Optimal. Leaf size=119 \[ \frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
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Rubi [A]
time = 0.07, antiderivative size = 119, 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 = {871, 837, 12,
272, 65, 214} \begin {gather*} \frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-5 d e^2+4 e^3 x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^3 e^4+8 d^2 e^5 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^5 e^6}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^5}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}-\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^5 e^2}\\ &=\frac {5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-8 e x}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 118, normalized size = 0.99 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (23 d^4+8 d^3 e x-27 d^2 e^2 x^2-7 d e^3 x^3+8 e^4 x^4\right )}{(d-e x)^2 (d+e x)^3}+30 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{15 d^6} \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. \(254\) vs.
\(2(105)=210\).
time = 0.06, size = 255, normalized size = 2.14
method | result | size |
default | \(-\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{d}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d}\) | \(255\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs.
\(2 (104) = 208\).
time = 1.94, size = 224, normalized size = 1.88 \begin {gather*} \frac {23 \, x^{5} e^{5} + 23 \, d x^{4} e^{4} - 46 \, d^{2} x^{3} e^{3} - 46 \, d^{3} x^{2} e^{2} + 23 \, d^{4} x e + 23 \, d^{5} + 15 \, {\left (x^{5} e^{5} + d x^{4} e^{4} - 2 \, d^{2} x^{3} e^{3} - 2 \, d^{3} x^{2} e^{2} + d^{4} x e + d^{5}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (8 \, x^{4} e^{4} - 7 \, d x^{3} e^{3} - 27 \, d^{2} x^{2} e^{2} + 8 \, d^{3} x e + 23 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{6} x^{5} e^{5} + d^{7} x^{4} e^{4} - 2 \, d^{8} x^{3} e^{3} - 2 \, d^{9} x^{2} e^{2} + d^{10} x e + d^{11}\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 \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{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\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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