Optimal. Leaf size=163 \[ \frac {5}{12} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (-2 a x^2-9\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{24 b}+\frac {3 \tanh ^{-1}\left (\sqrt {2} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{8 \sqrt {2} b} \]
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Rubi [F] time = 0.43, 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 \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}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \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}}} \, dx &=\int \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}}} \, dx\\ \end {align*}
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Mathematica [B] time = 4.60, size = 714, normalized size = 4.38 \begin {gather*} \frac {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^6 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right )^2 \left (12 \sqrt {a} x \left (4 a^2 x^3+a x \left (4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-3\right )-b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt {a}}\right )+\sqrt {2} x \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (32 a^3 x^5+16 a^2 x^3 \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-5\right )-a x \left (64 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-37\right )+9 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right )-3 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}\right )\right )}{24 \sqrt {2} a^2 b^2 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (4096 a^7 x^{13}+1024 a^6 x^{11} \left (4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-256 a^5 x^9 \left (44 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-65\right )+768 a^4 x^7 \left (15 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-224 a^3 x^5 \left (24 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )+28 a^2 x^3 \left (40 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-a x \left (84 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )+b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.27, size = 223, normalized size = 1.37 \begin {gather*} \frac {\left (-9-2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{24 b}+\frac {5}{12} x \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}}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{16 \sqrt {2} b}-\frac {3 \log \left (-b+\sqrt {2} b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{16 \sqrt {2} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 21.29, size = 168, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (2 \, a x^{2} - 10 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 9\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} - 9 \, \sqrt {2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right )}{96 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \[\int \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]
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 \sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}\,\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 \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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