Optimal. Leaf size=117 \[ -\frac {4 \left (3 b c+32 a c x-8 c^3 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^2}+\frac {4 \left (15 b+20 a x+4 c^2 x\right ) \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^3} \]
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Rubi [F]
time = 2.55, 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 \sqrt {-b x+a x^2}} \, 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 \sqrt {-b x+a x^2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x}\right ) \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^{7/2} \sqrt {-b+a x}} \, dx}{\sqrt {-b x+a x^2}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {c x^4-x^2 \sqrt {-b x^2+a x^4}}}{x^6 \sqrt {-b+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^2}}\\ \end {align*}
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Mathematica [F]
time = 14.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3} \sqrt {a \,x^{2}-b x}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 78, normalized size = 0.67 \begin {gather*} -\frac {4 \, {\left (3 \, b c x - 8 \, {\left (c^{3} - 4 \, a c\right )} x^{2} - \sqrt {a x^{2} - b x} {\left (4 \, {\left (c^{2} + 5 \, a\right )} x + 15 \, b\right )}\right )} \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{105 \, b^{2} x^{3}} \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} \sqrt {x \left (a x - b\right )}}\, dx \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*} \text {could not integrate} \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\,\sqrt {a\,x^2-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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