Optimal. Leaf size=117 \[ \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 117, 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 = {813, 815, 844, 217, 203, 266, 63, 208} \[ \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx &=-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {\left (-2 d^2 e+6 d e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx\\ &=\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac {\int \frac {4 d^4 e^3-6 d^3 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\left (d^4 e\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (3 d^3 e^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac {1}{2} \left (d^4 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (3 d^3 e^2\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 {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^4 \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 )}{e}\\ &=\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 124, normalized size = 1.06 \[ -\frac {d^5 \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x \sqrt {d^2-e^2 x^2}}-\frac {1}{3} e \left (\sqrt {d^2-e^2 x^2} \left (e^2 x^2-4 d^2\right )+3 d^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 124, normalized size = 1.06 \[ \frac {18 \, d^{3} e x \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 6 \, d^{3} e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 8 \, d^{3} e x - {\left (2 \, e^{3} x^{3} + 3 \, d e^{2} x^{2} - 8 \, d^{2} e x + 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 157, normalized size = 1.34 \[ -\frac {3}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e \mathrm {sgn}\relax (d) - d^{3} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {d^{3} x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-1\right )}}{2 \, x} + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (8 \, d^{2} e - {\left (2 \, x e^{3} + 3 \, d e^{2}\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 182, normalized size = 1.56 \[ -\frac {d^{4} e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-\frac {3 d^{3} e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{2} x}{2}+\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2} x}{d}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{3}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 129, normalized size = 1.10 \[ -\frac {3}{2} \, d^{3} e \arcsin \left (\frac {e x}{d}\right ) - d^{3} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.51, size = 114, normalized size = 0.97 \[ \frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{3}+d^2\,e\,\sqrt {d^2-e^2\,x^2}-d^3\,e\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )-\frac {d^3\,\sqrt {d^2-e^2\,x^2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ \frac {e^2\,x^2}{d^2}\right )}{x\,\sqrt {1-\frac {e^2\,x^2}{d^2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 8.18, size = 386, normalized size = 3.30 \[ d^{3} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________