Optimal. Leaf size=41 \[ -\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 660, 37}
\begin {gather*} -\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 660
Rule 1125
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {\left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 a x^8}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a^3+4 a^2 b x^2+6 a b^2 x^4+4 b^3 x^6\right )}{8 x^8 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 56, normalized size = 1.37
method | result | size |
gosper | \(-\frac {\left (4 b^{3} x^{6}+6 a \,b^{2} x^{4}+4 a^{2} b \,x^{2}+a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{8 x^{8} \left (b \,x^{2}+a \right )^{3}}\) | \(56\) |
default | \(-\frac {\left (4 b^{3} x^{6}+6 a \,b^{2} x^{4}+4 a^{2} b \,x^{2}+a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{8 x^{8} \left (b \,x^{2}+a \right )^{3}}\) | \(56\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {1}{2} b^{3} x^{6}-\frac {3}{4} a \,b^{2} x^{4}-\frac {1}{2} a^{2} b \,x^{2}-\frac {1}{8} a^{3}\right )}{\left (b \,x^{2}+a \right ) x^{8}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 35, normalized size = 0.85 \begin {gather*} -\frac {b^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2}}{4 \, x^{4}} - \frac {a^{2} b}{2 \, x^{6}} - \frac {a^{3}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 35, normalized size = 0.85 \begin {gather*} -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \end {gather*}
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 \begin {gather*} \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (28) = 56\).
time = 4.01, size = 68, normalized size = 1.66 \begin {gather*} -\frac {4 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.24, size = 151, normalized size = 3.68 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^8\,\left (b\,x^2+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^2\,\left (b\,x^2+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^4\,\left (b\,x^2+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^6\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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