Optimal. Leaf size=248 \[ \frac{b^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a b^4 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.058232, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a b^4 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^8} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^5}{x^8} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a^5 b^5}{x^8}+\frac{5 a^4 b^6}{x^5}+\frac{10 a^3 b^7}{x^2}+10 a^2 b^8 x+5 a b^9 x^4+b^{10} x^7\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{a b^4 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0202955, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (280 a^2 b^3 x^9-560 a^3 b^2 x^6-70 a^4 b x^3-8 a^5+56 a b^4 x^{12}+7 b^5 x^{15}\right )}{56 x^7 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-7\,{b}^{5}{x}^{15}-56\,a{b}^{4}{x}^{12}-280\,{a}^{2}{b}^{3}{x}^{9}+560\,{a}^{3}{b}^{2}{x}^{6}+70\,{a}^{4}b{x}^{3}+8\,{a}^{5}}{56\,{x}^{7} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07002, size = 80, normalized size = 0.32 \begin{align*} \frac{7 \, b^{5} x^{15} + 56 \, a b^{4} x^{12} + 280 \, a^{2} b^{3} x^{9} - 560 \, a^{3} b^{2} x^{6} - 70 \, a^{4} b x^{3} - 8 \, a^{5}}{56 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77122, size = 132, normalized size = 0.53 \begin{align*} \frac{7 \, b^{5} x^{15} + 56 \, a b^{4} x^{12} + 280 \, a^{2} b^{3} x^{9} - 560 \, a^{3} b^{2} x^{6} - 70 \, a^{4} b x^{3} - 8 \, a^{5}}{56 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10424, size = 144, normalized size = 0.58 \begin{align*} \frac{1}{8} \, b^{5} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) + a b^{4} x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{280 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 35 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 4 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{28 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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