left (a+b \sqrt{x}\right )^{16}}{32890 a^7 x^{10}}+\frac{6 b^7 \left (a+b \sqrt{x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac{b^8 \left (a+b \sqrt{x}\right )^{16}}{312455 a^9 x^9}+\frac{2 b^9 \left (a+b \sqrt{x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac{b^{10} \left (a+b \sqrt{x}\right )^{16}}{42493880 a^{11} x^8}\\ \end{align*}
Mathematica [A] time = 0.107446, size = 207, normalized size = 0.77 \[ -\frac{35 a^{13} b^2}{4 x^{12}}-\frac{910 a^{12} b^3}{23 x^{23/2}}-\frac{1365 a^{11} b^4}{11 x^{11}}-\frac{286 a^{10} b^5}{x^{21/2}}-\frac{1001 a^9 b^6}{2 x^{10}}-\frac{12870 a^8 b^7}{19 x^{19/2}}-\frac{715 a^7 b^8}{x^9}-\frac{10010 a^6 b^9}{17 x^{17/2}}-\frac{3003 a^5 b^{10}}{8 x^8}-\frac{182 a^4 b^{11}}{x^{15/2}}-\frac{65 a^3 b^{12}}{x^7}-\frac{210 a^2 b^{13}}{13 x^{13/2}}-\frac{6 a^{14} b}{5 x^{25/2}}-\frac{a^{15}}{13 x^{13}}-\frac{5 a b^{14}}{2 x^6}-\frac{2 b^{15}}{11 x^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 168, normalized size = 0.6 \begin{align*} -{\frac{2\,{b}^{15}}{11}{x}^{-{\frac{11}{2}}}}-{\frac{5\,a{b}^{14}}{2\,{x}^{6}}}-{\frac{210\,{a}^{2}{b}^{13}}{13}{x}^{-{\frac{13}{2}}}}-65\,{\frac{{a}^{3}{b}^{12}}{{x}^{7}}}-182\,{\frac{{a}^{4}{b}^{11}}{{x}^{15/2}}}-{\frac{3003\,{a}^{5}{b}^{10}}{8\,{x}^{8}}}-{\frac{10010\,{a}^{6}{b}^{9}}{17}{x}^{-{\frac{17}{2}}}}-715\,{\frac{{a}^{7}{b}^{8}}{{x}^{9}}}-{\frac{12870\,{a}^{8}{b}^{7}}{19}{x}^{-{\frac{19}{2}}}}-{\frac{1001\,{a}^{9}{b}^{6}}{2\,{x}^{10}}}-286\,{\frac{{a}^{10}{b}^{5}}{{x}^{21/2}}}-{\frac{1365\,{a}^{11}{b}^{4}}{11\,{x}^{11}}}-{\frac{910\,{a}^{12}{b}^{3}}{23}{x}^{-{\frac{23}{2}}}}-{\frac{35\,{a}^{13}{b}^{2}}{4\,{x}^{12}}}-{\frac{6\,{a}^{14}b}{5}{x}^{-{\frac{25}{2}}}}-{\frac{{a}^{15}}{13\,{x}^{13}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04356, size = 225, normalized size = 0.83 \begin{align*} -\frac{7726160 \, b^{15} x^{\frac{15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac{13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac{11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac{9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac{7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac{5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac{3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt{x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32377, size = 547, normalized size = 2.03 \begin{align*} -\frac{106234700 \, a b^{14} x^{7} + 2762102200 \, a^{3} b^{12} x^{6} + 15951140205 \, a^{5} b^{10} x^{5} + 30383124200 \, a^{7} b^{8} x^{4} + 21268186940 \, a^{9} b^{6} x^{3} + 5273104200 \, a^{11} b^{4} x^{2} + 371821450 \, a^{13} b^{2} x + 3268760 \, a^{15} + 16 \,{\left (482885 \, b^{15} x^{7} + 42902475 \, a^{2} b^{13} x^{6} + 483367885 \, a^{4} b^{11} x^{5} + 1563837275 \, a^{6} b^{9} x^{4} + 1799000775 \, a^{8} b^{7} x^{3} + 759578105 \, a^{10} b^{5} x^{2} + 105079975 \, a^{12} b^{3} x + 3187041 \, a^{14} b\right )} \sqrt{x}}{42493880 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.4415, size = 212, normalized size = 0.79 \begin{align*} - \frac{a^{15}}{13 x^{13}} - \frac{6 a^{14} b}{5 x^{\frac{25}{2}}} - \frac{35 a^{13} b^{2}}{4 x^{12}} - \frac{910 a^{12} b^{3}}{23 x^{\frac{23}{2}}} - \frac{1365 a^{11} b^{4}}{11 x^{11}} - \frac{286 a^{10} b^{5}}{x^{\frac{21}{2}}} - \frac{1001 a^{9} b^{6}}{2 x^{10}} - \frac{12870 a^{8} b^{7}}{19 x^{\frac{19}{2}}} - \frac{715 a^{7} b^{8}}{x^{9}} - \frac{10010 a^{6} b^{9}}{17 x^{\frac{17}{2}}} - \frac{3003 a^{5} b^{10}}{8 x^{8}} - \frac{182 a^{4} b^{11}}{x^{\frac{15}{2}}} - \frac{65 a^{3} b^{12}}{x^{7}} - \frac{210 a^{2} b^{13}}{13 x^{\frac{13}{2}}} - \frac{5 a b^{14}}{2 x^{6}} - \frac{2 b^{15}}{11 x^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12144, size = 225, normalized size = 0.83 \begin{align*} -\frac{7726160 \, b^{15} x^{\frac{15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac{13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac{11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac{9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac{7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac{5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac{3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt{x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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