Optimal. Leaf size=158 \[ -\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 158, 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 = {1112, 270} \[ -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{x^6} \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (b^6+\frac {a^3 b^3}{x^6}+\frac {3 a^2 b^4}{x^4}+\frac {3 a b^5}{x^2}\right ) \, dx}{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}}{5 x^5 \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {b^3 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 0.37 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a^3+5 a^2 b x^2+15 a b^2 x^4-5 b^3 x^6\right )}{5 x^5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 37, normalized size = 0.23 \[ \frac {5 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} - 5 \, a^{2} b x^{2} - a^{3}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 66, normalized size = 0.42 \[ b^{3} x \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {15 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 5 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.35 \[ -\frac {\left (-5 b^{3} x^{6}+15 a \,b^{2} x^{4}+5 a^{2} b \,x^{2}+a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{5 \left (b \,x^{2}+a \right )^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 32, normalized size = 0.20 \[ b^{3} x - \frac {3 \, a b^{2}}{x} - \frac {a^{2} b}{x^{3}} - \frac {a^{3}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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