Optimal. Leaf size=80 \[ -\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1807, 807, 266, 63, 208} \[ -\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {\int \frac {-4 d^3 e-3 d^2 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{2} \left (3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{4} \left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3}{2} \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 {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 122, normalized size = 1.52 \[ \frac {e \left (-\frac {4 d \sqrt {d^2-e^2 x^2}}{x}-2 d e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-e \sqrt {d^2-e^2 x^2} \left (\frac {d^2}{e^2 x^2}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 63, normalized size = 0.79 \[ \frac {3 \, e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \sqrt {-e^{2} x^{2} + d^{2}} {\left (4 \, e x + d\right )}}{2 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 170, normalized size = 2.12 \[ -\frac {3 \, e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d} + \frac {x^{2} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d} - \frac {{\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{8}}{x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 86, normalized size = 1.08 \[ -\frac {3 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}-\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}\, e}{d x}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 83, normalized size = 1.04 \[ -\frac {3 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} e}{d x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x^{2}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2}{x^3\,\sqrt {d^2-e^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.72, size = 214, normalized size = 2.68 \[ d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {d}{e x} \right )}}{d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {d}{e x} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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