Optimal. Leaf size=126 \[ \frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4}-\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{10 b}{a^6 \sqrt{x}}-\frac{1}{a^5 x} \]
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Rubi [A] time = 0.0853851, antiderivative size = 126, 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, 44} \[ \frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4}-\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{10 b}{a^6 \sqrt{x}}-\frac{1}{a^5 x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^5 x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^5 x^3}-\frac{5 b}{a^6 x^2}+\frac{15 b^2}{a^7 x}-\frac{b^3}{a^3 (a+b x)^5}-\frac{3 b^3}{a^4 (a+b x)^4}-\frac{6 b^3}{a^5 (a+b x)^3}-\frac{10 b^3}{a^6 (a+b x)^2}-\frac{15 b^3}{a^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}-\frac{1}{a^5 x}+\frac{10 b}{a^6 \sqrt{x}}-\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.115404, size = 104, normalized size = 0.83 \[ \frac{\frac{a \left (260 a^2 b^3 x^{3/2}+125 a^3 b^2 x+12 a^4 b \sqrt{x}-2 a^5+210 a b^4 x^2+60 b^5 x^{5/2}\right )}{x \left (a+b \sqrt{x}\right )^4}-60 b^2 \log \left (a+b \sqrt{x}\right )+30 b^2 \log (x)}{2 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 113, normalized size = 0.9 \begin{align*} -{\frac{1}{x{a}^{5}}}+15\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+10\,{\frac{b}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{2}}{2\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-4}}+2\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\sqrt{x} \right ) ^{3}}}+6\,{\frac{{b}^{2}}{{a}^{5} \left ( a+b\sqrt{x} \right ) ^{2}}}+20\,{\frac{{b}^{2}}{{a}^{6} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01853, size = 176, normalized size = 1.4 \begin{align*} \frac{60 \, b^{5} x^{\frac{5}{2}} + 210 \, a b^{4} x^{2} + 260 \, a^{2} b^{3} x^{\frac{3}{2}} + 125 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt{x} - 2 \, a^{5}}{2 \,{\left (a^{6} b^{4} x^{3} + 4 \, a^{7} b^{3} x^{\frac{5}{2}} + 6 \, a^{8} b^{2} x^{2} + 4 \, a^{9} b x^{\frac{3}{2}} + a^{10} x\right )}} - \frac{30 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{2} \log \left (x\right )}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36896, size = 575, normalized size = 4.56 \begin{align*} -\frac{30 \, a^{2} b^{8} x^{4} - 105 \, a^{4} b^{6} x^{3} + 130 \, a^{6} b^{4} x^{2} - 65 \, a^{8} b^{2} x + 2 \, a^{10} + 60 \,{\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 60 \,{\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (15 \, a b^{9} x^{4} - 55 \, a^{3} b^{7} x^{3} + 73 \, a^{5} b^{5} x^{2} - 40 \, a^{7} b^{3} x + 5 \, a^{9} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{8} x^{5} - 4 \, a^{9} b^{6} x^{4} + 6 \, a^{11} b^{4} x^{3} - 4 \, a^{13} b^{2} x^{2} + a^{15} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.7859, size = 1229, normalized size = 9.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11892, size = 136, normalized size = 1.08 \begin{align*} -\frac{30 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 210 \, a^{2} b^{4} x^{2} + 260 \, a^{3} b^{3} x^{\frac{3}{2}} + 125 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt{x} - 2 \, a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} a^{7} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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