Optimal. Leaf size=233 \[ \frac {\left (384 a^2 x^4-136 a x^2-255\right ) \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{1920 a^2}+\frac {\left (-384 a^2 x^5+568 a x^3+85 x\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{1920 a b}+\frac {17 \log \left (b \left (-\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}\right )+\sqrt {2} \sqrt {a} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-a x\right )}{128 \sqrt {2} a^{3/2} b} \]
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Rubi [F] time = 2.10, 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 {x^3 \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 \frac {x^3 \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 \frac {x^3 \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 [C] time = 12.15, size = 733, normalized size = 3.15 \begin {gather*} \frac {b x^2 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (\frac {2 x^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (-\sqrt {2} a^5 \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \, _2F_1\left (-\frac {9}{2},-\frac {3}{2};-\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-3 \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^6\right )}{b}-\frac {10 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^2 \left (-\sqrt {2} a^3 \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-\left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^4\right )}{a b}-\frac {2 \sqrt {2} a^5 x^3 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \left (15 \left (8 a^2 x^4+8 a x^2 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-1\right )-4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+1\right ) \, _2F_1\left (-\frac {7}{2},-\frac {1}{2};\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )+3 \, _2F_1\left (-\frac {7}{2},-\frac {5}{2};-\frac {3}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-10 \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right ) \, _2F_1\left (-\frac {7}{2},-\frac {3}{2};-\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )\right )}{b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1}\right )}{60 a \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.51, size = 233, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \left (-255-136 a x^2+384 a^2 x^4\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a^2}+\frac {\left (85 x+568 a x^3-384 a^2 x^5\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a b}+\frac {17 \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{128 \sqrt {2} a^{3/2} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 40.47, size = 359, normalized size = 1.54 \begin {gather*} \left [\frac {255 \, \sqrt {2} \sqrt {a} \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \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}}}} + a\right ) - 4 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, 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}}}}}{7680 \, a^{2} b}, \frac {255 \, \sqrt {2} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, 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}}}}}{3840 \, a^{2} b}\right ] \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 = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]
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 \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x^{3}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}}\,{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 = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{\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 {x^{3} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{\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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