Optimal. Leaf size=167 \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0401521, antiderivative size = 167, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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 )^{3/2}}{x^{17}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^3}{x^{17}} \, dx}{b^2 \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^3 b^3}{x^{17}}+\frac{3 a^2 b^4}{x^{14}}+\frac{3 a b^5}{x^{11}}+\frac{b^6}{x^8}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.01461, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (1680 a^2 b x^3+455 a^3+2184 a b^2 x^6+1040 b^3 x^9\right )}{7280 x^{16} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 58, normalized size = 0.4 \begin{align*} -{\frac{1040\,{b}^{3}{x}^{9}+2184\,a{b}^{2}{x}^{6}+1680\,{a}^{2}b{x}^{3}+455\,{a}^{3}}{7280\,{x}^{16} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02071, size = 50, normalized size = 0.3 \begin{align*} -\frac{1040 \, b^{3} x^{9} + 2184 \, a b^{2} x^{6} + 1680 \, a^{2} b x^{3} + 455 \, a^{3}}{7280 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76863, size = 99, normalized size = 0.59 \begin{align*} -\frac{1040 \, b^{3} x^{9} + 2184 \, a b^{2} x^{6} + 1680 \, a^{2} b x^{3} + 455 \, a^{3}}{7280 \, x^{16}} \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{3}{2}}}{x^{17}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09777, size = 93, normalized size = 0.56 \begin{align*} -\frac{1040 \, b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 2184 \, a b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 1680 \, a^{2} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 455 \, a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{7280 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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