Optimal. Leaf size=161 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {3 a b^2 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 161, 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} \[ -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {3 a b^2 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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^4} \, 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^4} \, 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 b^5+\frac {a^3 b^3}{x^4}+\frac {3 a^2 b^4}{x^2}+b^6 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}}{3 x^3 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {3 a b^2 x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 0.37 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a^3+9 a^2 b x^2-9 a b^2 x^4-b^3 x^6\right )}{3 x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 36, normalized size = 0.22 \[ \frac {b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 67, normalized size = 0.42 \[ \frac {1}{3} \, b^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, a b^{2} x \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {9 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{3 \, x^{3}} \]
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 \[ -\frac {\left (-b^{3} x^{6}-9 a \,b^{2} x^{4}+9 a^{2} b \,x^{2}+a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{3 \left (b \,x^{2}+a \right )^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 33, normalized size = 0.20 \[ \frac {1}{3} \, b^{3} x^{3} + 3 \, a b^{2} x - \frac {3 \, a^{2} b}{x} - \frac {a^{3}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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