Optimal. Leaf size=251 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{b^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.069104, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{b^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{16}} \, 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^{16}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^5}{x^6} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \left (\frac{a^5 b^5}{x^6}+\frac{5 a^4 b^6}{x^5}+\frac{10 a^3 b^7}{x^4}+\frac{10 a^2 b^8}{x^3}+\frac{5 a b^9}{x^2}+\frac{b^{10}}{x}\right ) \, dx,x,x^3\right )}{3 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}}{15 x^{15} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0269838, size = 85, normalized size = 0.34 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a \left (200 a^2 b^2 x^6+75 a^3 b x^3+12 a^4+300 a b^3 x^9+300 b^4 x^{12}\right )-180 b^5 x^{15} \log (x)\right )}{180 x^{15} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 82, normalized size = 0.3 \begin{align*}{\frac{180\,{b}^{5}\ln \left ( x \right ){x}^{15}-300\,a{b}^{4}{x}^{12}-300\,{a}^{2}{b}^{3}{x}^{9}-200\,{a}^{3}{b}^{2}{x}^{6}-75\,{a}^{4}b{x}^{3}-12\,{a}^{5}}{180\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{15}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6705, size = 150, normalized size = 0.6 \begin{align*} \frac{180 \, b^{5} x^{15} \log \left (x\right ) - 300 \, a b^{4} x^{12} - 300 \, a^{2} b^{3} x^{9} - 200 \, a^{3} b^{2} x^{6} - 75 \, a^{4} b x^{3} - 12 \, a^{5}}{180 \, x^{15}} \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^{16}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11427, size = 166, normalized size = 0.66 \begin{align*} b^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{137 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 300 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 300 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 200 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 75 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 12 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{180 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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