Optimal. Leaf size=255 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0580603, antiderivative size = 255, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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^{23}} \, 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^{23}} \, 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^{23}}+\frac{5 a^4 b^6}{x^{20}}+\frac{10 a^3 b^7}{x^{17}}+\frac{10 a^2 b^8}{x^{14}}+\frac{5 a b^9}{x^{11}}+\frac{b^{10}}{x^8}\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}}{22 x^{22} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{b^5 \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 = 0.0171454, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (117040 a^2 b^3 x^9+95095 a^3 b^2 x^6+40040 a^4 b x^3+6916 a^5+76076 a b^4 x^{12}+21736 b^5 x^{15}\right )}{152152 x^{22} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 80, normalized size = 0.3 \begin{align*} -{\frac{21736\,{b}^{5}{x}^{15}+76076\,a{b}^{4}{x}^{12}+117040\,{a}^{2}{b}^{3}{x}^{9}+95095\,{a}^{3}{b}^{2}{x}^{6}+40040\,{a}^{4}b{x}^{3}+6916\,{a}^{5}}{152152\,{x}^{22} \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 = 1.00452, size = 80, normalized size = 0.31 \begin{align*} -\frac{21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65463, size = 165, normalized size = 0.65 \begin{align*} -\frac{21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \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^{23}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12912, size = 144, normalized size = 0.56 \begin{align*} -\frac{21736 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 76076 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 117040 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 95095 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 40040 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 6916 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{152152 \, x^{22}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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