Optimal. Leaf size=89 \[ -\frac {2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]
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Rubi [A] time = 0.23, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1574, 933, 168, 538, 537} \[ -\frac {2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 537
Rule 538
Rule 933
Rule 1574
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx &=\frac {\sqrt {-\frac {1}{c^2}+x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{\sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{\sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {\left (2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 188, normalized size = 2.11 \[ -\frac {2 i (d+e x) \sqrt {\frac {e (c x-1)}{c (d+e x)}} \sqrt {\frac {c e x+e}{c d+c e x}} \left (F\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {c d+e}{c}}}{\sqrt {d+e x}}\right )|\frac {c d-e}{c d+e}\right )-\Pi \left (\frac {c d}{c d+e};i \sinh ^{-1}\left (\frac {\sqrt {-\frac {c d+e}{c}}}{\sqrt {d+e x}}\right )|\frac {c d-e}{c d+e}\right )\right )}{d x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {-\frac {c d+e}{c}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\sqrt {e x + d} x^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 148, normalized size = 1.66 \[ -\frac {2 \left (c d -e \right ) \sqrt {-\frac {\left (c x +1\right ) e}{c d -e}}\, \sqrt {-\frac {\left (c x -1\right ) e}{c d +e}}\, \sqrt {\frac {\left (e x +d \right ) c}{c d -e}}\, \EllipticPi \left (\sqrt {\frac {\left (e x +d \right ) c}{c d -e}}, \frac {c d -e}{c d}, \sqrt {\frac {c d -e}{c d +e}}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \sqrt {e x +d}\, c d x} \]
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 \[ \int \frac {1}{\sqrt {e x + d} x^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {1-\frac {1}{c^2\,x^2}}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {- \left (-1 + \frac {1}{c x}\right ) \left (1 + \frac {1}{c x}\right )} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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