Optimal. Leaf size=64 \[ \frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}-\frac {4 \left (d^2-e^2 x^2\right )^{3/2}}{15 d e^2 (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {807, 665}
\begin {gather*} \frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}-\frac {4 \left (d^2-e^2 x^2\right )^{3/2}}{15 d e^2 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 665
Rule 807
Rubi steps
\begin {align*} \int \frac {x \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}+\frac {4 \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^3} \, dx}{5 e}\\ &=\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4}-\frac {4 \left (d^2-e^2 x^2\right )^{3/2}}{15 d e^2 (d+e x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.27, size = 52, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-d^2-3 d e x+4 e^2 x^2\right )}{15 d e^2 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs.
\(2(56)=112\).
time = 0.06, size = 141, normalized size = 2.20
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (4 e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 \left (e x +d \right )^{3} d \,e^{2}}\) | \(42\) |
trager | \(-\frac {\left (-4 e^{2} x^{2}+3 d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d \left (e x +d \right )^{3} e^{2}}\) | \(47\) |
default | \(-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{5} d \left (x +\frac {d}{e}\right )^{3}}-\frac {d \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{5}}\) | \(141\) |
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. 113 vs.
\(2 (54) = 108\).
time = 0.42, size = 113, normalized size = 1.77 \begin {gather*} \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{5 \, {\left (x^{3} e^{5} + 3 \, d x^{2} e^{4} + 3 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} - \frac {11 \, \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{2}\right )}} + \frac {4 \, \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d x e^{3} + d^{2} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.83, size = 96, normalized size = 1.50 \begin {gather*} -\frac {x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3} - {\left (4 \, x^{2} e^{2} - 3 \, d x e - d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d x^{3} e^{5} + 3 \, d^{2} x^{2} e^{4} + 3 \, d^{3} x e^{3} + d^{4} e^{2}\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 {x \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. 128 vs.
\(2 (54) = 108\).
time = 1.20, size = 128, normalized size = 2.00 \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 {5 \, {\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}} + 1\right )} e^{\left (-2\right )}}{15 \, d {\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 = 2.90, size = 46, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+3\,d\,e\,x-4\,e^2\,x^2\right )}{15\,d\,e^2\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________