Optimal. Leaf size=164 \[ \frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 164, 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 = {1112, 266, 43} \begin {gather*} \frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, 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^3} \, dx}{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 \frac {\left (a b+b^2 x\right )^3}{x^2} \, 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 (3 a b^5+\frac {a^3 b^3}{x^2}+\frac {3 a^2 b^4}{x}+b^6 x\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}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.38 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (-2 a^3+12 a^2 b x^2 \log (x)+6 a b^2 x^4+b^3 x^6\right )}{4 x^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 320, normalized size = 1.95 \begin {gather*} -\frac {3}{4} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x^2+b^2 x^4}-a-\sqrt {b^2} x^2\right )-\frac {3}{4} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x^2+b^2 x^4}+a-\sqrt {b^2} x^2\right )+\frac {3}{2} a^2 b \tanh ^{-1}\left (\frac {\sqrt {b^2} x^2}{a}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{a}\right )+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-8 a^3 b-21 a^2 b^2 x^2+24 a b^3 x^4+4 b^4 x^6\right )+\sqrt {b^2} \left (8 a^4+29 a^3 b x^2-3 a^2 b^2 x^4-28 a b^3 x^6-4 b^4 x^8\right )}{16 x^2 \left (a b+b^2 x^2\right )-16 \sqrt {b^2} x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 38, normalized size = 0.23 \begin {gather*} \frac {b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \relax (x) - 2 \, a^{3}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 87, normalized size = 0.53 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a b^{2} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {3 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.36 \begin {gather*} \frac {\left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{6}+6 a \,b^{2} x^{4}+12 a^{2} b \,x^{2} \ln \relax (x )-2 a^{3}\right )}{4 \left (b \,x^{2}+a \right )^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 34, normalized size = 0.21 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b \log \relax (x) - \frac {a^{3}}{2 \, x^{2}} \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^3} \,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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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