Optimal. Leaf size=107 \[ -\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
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Rubi [A] time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1807, 835, 807, 266, 63, 208} \[ -\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {\int \frac {-6 d^3 e-5 d^2 e^2 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}+\frac {\int \frac {10 d^4 e^2+6 d^3 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {e^3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {e^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e \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 )}{d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 87, normalized size = 0.81 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {d \left (d^2+3 d e x+5 e^2 x^2\right )}{x^3}-\frac {3 e^3 \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 74, normalized size = 0.69 \[ \frac {3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (5 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, d^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 239, normalized size = 2.23 \[ \frac {x^{3} {\left (\frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2}} - \frac {e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{2}} - \frac {{\left (\frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{16}}{x} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{14}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 114, normalized size = 1.07 \[ -\frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2}}{3 d^{2} x}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e}{d \,x^{2}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 108, normalized size = 1.01 \[ -\frac {e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{3 \, d^{2} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e}{d x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2}{x^4\,\sqrt {d^2-e^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.09, size = 303, normalized size = 2.83 \[ d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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