Optimal. Leaf size=127 \[ -\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+210 a^4 b^6 \log (x)-\frac{4 a^9 b}{x^{5/2}}-\frac{a^{10}}{3 x^3}+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]
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Rubi [A] time = 0.0689496, antiderivative size = 127, 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} \[ -\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+210 a^4 b^6 \log (x)-\frac{4 a^9 b}{x^{5/2}}-\frac{a^{10}}{3 x^3}+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (120 a^3 b^7+\frac{a^{10}}{x^7}+\frac{10 a^9 b}{x^6}+\frac{45 a^8 b^2}{x^5}+\frac{120 a^7 b^3}{x^4}+\frac{210 a^6 b^4}{x^3}+\frac{252 a^5 b^5}{x^2}+\frac{210 a^4 b^6}{x}+45 a^2 b^8 x+10 a b^9 x^2+b^{10} x^3\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{10}}{3 x^3}-\frac{4 a^9 b}{x^{5/2}}-\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2}+210 a^4 b^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.069649, size = 124, normalized size = 0.98 \[ 210 a^4 b^6 \log (x)-\frac{480 a^7 b^3 x^{3/2}+1260 a^6 b^4 x^2+3024 a^5 b^5 x^{5/2}-1440 a^3 b^7 x^{7/2}-270 a^2 b^8 x^4+135 a^8 b^2 x+24 a^9 b \sqrt{x}+2 a^{10}-40 a b^9 x^{9/2}-3 b^{10} x^5}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{3\,{x}^{3}}}-4\,{\frac{{a}^{9}b}{{x}^{5/2}}}-{\frac{45\,{a}^{8}{b}^{2}}{2\,{x}^{2}}}-80\,{\frac{{a}^{7}{b}^{3}}{{x}^{3/2}}}-210\,{\frac{{a}^{6}{b}^{4}}{x}}+45\,{a}^{2}{b}^{8}x+{\frac{20\,a{b}^{9}}{3}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{10}{x}^{2}}{2}}+210\,{a}^{4}{b}^{6}\ln \left ( x \right ) -504\,{\frac{{a}^{5}{b}^{5}}{\sqrt{x}}}+240\,{a}^{3}{b}^{7}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991752, size = 149, normalized size = 1.17 \begin{align*} \frac{1}{2} \, b^{10} x^{2} + \frac{20}{3} \, a b^{9} x^{\frac{3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6} \log \left (x\right ) + 240 \, a^{3} b^{7} \sqrt{x} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac{3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt{x} + 2 \, a^{10}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57971, size = 274, normalized size = 2.16 \begin{align*} \frac{3 \, b^{10} x^{5} + 270 \, a^{2} b^{8} x^{4} + 2520 \, a^{4} b^{6} x^{3} \log \left (\sqrt{x}\right ) - 1260 \, a^{6} b^{4} x^{2} - 135 \, a^{8} b^{2} x - 2 \, a^{10} + 8 \,{\left (5 \, a b^{9} x^{4} + 180 \, a^{3} b^{7} x^{3} - 378 \, a^{5} b^{5} x^{2} - 60 \, a^{7} b^{3} x - 3 \, a^{9} b\right )} \sqrt{x}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.68061, size = 128, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{3 x^{3}} - \frac{4 a^{9} b}{x^{\frac{5}{2}}} - \frac{45 a^{8} b^{2}}{2 x^{2}} - \frac{80 a^{7} b^{3}}{x^{\frac{3}{2}}} - \frac{210 a^{6} b^{4}}{x} - \frac{504 a^{5} b^{5}}{\sqrt{x}} + 210 a^{4} b^{6} \log{\left (x \right )} + 240 a^{3} b^{7} \sqrt{x} + 45 a^{2} b^{8} x + \frac{20 a b^{9} x^{\frac{3}{2}}}{3} + \frac{b^{10} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08797, size = 150, normalized size = 1.18 \begin{align*} \frac{1}{2} \, b^{10} x^{2} + \frac{20}{3} \, a b^{9} x^{\frac{3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6} \log \left ({\left | x \right |}\right ) + 240 \, a^{3} b^{7} \sqrt{x} - \frac{3024 \, a^{5} b^{5} x^{\frac{5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac{3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt{x} + 2 \, a^{10}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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