Optimal. Leaf size=167 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 167, normalized size of antiderivative = 1.00, 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 = {1111, 646, 43} \[ -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{17}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^9} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (\frac {a^3 b^3}{x^9}+\frac {3 a^2 b^4}{x^8}+\frac {3 a b^5}{x^7}+\frac {b^6}{x^6}\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 61, normalized size = 0.37 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (35 a^3+120 a^2 b x^2+140 a b^2 x^4+56 b^3 x^6\right )}{560 x^{16} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 37, normalized size = 0.22 \[ -\frac {56 \, b^{3} x^{6} + 140 \, a b^{2} x^{4} + 120 \, a^{2} b x^{2} + 35 \, a^{3}}{560 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 0.41 \[ -\frac {56 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 140 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 120 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 35 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{560 \, x^{16}} \]
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 \[ -\frac {\left (56 b^{3} x^{6}+140 a \,b^{2} x^{4}+120 a^{2} b \,x^{2}+35 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{560 \left (b \,x^{2}+a \right )^{3} x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 35, normalized size = 0.21 \[ -\frac {b^{3}}{10 \, x^{10}} - \frac {a b^{2}}{4 \, x^{12}} - \frac {3 \, a^{2} b}{14 \, x^{14}} - \frac {a^{3}}{16 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 151, normalized size = 0.90 \[ -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{16\,x^{16}\,\left (b\,x^2+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{10\,x^{10}\,\left (b\,x^2+a\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^{12}\,\left (b\,x^2+a\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )} \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{17}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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