Optimal. Leaf size=114 \[ -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 849, 821,
272, 65, 214} \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 864
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx &=\int \frac {d-e x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {\int \frac {3 d^2 e-2 d e^2 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}+\frac {\int \frac {4 d^3 e^2-3 d^2 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 131, normalized size = 1.15 \begin {gather*} \frac {\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}-3 e^3 x^3 \log \left (d^3 \left (-d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\right )+3 e^3 x^3 \log \left (d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{6 d^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs.
\(2(98)=196\).
time = 0.08, size = 351, normalized size = 3.08
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{3} x^{3}}+\frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\) | \(88\) |
default | \(\frac {e^{3} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{4}}-\frac {e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{3} x^{3}}-\frac {e^{3} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{4}}\) | \(351\) |
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 = 4.27, size = 72, normalized size = 0.63 \begin {gather*} -\frac {3 \, x^{3} e^{3} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (4 \, x^{2} e^{2} - 3 \, d x e + 2 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6 \, d^{3} 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 {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs.
\(2 (93) = 186\).
time = 1.20, size = 239, normalized size = 2.10 \begin {gather*} -\frac {x^{3} {\left (\frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e}{x} - \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} - e^{3}\right )} e^{6}}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3}} + \frac {e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d^{3}} - \frac {\frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6} e}{x} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{\left (-1\right )}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{6} e^{\left (-3\right )}}{x^{3}}}{24 \, d^{9}} \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 {d^2-e^2\,x^2}}{x^4\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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