Optimal. Leaf size=140 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2}+\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2} \]
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Rubi [A] time = 0.18, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {852, 1807, 835, 807, 266, 47, 63, 208} \[ \frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}-\frac {e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^6} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}-\frac {\int \frac {\left (10 d^3 e-7 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{5 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}+\frac {\int \frac {\left (28 d^4 e^2-10 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{20 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{2 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{4 d}\\ &=\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}+\frac {e^5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{8 d}\\ &=\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \operatorname {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 )}{4 d}\\ &=\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 106, normalized size = 0.76 \[ \frac {-15 e^5 x^5 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-12 d^4+30 d^3 e x-16 d^2 e^2 x^2-15 d e^3 x^3+28 e^4 x^4\right )+15 e^5 x^5 \log (x)}{60 d^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 97, normalized size = 0.69 \[ \frac {15 \, e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (28 \, e^{4} x^{4} - 15 \, d e^{3} x^{3} - 16 \, d^{2} e^{2} x^{2} + 30 \, d^{3} e x - 12 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{60 \, d^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 541, normalized size = 3.86 \[ -\frac {e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{4 \sqrt {d^{2}}\, d}+\frac {23 e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{2}}-\frac {23 e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{2}}+\frac {23 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{6} x}{8 d^{4}}-\frac {23 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x}{8 d^{4}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}{4 d^{3}}+\frac {23 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{6} x}{12 d^{6}}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6} x}{12 d^{6}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}{12 d^{5}}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6} x}{15 d^{8}}+\frac {23 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{5}}{15 d^{7}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}{20 d^{7}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{3}}{3 \left (x +\frac {d}{e}\right )^{2} d^{7}}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{15 d^{8} x}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{4 d^{7} x^{2}}-\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{15 d^{6} x^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{2 d^{5} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 155, normalized size = 1.11 \[ -\frac {e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{4 \, d^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{5}}{4 \, d^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{2}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{15 \, d^{2} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{2 \, d x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{5 \, x^{5}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.42, size = 660, normalized size = 4.71 \[ d^{2} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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