Optimal. Leaf size=247 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0601738, antiderivative size = 247, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
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^{12}} \, 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^{12}} \, 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 (5 a b^9+\frac{a^5 b^5}{x^{12}}+\frac{5 a^4 b^6}{x^9}+\frac{10 a^3 b^7}{x^6}+\frac{10 a^2 b^8}{x^3}+b^{10} x^3\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}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0160328, size = 83, normalized size = 0.34 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (440 a^2 b^3 x^9+176 a^3 b^2 x^6+55 a^4 b x^3+8 a^5-440 a b^4 x^{12}-22 b^5 x^{15}\right )}{88 x^{11} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-22\,{b}^{5}{x}^{15}-440\,a{b}^{4}{x}^{12}+440\,{a}^{2}{b}^{3}{x}^{9}+176\,{a}^{3}{b}^{2}{x}^{6}+55\,{a}^{4}b{x}^{3}+8\,{a}^{5}}{88\,{x}^{11} \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 = 1.01425, size = 80, normalized size = 0.32 \begin{align*} \frac{22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7216, size = 136, normalized size = 0.55 \begin{align*} \frac{22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \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^{12}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14235, size = 143, normalized size = 0.58 \begin{align*} \frac{1}{4} \, b^{5} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a b^{4} x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{440 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 176 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 55 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 8 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{88 \, x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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