Optimal. Leaf size=51 \[ \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1150, 390, 208} \[ \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 390
Rule 1150
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^4}{d^2-e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^3}{d-e x^2} \, dx\\ &=\int \left (-7 d^2-4 d e x^2-e^2 x^4+\frac {8 d^3}{d-e x^2}\right ) \, dx\\ &=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\left (8 d^3\right ) \int \frac {1}{d-e x^2} \, dx\\ &=-7 d^2 x-\frac {4}{3} d e x^3-\frac {e^2 x^5}{5}+\frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ \frac {8 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-7 d^2 x-\frac {4}{3} d e x^3-\frac {1}{5} e^2 x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 116, normalized size = 2.27 \[ \left [-\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} + 4 \, d^{2} \sqrt {\frac {d}{e}} \log \left (\frac {e x^{2} + 2 \, e x \sqrt {\frac {d}{e}} + d}{e x^{2} - d}\right ) - 7 \, d^{2} x, -\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} - 8 \, d^{2} \sqrt {-\frac {d}{e}} \arctan \left (\frac {e x \sqrt {-\frac {d}{e}}}{d}\right ) - 7 \, d^{2} x\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 144, normalized size = 2.82 \[ 4 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {11}{2}} - {\left (d^{2}\right )}^{\frac {1}{4}} d {\left | d \right |} e^{\frac {11}{2}}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{{\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-6\right )} + 2 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {15}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} d e^{\frac {15}{2}}\right )} e^{\left (-8\right )} \log \left ({\left | {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - 2 \, {\left ({\left (d^{2}\right )}^{\frac {1}{4}} d^{2} e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {1}{4}} d {\left | d \right |} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right ) - \frac {1}{15} \, {\left (3 \, x^{5} e^{12} + 20 \, d x^{3} e^{11} + 105 \, d^{2} x e^{10}\right )} e^{\left (-10\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 42, normalized size = 0.82 \[ -\frac {e^{2} x^{5}}{5}-\frac {4 d e \,x^{3}}{3}+\frac {8 d^{3} \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-7 d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.25, size = 56, normalized size = 1.10 \[ -\frac {1}{5} \, e^{2} x^{5} - \frac {4}{3} \, d e x^{3} - \frac {4 \, d^{3} \log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{\sqrt {d e}} - 7 \, d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 42, normalized size = 0.82 \[ -7\,d^2\,x-\frac {e^2\,x^5}{5}-\frac {4\,d\,e\,x^3}{3}-\frac {d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x\,1{}\mathrm {i}}{\sqrt {d}}\right )\,8{}\mathrm {i}}{\sqrt {e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 75, normalized size = 1.47 \[ - 7 d^{2} x - \frac {4 d e x^{3}}{3} - \frac {e^{2} x^{5}}{5} - 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x - \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt {\frac {d^{5}}{e}} \log {\left (x + \frac {\sqrt {\frac {d^{5}}{e}}}{d^{2}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________