Optimal. Leaf size=167 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 167, normalized size of antiderivative = 1.00, 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 270
Rule 1355
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*}
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Mathematica [A] time = 0.01, size = 61, normalized size = 0.37 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (455 a^3+1680 a^2 b x^3+2184 a b^2 x^6+1040 b^3 x^9\right )}{7280 x^{16} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 37, normalized size = 0.22 \[ -\frac {1040 \, b^{3} x^{9} + 2184 \, a b^{2} x^{6} + 1680 \, a^{2} b x^{3} + 455 \, a^{3}}{7280 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 69, normalized size = 0.41 \[ -\frac {1040 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 2184 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 1680 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 455 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{7280 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.35 \[ -\frac {\left (1040 b^{3} x^{9}+2184 a \,b^{2} x^{6}+1680 a^{2} b \,x^{3}+455 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}}}{7280 \left (b \,x^{3}+a \right )^{3} x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 37, normalized size = 0.22 \[ -\frac {1040 \, b^{3} x^{9} + 2184 \, a b^{2} x^{6} + 1680 \, a^{2} b x^{3} + 455 \, a^{3}}{7280 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 151, normalized size = 0.90 \[ -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{16\,x^{16}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{10\,x^{10}\,\left (b\,x^3+a\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{13\,x^{13}\,\left (b\,x^3+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{17}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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