Optimal. Leaf size=102 \[ \frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}+e^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {852, 1807, 811, 844, 217, 203, 266, 63, 208} \begin {gather*} \frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}+e^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 811
Rule 844
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^4} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}-\frac {\int \frac {\left (6 d^3 e-3 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{3 d^2}\\ &=\frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}+\frac {\int \frac {12 d^5 e^3-12 d^4 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4}\\ &=\frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}+\left (d e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^4 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{2} \left (d e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^4 \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 (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}-e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-(d 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 )\\ &=\frac {e (d-e x) \sqrt {d^2-e^2 x^2}}{x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 x^3}-e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 96, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (d^2-3 d e x+2 e^2 x^2\right )}{3 x^3}-e^3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+e^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+e^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 129, normalized size = 1.26 \begin {gather*} -\sqrt {-e^2} e^2 \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )+\frac {\left (-d^2+3 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 x^3}+2 e^3 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 106, normalized size = 1.04 \begin {gather*} \frac {6 \, e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (2 \, e^{2} x^{2} - 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, x^{3}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 479, normalized size = 4.70 \begin {gather*} -\frac {d \,e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {17 e^{4} \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}}}-\frac {25 e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {17 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{4} x}{8 d^{2}}-\frac {25 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4} x}{8 d^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}{d}+\frac {17 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{4} x}{12 d^{4}}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4} x}{12 d^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}{3 d^{3}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4} x}{3 d^{6}}+\frac {17 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{3}}{15 d^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}{5 d^{5}}+\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 \left (x +\frac {d}{e}\right )^{2} d^{5}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{3 d^{6} x}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{d^{5} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 134, normalized size = 1.31 \begin {gather*} -e^{3} \arcsin \left (\frac {e x}{d}\right ) - 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^{3}}{d} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{x} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^4\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.73, size = 338, normalized size = 3.31 \begin {gather*} d^{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 ) - 2 d 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 ) + 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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