Optimal. Leaf size=108 \[ \frac {4 \left (3 a^2 x+b\right ) \sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{3 b^2 x^2}-\frac {4 \left (3 a^3 x+5 a b\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{3 b^2 x} \]
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Rubi [F] time = 3.59, 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 {\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 {\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 {\sqrt {x} \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^2 \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 = 1.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 = 4.92, size = 108, normalized size = 1.00 \begin {gather*} -\frac {4 \left (5 a b+3 a^3 x\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x}+\frac {4 \left (b+3 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 70, normalized size = 0.65 \begin {gather*} -\frac {4 \, {\left (3 \, a^{3} x^{2} + 5 \, a b x - \sqrt {a^{2} x^{2} - b x} {\left (3 \, a^{2} x + b\right )}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{3 \, b^{2} x^{2}} \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}}{{\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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b x}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]
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 {a^{2} x^{2} - b x}}{{\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 = -0.01 \begin {gather*} \int \frac {\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 {\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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