Optimal. Leaf size=186 \[ \frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8} \]
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Rubi [A]
time = 0.11, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {871, 837, 849,
821, 272, 65, 214} \begin {gather*} \frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}+\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 849
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-7 d e^2+6 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-35 d^3 e^4+24 d^2 e^5 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-105 d^5 e^6+48 d^4 e^7 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {\int \frac {-96 d^6 e^7+105 d^5 e^8 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^{12} e^6}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}+\frac {\left (7 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^7}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}+\frac {\left (7 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^7}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 \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^7}\\ &=\frac {7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d-24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 146, normalized size = 0.78 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (-15 d^6+15 d^5 e x+176 d^4 e^2 x^2-4 d^3 e^3 x^3-249 d^2 e^4 x^4-9 d e^5 x^5+96 e^6 x^6\right )}{x^2 (d-e x)^2 (d+e x)^3}+210 e^2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{30 d^8} \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. \(458\) vs.
\(2(162)=324\).
time = 0.11, size = 459, normalized size = 2.47
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{8} x^{2}}+\frac {673 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 d^{8} \left (x +\frac {d}{e}\right )}-\frac {29 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{48 d^{8} \left (x -\frac {d}{e}\right )}+\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{24 d^{7} \left (x -\frac {d}{e}\right )^{2}}-\frac {7 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{7} \sqrt {d^{2}}}+\frac {11 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{30 d^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{6} e \left (x +\frac {d}{e}\right )^{3}}\) | \(296\) |
default | \(-\frac {e^{2} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{3}}+\frac {-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 e^{2} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{2 d^{2}}}{d}-\frac {e \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{2}}+\frac {e^{2} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{3}}\) | \(459\) |
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 = 2.46, size = 265, normalized size = 1.42 \begin {gather*} \frac {116 \, x^{7} e^{7} + 116 \, d x^{6} e^{6} - 232 \, d^{2} x^{5} e^{5} - 232 \, d^{3} x^{4} e^{4} + 116 \, d^{4} x^{3} e^{3} + 116 \, d^{5} x^{2} e^{2} + 105 \, {\left (x^{7} e^{7} + d x^{6} e^{6} - 2 \, d^{2} x^{5} e^{5} - 2 \, d^{3} x^{4} e^{4} + d^{4} x^{3} e^{3} + d^{5} x^{2} e^{2}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (96 \, x^{6} e^{6} - 9 \, d x^{5} e^{5} - 249 \, d^{2} x^{4} e^{4} - 4 \, d^{3} x^{3} e^{3} + 176 \, d^{4} x^{2} e^{2} + 15 \, d^{5} x e - 15 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (d^{8} x^{7} e^{5} + d^{9} x^{6} e^{4} - 2 \, d^{10} x^{5} e^{3} - 2 \, d^{11} x^{4} e^{2} + d^{12} x^{3} e + d^{13} x^{2}\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^{3} \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^3\,{\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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