Optimal. Leaf size=158 \[ \frac {3 a^2 b x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {a b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
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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} \begin {gather*} \frac {b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {a b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {3 a^2 b 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}}{x \left (a+b x^2\right )} \end {gather*}
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^2} \, 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^2} \, 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 (3 a^2 b^4+\frac {a^3 b^3}{x^2}+3 a b^5 x^2+b^6 x^4\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}}{x \left (a+b x^2\right )}+\frac {3 a^2 b x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {a b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.38 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (-5 a^3+15 a^2 b x^2+5 a b^2 x^4+b^3 x^6\right )}{5 x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.71, size = 60, normalized size = 0.38 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (-5 a^3+15 a^2 b x^2+5 a b^2 x^4+b^3 x^6\right )}{5 x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 36, normalized size = 0.23 \begin {gather*} \frac {b^{3} x^{6} + 5 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - 5 \, a^{3}}{5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 64, normalized size = 0.41 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + a b^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a^{2} b x \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.37 \begin {gather*} -\frac {\left (-b^{3} x^{6}-5 a \,b^{2} x^{4}-15 a^{2} b \,x^{2}+5 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{5 \left (b \,x^{2}+a \right )^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 32, normalized size = 0.20 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + a b^{2} x^{3} + 3 \, a^{2} b x - \frac {a^{3}}{x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x^2} \,d x \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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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