Optimal. Leaf size=80 \[ \frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1164, 425, 541,
12, 385, 214} \begin {gather*} \frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 214
Rule 385
Rule 425
Rule 541
Rule 1164
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {-5 d e+2 e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{6 d^2 e}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {3 d^2 e^2}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{12 d^4 e^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\text {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{4 d^2}\\ &=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 80, normalized size = 1.00 \begin {gather*} \frac {\frac {2 \left (9 d x+7 e x^3\right )}{\left (d+e x^2\right )^{3/2}}+\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {e}}}{24 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(864\) vs.
\(2(60)=120\).
time = 0.21, size = 865, normalized size = 10.81
method | result | size |
default | \(-\frac {e \left (\frac {1}{3 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right ) \sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}+\frac {2 e \left (x +\frac {\sqrt {-d e}}{e}\right )-2 \sqrt {-d e}}{3 \sqrt {-d e}\, d \sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}-\frac {e \left (\frac {1}{2 d \sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {d e}\, \left (2 e \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {d e}\right )}{4 d^{2} e \sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}{x -\frac {\sqrt {d e}}{e}}\right )}{4 d^{\frac {3}{2}}}\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (\frac {1}{2 d \sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}+\frac {\sqrt {d e}\, \left (2 e \left (x +\frac {\sqrt {d e}}{e}\right )-2 \sqrt {d e}\right )}{4 d^{2} e \sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}{x +\frac {\sqrt {d e}}{e}}\right )}{4 d^{\frac {3}{2}}}\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (-\frac {1}{3 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right ) \sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}-\frac {2 e \left (x -\frac {\sqrt {-d e}}{e}\right )+2 \sqrt {-d e}}{3 \sqrt {-d e}\, d \sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}\) | \(865\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (61) = 122\).
time = 0.35, size = 151, normalized size = 1.89 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{4} e^{2} + 2 \, d x^{2} e + d^{2}\right )} e^{\frac {1}{2}} \log \left (\frac {17 \, x^{4} e^{2} + 14 \, d x^{2} e + 4 \, \sqrt {2} {\left (3 \, x^{3} e + d x\right )} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + d^{2}}{x^{4} e^{2} - 2 \, d x^{2} e + d^{2}}\right ) + 8 \, {\left (7 \, x^{3} e^{2} + 9 \, d x e\right )} \sqrt {x^{2} e + d}}{96 \, {\left (d^{3} x^{4} e^{3} + 2 \, d^{4} x^{2} 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 {1}{- d^{3} \sqrt {d + e x^{2}} - d^{2} e x^{2} \sqrt {d + e x^{2}} + d e^{2} x^{4} \sqrt {d + e x^{2}} + e^{3} x^{6} \sqrt {d + e x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.88, size = 114, normalized size = 1.42 \begin {gather*} \frac {x {\left (\frac {7 \, x^{2} e}{d^{3}} + \frac {9}{d^{2}}\right )}}{12 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}}} + \frac {\sqrt {2} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{16 \, d^{2} {\left | d \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (d^2-e^2\,x^4\right )\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________