Optimal. Leaf size=98 \[ -\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e} \]
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Rubi [A]
time = 0.07, antiderivative size = 98, 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 = {677, 675, 214}
\begin {gather*} \frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e}-\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 677
Rubi steps
\begin {align*} \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx &=-\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-\frac {1}{2} c \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}-(c e) \text {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 {\sqrt {c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {2} \sqrt {d} e}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 101, normalized size = 1.03 \begin {gather*} \frac {\sqrt {c \left (d^2-e^2 x^2\right )} \left (-\frac {2}{(d+e x)^{3/2}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d} \sqrt {d^2-e^2 x^2}}\right )}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 117, normalized size = 1.19
method | result | size |
default | \(\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (\sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c e x +c d \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right )-2 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\right )}{2 \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\) | \(117\) |
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 = 2.93, size = 289, normalized size = 2.95 \begin {gather*} \left [\frac {\sqrt {\frac {1}{2}} {\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c x^{2} e^{2} - 2 \, c d x e - 3 \, c d^{2} - 4 \, \sqrt {\frac {1}{2}} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d} d \sqrt {\frac {c}{d}}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}}, \frac {\sqrt {\frac {1}{2}} {\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d} d \sqrt {-\frac {c}{d}}}{c x^{2} e^{2} - c d^{2}}\right ) - \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e}\right ] \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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 70, normalized size = 0.71 \begin {gather*} -\frac {1}{2} \, {\left (\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{x e + d}\right )} e^{\left (-1\right )} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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