Optimal. Leaf size=102 \[ -\frac {4 \left (-3 a x+3 b+2 c^2 x\right ) \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{15 b x^2}-\frac {4 c \sqrt {a x^2-b x} \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{15 b x^2} \]
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Rubi [F] time = 0.34, 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 {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx &=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 7.07, size = 102, normalized size = 1.00 \begin {gather*} -\frac {4 \left (3 b-3 a x+2 c^2 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2}-\frac {4 c \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 60, normalized size = 0.59 \begin {gather*} -\frac {4 \, \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x} {\left ({\left (2 \, c^{2} - 3 \, a\right )} x + \sqrt {a x^{2} - b x} c + 3 \, b\right )}}{15 \, b 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 {c x^{2} - \sqrt {a x^{2} - b x} 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 \[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3}}\, dx\]
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 {c x^{2} - \sqrt {a x^{2} - b x} 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 {\sqrt {c\,x^2-x\,\sqrt {a\,x^2-b\,x}}}{x^3} \,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 (c x - \sqrt {a x^{2} - b x}\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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