Optimal. Leaf size=108 \[ -\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac {5 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d} \]
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Rubi [A] time = 0.15, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {852, 1807, 807, 266, 47, 63, 208} \[ -\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{8 x^2}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {5 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^5} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\int \frac {\left (8 d^3 e-5 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{4 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac {1}{4} \left (5 e^2\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac {1}{8} \left (5 e^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}-\frac {1}{16} \left (5 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac {1}{8} \left (5 e^2\right ) \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 )\\ &=-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^3}+\frac {5 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 95, normalized size = 0.88 \[ -\frac {-15 e^4 x^4 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (6 d^3-16 d^2 e x+9 d e^2 x^2+16 e^3 x^3\right )+15 e^4 x^4 \log (x)}{24 d x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 86, normalized size = 0.80 \[ -\frac {15 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{3} x^{3} + 9 \, d e^{2} x^{2} - 16 \, d^{2} e x + 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, d x^{4}} \]
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 = 513, normalized size = 4.75 \[ -\frac {5 e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{2 \sqrt {e^{2}}\, d}+\frac {5 e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, d}+\frac {5 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}-\frac {5 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{5} x}{2 d^{3}}+\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5} x}{2 d^{3}}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4}}{8 d^{2}}-\frac {5 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{5} x}{3 d^{5}}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5} x}{3 d^{5}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}{24 d^{4}}+\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5} x}{3 d^{7}}-\frac {4 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{4}}{3 d^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}{8 d^{6}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{2}}{3 \left (x +\frac {d}{e}\right )^{2} d^{6}}+\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{3 d^{7} x}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{8 d^{6} x^{2}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{3 d^{5} x^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 130, normalized size = 1.20 \[ \frac {5 \, e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{8 \, d^{2}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d^{2} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{3 \, d x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{4 \, x^{4}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^5\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.36, size = 422, normalized size = 3.91 \[ d^{2} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {d^{2}}{2 e x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e}{2 x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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