Optimal. Leaf size=73 \[ \frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {811, 655, 223,
209, 651} \begin {gather*} -\frac {d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}+\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 651
Rule 655
Rule 811
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}+\frac {d^2 \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 79, normalized size = 1.08 \begin {gather*} \frac {(-2 d+e x) \sqrt {d^2-e^2 x^2}}{e^3 (-d+e x)}-\frac {d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\left (-e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 103, normalized size = 1.41
method | result | size |
risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{3}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}-\frac {d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{e^{4} \left (x -\frac {d}{e}\right )}\) | \(99\) |
default | \(e \left (-\frac {x^{2}}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 d^{2}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )+d \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 72, normalized size = 0.99 \begin {gather*} -d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} - \frac {x^{2} e^{\left (-1\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} + \frac {d x e^{\left (-2\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} + \frac {2 \, d^{2} e^{\left (-3\right )}}{\sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.34, size = 85, normalized size = 1.16 \begin {gather*} \frac {2 \, d x e - 2 \, d^{2} + 2 \, {\left (d x e - d^{2}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e - 2 \, d\right )}}{x e^{4} - d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.06, size = 163, normalized size = 2.23 \begin {gather*} d \left (\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{e^{3}} - \frac {i x}{d e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {e x}{d} \right )}}{e^{3}} + \frac {x}{d e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \tilde {\infty } x^{4} & \text {for}\: d = 0 \wedge e = 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {3}{2}}} & \text {for}\: e = 0 \\\tilde {\infty } x^{4} & \text {for}\: d = - \sqrt {e^{2} x^{2}} \vee d = \sqrt {e^{2} x^{2}} \\\frac {2 d^{2}}{e^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {x^{2}}{e^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 68, normalized size = 0.93 \begin {gather*} -d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \sqrt {-x^{2} e^{2} + d^{2}} e^{\left (-3\right )} + \frac {2 \, d e^{\left (-3\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.96, size = 87, normalized size = 1.19 \begin {gather*} \frac {2\,d^2-e^2\,x^2}{e^3\,\sqrt {d^2-e^2\,x^2}}+\frac {d\,\ln \left (x\,\sqrt {-e^2}+\sqrt {d^2-e^2\,x^2}\right )}{{\left (-e^2\right )}^{3/2}}+\frac {d\,x}{e^2\,\sqrt {d^2-e^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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