Optimal. Leaf size=190 \[ 4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+1365 a^{11} b^4 \log (x)-\frac{10 a^{14} b}{x^{3/2}}-\frac{a^{15}}{2 x^2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]
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Rubi [A] time = 0.115418, antiderivative size = 190, 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} \[ 4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+1365 a^{11} b^4 \log (x)-\frac{10 a^{14} b}{x^{3/2}}-\frac{a^{15}}{2 x^2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]
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^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (3003 a^{10} b^5+\frac{a^{15}}{x^5}+\frac{15 a^{14} b}{x^4}+\frac{105 a^{13} b^2}{x^3}+\frac{455 a^{12} b^3}{x^2}+\frac{1365 a^{11} b^4}{x}+5005 a^9 b^6 x+6435 a^8 b^7 x^2+6435 a^7 b^8 x^3+5005 a^6 b^9 x^4+3003 a^5 b^{10} x^5+1365 a^4 b^{11} x^6+455 a^3 b^{12} x^7+105 a^2 b^{13} x^8+15 a b^{14} x^9+b^{15} x^{10}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{2 x^2}-\frac{10 a^{14} b}{x^{3/2}}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2}+1365 a^{11} b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0998445, size = 190, normalized size = 1. \[ 4290 a^8 b^7 x^{3/2}+\frac{6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac{455}{4} a^3 b^{12} x^4+\frac{70}{3} a^2 b^{13} x^{9/2}-\frac{105 a^{13} b^2}{x}-\frac{910 a^{12} b^3}{\sqrt{x}}+6006 a^{10} b^5 \sqrt{x}+5005 a^9 b^6 x+1365 a^{11} b^4 \log (x)-\frac{10 a^{14} b}{x^{3/2}}-\frac{a^{15}}{2 x^2}+3 a b^{14} x^5+\frac{2}{11} b^{15} x^{11/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 165, normalized size = 0.9 \begin{align*} -{\frac{{a}^{15}}{2\,{x}^{2}}}-10\,{\frac{{a}^{14}b}{{x}^{3/2}}}-105\,{\frac{{a}^{13}{b}^{2}}{x}}+5005\,{a}^{9}{b}^{6}x+4290\,{a}^{8}{b}^{7}{x}^{3/2}+{\frac{6435\,{a}^{7}{b}^{8}{x}^{2}}{2}}+2002\,{a}^{6}{b}^{9}{x}^{5/2}+1001\,{a}^{5}{b}^{10}{x}^{3}+390\,{a}^{4}{b}^{11}{x}^{7/2}+{\frac{455\,{a}^{3}{b}^{12}{x}^{4}}{4}}+{\frac{70\,{a}^{2}{b}^{13}}{3}{x}^{{\frac{9}{2}}}}+3\,a{b}^{14}{x}^{5}+{\frac{2\,{b}^{15}}{11}{x}^{{\frac{11}{2}}}}+1365\,{a}^{11}{b}^{4}\ln \left ( x \right ) -910\,{\frac{{a}^{12}{b}^{3}}{\sqrt{x}}}+6006\,{a}^{10}{b}^{5}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959836, size = 220, normalized size = 1.16 \begin{align*} \frac{2}{11} \, b^{15} x^{\frac{11}{2}} + 3 \, a b^{14} x^{5} + \frac{70}{3} \, a^{2} b^{13} x^{\frac{9}{2}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac{7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac{5}{2}} + \frac{6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac{3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left (x\right ) + 6006 \, a^{10} b^{5} \sqrt{x} - \frac{1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt{x} + a^{15}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30779, size = 451, normalized size = 2.37 \begin{align*} \frac{396 \, a b^{14} x^{7} + 15015 \, a^{3} b^{12} x^{6} + 132132 \, a^{5} b^{10} x^{5} + 424710 \, a^{7} b^{8} x^{4} + 660660 \, a^{9} b^{6} x^{3} + 360360 \, a^{11} b^{4} x^{2} \log \left (\sqrt{x}\right ) - 13860 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \,{\left (3 \, b^{15} x^{7} + 385 \, a^{2} b^{13} x^{6} + 6435 \, a^{4} b^{11} x^{5} + 33033 \, a^{6} b^{9} x^{4} + 70785 \, a^{8} b^{7} x^{3} + 99099 \, a^{10} b^{5} x^{2} - 15015 \, a^{12} b^{3} x - 165 \, a^{14} b\right )} \sqrt{x}}{132 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.85807, size = 196, normalized size = 1.03 \begin{align*} - \frac{a^{15}}{2 x^{2}} - \frac{10 a^{14} b}{x^{\frac{3}{2}}} - \frac{105 a^{13} b^{2}}{x} - \frac{910 a^{12} b^{3}}{\sqrt{x}} + 1365 a^{11} b^{4} \log{\left (x \right )} + 6006 a^{10} b^{5} \sqrt{x} + 5005 a^{9} b^{6} x + 4290 a^{8} b^{7} x^{\frac{3}{2}} + \frac{6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac{5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac{7}{2}} + \frac{455 a^{3} b^{12} x^{4}}{4} + \frac{70 a^{2} b^{13} x^{\frac{9}{2}}}{3} + 3 a b^{14} x^{5} + \frac{2 b^{15} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10221, size = 221, normalized size = 1.16 \begin{align*} \frac{2}{11} \, b^{15} x^{\frac{11}{2}} + 3 \, a b^{14} x^{5} + \frac{70}{3} \, a^{2} b^{13} x^{\frac{9}{2}} + \frac{455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac{7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac{5}{2}} + \frac{6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac{3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left ({\left | x \right |}\right ) + 6006 \, a^{10} b^{5} \sqrt{x} - \frac{1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt{x} + a^{15}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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