Optimal. Leaf size=225 \[ \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (96 a^4 x^2-104 a^2 b x-15 b^2\right )}{120 a^2 b^2 x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {\sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{8 \sqrt {2} a^{5/2} x}+\frac {-96 a^4 x^2+152 a^2 b x+5 b^2}{120 a b^2}\right ) \]
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Rubi [F] time = 4.40, 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^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {-b x+a^2 x^2} \int \frac {x^{5/2} \sqrt {-b+a^2 x}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx}{\sqrt {x} \sqrt {-b+a^2 x}}\\ &=\frac {\left (2 \sqrt {-b x+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {-b+a^2 x^2}}{\left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a^2 x}}\\ \end {align*}
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Mathematica [F] time = 1.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 7.04, size = 225, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (96 a^4 x^2-104 a^2 b x-15 b^2\right )}{120 a^2 b^2 x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {\sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{8 \sqrt {2} a^{5/2} x}+\frac {-96 a^4 x^2+152 a^2 b x+5 b^2}{120 a b^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 367, normalized size = 1.63 \begin {gather*} \left [\frac {15 \, \sqrt {2} \sqrt {a} b^{3} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (96 \, a^{6} x^{3} - 152 \, a^{4} b x^{2} - 5 \, a^{2} b^{2} x - {\left (96 \, a^{5} x^{2} - 104 \, a^{3} b x - 15 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{480 \, a^{3} b^{2} x}, \frac {15 \, \sqrt {2} \sqrt {-a} b^{3} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (96 \, a^{6} x^{3} - 152 \, a^{4} b x^{2} - 5 \, a^{2} b^{2} x - {\left (96 \, a^{5} x^{2} - 104 \, a^{3} b x - 15 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{240 \, a^{3} b^{2} x}\right ] \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 {\sqrt {a^{2} x^{2} - b x} x^{2}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {a^{2} x^{2}-b x}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 {a^{2} x^{2} - b x} x^{2}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{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 = -0.00 \begin {gather*} \int \frac {x^2\,\sqrt {a^2\,x^2-b\,x}}{{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,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^{2} \sqrt {x \left (a^{2} x - b\right )}}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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