Optimal. Leaf size=99 \[ \frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \]
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Rubi [A] time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {665, 661, 208} \begin {gather*} \frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 665
Rubi steps
\begin {align*} \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+(2 c d) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+(4 c d e) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 98, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {1}{\sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d^2-e^2 x^2}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 127, normalized size = 1.28 \begin {gather*} \frac {2 \sqrt {2 c d (d+e x)-c (d+e x)^2}}{e \sqrt {d+e x}}+\frac {4 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c} (d+e x)-\sqrt {2 c d (d+e x)-c (d+e x)^2}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 237, normalized size = 2.39 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {c d} {\left (e x + d\right )} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{e^{2} x + d e}, -\frac {2 \, {\left (\sqrt {2} \sqrt {-c d} {\left (e x + d\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) - \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}\right )}}{e^{2} x + d e}\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 {-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 97, normalized size = 0.98 \begin {gather*} -\frac {2 \sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (\sqrt {2}\, c d \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )-\sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\right )}{\sqrt {e x +d}\, \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, e} \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 {-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\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 {c\,d^2-c\,e^2\,x^2}}{{\left (d+e\,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 {- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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