Optimal. Leaf size=111 \[ \frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{6 b^2}{a^5 x}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]
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Rubi [A] time = 0.0714815, antiderivative size = 111, 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} \[ \frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{6 b^2}{a^5 x}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]
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 )^3 x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^5}-\frac{3 b}{a^4 x^4}+\frac{6 b^2}{a^5 x^3}-\frac{10 b^3}{a^6 x^2}+\frac{15 b^4}{a^7 x}-\frac{b^5}{a^5 (a+b x)^3}-\frac{5 b^5}{a^6 (a+b x)^2}-\frac{15 b^5}{a^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}-\frac{1}{2 a^3 x^2}+\frac{2 b}{a^4 x^{3/2}}-\frac{6 b^2}{a^5 x}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.119479, size = 104, normalized size = 0.94 \[ \frac{\frac{a \left (20 a^2 b^3 x^{3/2}-5 a^3 b^2 x+2 a^4 b \sqrt{x}-a^5+90 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^2 \left (a+b \sqrt{x}\right )^2}-60 b^4 \log \left (a+b \sqrt{x}\right )+30 b^4 \log (x)}{2 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 100, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}{a}^{3}}}+2\,{\frac{b}{{a}^{4}{x}^{3/2}}}-6\,{\frac{{b}^{2}}{x{a}^{5}}}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+20\,{\frac{{b}^{3}}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{4}}{{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-2}}+10\,{\frac{{b}^{4}}{{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 = 1.01847, size = 149, normalized size = 1.34 \begin{align*} \frac{60 \, b^{5} x^{\frac{5}{2}} + 90 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} - 5 \, a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} - a^{5}}{2 \,{\left (a^{6} b^{2} x^{3} + 2 \, a^{7} b x^{\frac{5}{2}} + a^{8} x^{2}\right )}} - \frac{30 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{4} \log \left (x\right )}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37366, size = 392, normalized size = 3.53 \begin{align*} -\frac{30 \, a^{2} b^{6} x^{3} - 45 \, a^{4} b^{4} x^{2} + 10 \, a^{6} b^{2} x + a^{8} + 60 \,{\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) - 60 \,{\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (15 \, a b^{7} x^{3} - 25 \, a^{3} b^{5} x^{2} + 8 \, a^{5} b^{3} x + a^{7} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{4} x^{4} - 2 \, a^{9} b^{2} x^{3} + a^{11} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.80979, size = 612, normalized size = 5.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 a^{3} x^{2}} & \text{for}\: b = 0 \\- \frac{2}{7 b^{3} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{a^{6} \sqrt{x}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{2 a^{5} b x}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{5 a^{4} b^{2} x^{\frac{3}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{20 a^{3} b^{3} x^{2}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{90 a^{2} b^{4} x^{\frac{5}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{120 a b^{5} x^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 b^{6} x^{\frac{7}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 b^{6} x^{\frac{7}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11759, size = 136, normalized size = 1.23 \begin{align*} -\frac{30 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 90 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac{3}{2}} - 5 \, a^{4} b^{2} x + 2 \, a^{5} b \sqrt{x} - a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{2} a^{7} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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