Optimal. Leaf size=73 \[ \frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {797, 641, 217, 203, 637} \[ \frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 637
Rule 641
Rule 797
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}+\frac {d^2 \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ &=\frac {d (d+e x)}{e^3 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^3}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 77, normalized size = 1.05 \[ \frac {-d \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+2 d^2+d e x-e^2 x^2}{e^3 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 87, normalized size = 1.19 \[ \frac {2 \, d e x - 2 \, d^{2} + 2 \, {\left (d e x - d^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{e^{4} x - d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 66, normalized size = 0.90 \[ -d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\relax (d) - \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (2 \, d^{2} e^{\left (-3\right )} - {\left (x e^{\left (-1\right )} - d e^{\left (-2\right )}\right )} x\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 99, normalized size = 1.36 \[ -\frac {x^{2}}{\sqrt {-e^{2} x^{2}+d^{2}}\, e}+\frac {d x}{\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{2}}+\frac {2 d^{2}}{\sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 78, normalized size = 1.07 \[ -\frac {x^{2}}{\sqrt {-e^{2} x^{2} + d^{2}} e} + \frac {d x}{\sqrt {-e^{2} x^{2} + d^{2}} e^{2}} - \frac {d \arcsin \left (\frac {e x}{d}\right )}{e^{3}} + \frac {2 \, d^{2}}{\sqrt {-e^{2} x^{2} + d^{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.96, size = 87, normalized size = 1.19 \[ \frac {2\,d^2-e^2\,x^2}{e^3\,\sqrt {d^2-e^2\,x^2}}+\frac {d\,\ln \left (x\,\sqrt {-e^2}+\sqrt {d^2-e^2\,x^2}\right )}{{\left (-e^2\right )}^{3/2}}+\frac {d\,x}{e^2\,\sqrt {d^2-e^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 9.71, size = 184, normalized size = 2.52 \[ d \left (\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{e^{3}} - \frac {i x}{d e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {e x}{d} \right )}}{e^{3}} + \frac {x}{d e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \tilde {\infty } x^{4} & \text {for}\: \left (d = 0 \vee d = - \sqrt {e^{2} x^{2}} \vee d = \sqrt {e^{2} x^{2}}\right ) \wedge \left (d = - \sqrt {e^{2} x^{2}} \vee d = \sqrt {e^{2} x^{2}} \vee e = 0\right ) \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {2 d^{2}}{e^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {x^{2}}{e^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________