Optimal. Leaf size=120 \[ \frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]
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Rubi [A]
time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {871, 837, 821,
272, 65, 214} \begin {gather*} \frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-4 d e^2+3 e^3 x}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-8 d^3 e^4+3 d^2 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {\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 )}{d^4 e}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 112, normalized size = 0.93 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (3 d^3+7 d^2 e x-5 d e^2 x^2-8 e^3 x^3\right )}{x (-d+e x) (d+e x)^2}-6 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{3 d^5} \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. \(222\) vs.
\(2(106)=212\).
time = 0.08, size = 223, normalized size = 1.86
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{5} x}-\frac {17 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{12 d^{5} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{4 d^{5} \left (x -\frac {d}{e}\right )}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{4} \sqrt {d^{2}}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{6 e \,d^{4} \left (x +\frac {d}{e}\right )^{2}}\) | \(198\) |
default | \(\frac {e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{d^{2}}+\frac {-\frac {1}{d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 e^{2} x}{d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d}-\frac {e \left (\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\right )}{d^{2}}\) | \(223\) |
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 = 2.44, size = 173, normalized size = 1.44 \begin {gather*} -\frac {4 \, x^{4} e^{4} + 4 \, d x^{3} e^{3} - 4 \, d^{2} x^{2} e^{2} - 4 \, d^{3} x e + 3 \, {\left (x^{4} e^{4} + d x^{3} e^{3} - d^{2} x^{2} e^{2} - d^{3} x e\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (8 \, x^{3} e^{3} + 5 \, d x^{2} e^{2} - 7 \, d^{2} x e - 3 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d^{5} x^{4} e^{3} + d^{6} x^{3} e^{2} - d^{7} x^{2} e - d^{8} x\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 {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \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*} \text {could not integrate} \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 {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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