Optimal. Leaf size=123 \[ \frac {4 \left (a x^2-2\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{15 x^2}-\frac {2 b \left (2 a x^2-3\right ) \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{15 a x^3} \]
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Rubi [F] time = 0.41, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx &=\int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\\ \end {align*}
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Mathematica [A] time = 4.69, size = 98, normalized size = 0.80 \begin {gather*} -\frac {2 \left (a x^2-1\right ) \left (5 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+5 a x^2-3\right )}{15 x^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.98, size = 123, normalized size = 1.00 \begin {gather*} \frac {4 \left (-2+a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 x^2}-\frac {2 b \left (-3+2 a x^2\right ) \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 a x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 79, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (2 \, a^{2} x^{3} - 4 \, a x - {\left (2 \, a b x^{2} - 3 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{15 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}}\,{d 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 {1}{x^3\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^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 {1}{x^{3} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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