Optimal. Leaf size=196 \[ -\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+5005 a^9 b^6 \log (x)-\frac{6 a^{14} b}{x^{5/2}}-\frac{a^{15}}{3 x^3}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]
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Rubi [A] time = 0.115921, antiderivative size = 196, 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} \[ -\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+5005 a^9 b^6 \log (x)-\frac{6 a^{14} b}{x^{5/2}}-\frac{a^{15}}{3 x^3}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]
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^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (6435 a^8 b^7+\frac{a^{15}}{x^7}+\frac{15 a^{14} b}{x^6}+\frac{105 a^{13} b^2}{x^5}+\frac{455 a^{12} b^3}{x^4}+\frac{1365 a^{11} b^4}{x^3}+\frac{3003 a^{10} b^5}{x^2}+\frac{5005 a^9 b^6}{x}+6435 a^7 b^8 x+5005 a^6 b^9 x^2+3003 a^5 b^{10} x^3+1365 a^4 b^{11} x^4+455 a^3 b^{12} x^5+105 a^2 b^{13} x^6+15 a b^{14} x^7+b^{15} x^8\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{3 x^3}-\frac{6 a^{14} b}{x^{5/2}}-\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2}+5005 a^9 b^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.100847, size = 196, normalized size = 1. \[ -\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+5005 a^9 b^6 \log (x)-\frac{6 a^{14} b}{x^{5/2}}-\frac{a^{15}}{3 x^3}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 165, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{3\,{x}^{3}}}-6\,{\frac{{a}^{14}b}{{x}^{5/2}}}-{\frac{105\,{a}^{13}{b}^{2}}{2\,{x}^{2}}}-{\frac{910\,{a}^{12}{b}^{3}}{3}{x}^{-{\frac{3}{2}}}}-1365\,{\frac{{a}^{11}{b}^{4}}{x}}+6435\,{a}^{7}{b}^{8}x+{\frac{10010\,{a}^{6}{b}^{9}}{3}{x}^{{\frac{3}{2}}}}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{2}}{2}}+546\,{a}^{4}{b}^{11}{x}^{5/2}+{\frac{455\,{a}^{3}{b}^{12}{x}^{3}}{3}}+30\,{a}^{2}{b}^{13}{x}^{7/2}+{\frac{15\,a{b}^{14}{x}^{4}}{4}}+{\frac{2\,{b}^{15}}{9}{x}^{{\frac{9}{2}}}}+5005\,{a}^{9}{b}^{6}\ln \left ( x \right ) -6006\,{\frac{{a}^{10}{b}^{5}}{\sqrt{x}}}+12870\,{a}^{8}{b}^{7}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979495, size = 223, normalized size = 1.14 \begin{align*} \frac{2}{9} \, b^{15} x^{\frac{9}{2}} + \frac{15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac{7}{2}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac{5}{2}} + \frac{3003}{2} \, a^{5} b^{10} x^{2} + \frac{10010}{3} \, a^{6} b^{9} x^{\frac{3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6} \log \left (x\right ) + 12870 \, a^{8} b^{7} \sqrt{x} - \frac{36036 \, a^{10} b^{5} x^{\frac{5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt{x} + 2 \, a^{15}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32375, size = 439, normalized size = 2.24 \begin{align*} \frac{135 \, a b^{14} x^{7} + 5460 \, a^{3} b^{12} x^{6} + 54054 \, a^{5} b^{10} x^{5} + 231660 \, a^{7} b^{8} x^{4} + 360360 \, a^{9} b^{6} x^{3} \log \left (\sqrt{x}\right ) - 49140 \, a^{11} b^{4} x^{2} - 1890 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \,{\left (b^{15} x^{7} + 135 \, a^{2} b^{13} x^{6} + 2457 \, a^{4} b^{11} x^{5} + 15015 \, a^{6} b^{9} x^{4} + 57915 \, a^{8} b^{7} x^{3} - 27027 \, a^{10} b^{5} x^{2} - 1365 \, a^{12} b^{3} x - 27 \, a^{14} b\right )} \sqrt{x}}{36 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.10494, size = 201, normalized size = 1.03 \begin{align*} - \frac{a^{15}}{3 x^{3}} - \frac{6 a^{14} b}{x^{\frac{5}{2}}} - \frac{105 a^{13} b^{2}}{2 x^{2}} - \frac{910 a^{12} b^{3}}{3 x^{\frac{3}{2}}} - \frac{1365 a^{11} b^{4}}{x} - \frac{6006 a^{10} b^{5}}{\sqrt{x}} + 5005 a^{9} b^{6} \log{\left (x \right )} + 12870 a^{8} b^{7} \sqrt{x} + 6435 a^{7} b^{8} x + \frac{10010 a^{6} b^{9} x^{\frac{3}{2}}}{3} + \frac{3003 a^{5} b^{10} x^{2}}{2} + 546 a^{4} b^{11} x^{\frac{5}{2}} + \frac{455 a^{3} b^{12} x^{3}}{3} + 30 a^{2} b^{13} x^{\frac{7}{2}} + \frac{15 a b^{14} x^{4}}{4} + \frac{2 b^{15} x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12024, size = 224, normalized size = 1.14 \begin{align*} \frac{2}{9} \, b^{15} x^{\frac{9}{2}} + \frac{15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac{7}{2}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac{5}{2}} + \frac{3003}{2} \, a^{5} b^{10} x^{2} + \frac{10010}{3} \, a^{6} b^{9} x^{\frac{3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6} \log \left ({\left | x \right |}\right ) + 12870 \, a^{8} b^{7} \sqrt{x} - \frac{36036 \, a^{10} b^{5} x^{\frac{5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt{x} + 2 \, a^{15}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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