Optimal. Leaf size=120 \[ \frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {811, 813, 844, 217, 203, 266, 63, 208} \[ \frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 811
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx &=-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {\int \frac {\left (4 d^3 e^2+6 d^2 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{4 d^2}\\ &=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+\frac {\int \frac {-12 d^4 e^3+8 d^3 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^2}\\ &=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {1}{2} \left (3 d^2 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\left (d e^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {1}{4} \left (3 d^2 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\left (d e^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (3 d^2 e\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 {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 111, normalized size = 0.92 \[ -\frac {e^3 \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};1-\frac {e^2 x^2}{d^2}\right )}{5 d^4}-\frac {d^3 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 129, normalized size = 1.08 \[ -\frac {12 \, d e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 9 \, d e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 6 \, d e^{3} x^{3} + {\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 3 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 261, normalized size = 2.18 \[ d \arcsin \left (\frac {x e}{d}\right ) e^{3} \mathrm {sgn}\relax (d) + \frac {3}{2} \, d e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (d e^{8} + \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{6}}{x} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{4}}{x^{2}}\right )} x^{3} e}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3}} + \frac {1}{24} \, {\left (\frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{16}}{x} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{14}}{x^{2}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{12}}{x^{3}}\right )} e^{\left (-15\right )} - \sqrt {-x^{2} e^{2} + d^{2}} e^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 235, normalized size = 1.96 \[ \frac {3 d^{2} e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}+\frac {d \,e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{4} x}{d}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}{2}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4} x}{3 d^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}{2 d^{2}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}{3 d^{3} x}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{2 d^{2} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{3 d \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 184, normalized size = 1.53 \[ d e^{3} \arcsin \left (\frac {e x}{d}\right ) + \frac {3}{2} \, d e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x}{d} - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{2 \, d^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{3 \, d x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{2 \, d^{2} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{3 \, d x^{3}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.93, size = 457, normalized size = 3.81 \[ d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {d^{2}}{2 e x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e}{2 x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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