Optimal. Leaf size=67 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665}
\begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 665
Rule 673
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}+\frac {\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx}{5 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.37, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-4 d^2+3 d e x+e^2 x^2\right )}{15 d^2 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 93, normalized size = 1.39
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +4 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 \left (e x +d \right )^{3} d^{2} e}\) | \(43\) |
trager | \(-\frac {\left (-e^{2} x^{2}-3 d e x +4 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} \left (e x +d \right )^{3} e}\) | \(49\) |
default | \(\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{4}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs.
\(2 (57) = 114\).
time = 0.36, size = 117, normalized size = 1.75 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d x^{2} e^{3} + 2 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.78, size = 100, normalized size = 1.49 \begin {gather*} -\frac {4 \, x^{3} e^{3} + 12 \, d x^{2} e^{2} + 12 \, d^{2} x e + 4 \, d^{3} - {\left (x^{2} e^{2} + 3 \, d x e - 4 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x^{3} e^{4} + 3 \, d^{3} x^{2} e^{3} + 3 \, d^{4} x e^{2} + d^{5} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs.
\(2 (57) = 114\).
time = 1.55, size = 158, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {25 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 4\right )} e^{\left (-1\right )}}{15 \, d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.78, size = 47, normalized size = 0.70 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-4\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{15\,d^2\,e\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________