Optimal. Leaf size=164 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )} \]
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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 {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 \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^5} \, 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^5} \, 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^3} \, 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 (b^6+\frac {a^3 b^3}{x^3}+\frac {3 a^2 b^4}{x^2}+\frac {3 a b^5}{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}}{4 x^4 \left (a+b x^2\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {b^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 \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 = 61, normalized size = 0.37 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (a^3+6 a^2 b x^2-12 a b^2 x^4 \log (x)-2 b^3 x^6\right )}{4 x^4 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.55, size = 1170, normalized size = 7.13 \begin {gather*} -\frac {3 b^2 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a^5}{4 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}-\frac {3 b \sqrt {b^2} \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a^5}{4 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}+\frac {3 b^2 \left (\sqrt {b^2 x^4+2 a b x^2+a^2}-\sqrt {b^2} x^2\right )^2 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a^3}{2 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}+\frac {3 b \sqrt {b^2} \left (\sqrt {b^2 x^4+2 a b x^2+a^2}-\sqrt {b^2} x^2\right )^2 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a^3}{2 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}+\frac {3}{4} b^2 \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a-\frac {3}{4} b \sqrt {b^2} \log \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a-\frac {3 b^2 \left (\sqrt {b^2 x^4+2 a b x^2+a^2}-\sqrt {b^2} x^2\right )^4 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a}{4 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}-\frac {3 b \sqrt {b^2} \left (\sqrt {b^2 x^4+2 a b x^2+a^2}-\sqrt {b^2} x^2\right )^4 \log \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right ) a}{4 \left (-\sqrt {b^2} x^2-a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2 \left (-\sqrt {b^2} x^2+a+\sqrt {b^2 x^4+2 a b x^2+a^2}\right )^2}+\frac {\sqrt {b^2 x^4+2 a b x^2+a^2} \left (2 b^4 x^6+a b^3 x^4-6 a^2 b^2 x^2-a^3 b\right )+\sqrt {b^2} \left (-2 b^4 x^8-3 a b^3 x^6+5 a^2 b^2 x^4+7 a^3 b x^2+a^4\right )}{4 x^4 \left (b^2 x^2+a b\right )-4 \sqrt {b^2} x^4 \sqrt {b^2 x^4+2 a b x^2+a^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.48, size = 39, normalized size = 0.24 \begin {gather*} \frac {2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \relax (x) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 0.53 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a b^{2} \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {9 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.37 \begin {gather*} \frac {\left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} \left (2 b^{3} x^{6}+12 a \,b^{2} x^{4} \ln \relax (x )-6 a^{2} b \,x^{2}-a^{3}\right )}{4 \left (b \,x^{2}+a \right )^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 34, normalized size = 0.21 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} + 3 \, a b^{2} \log \relax (x) - \frac {3 \, a^{2} b}{2 \, x^{2}} - \frac {a^{3}}{4 \, x^{4}} \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^5} \,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^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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