Optimal. Leaf size=192 \[ \frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {27 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {852, 1635, 1815, 641, 217, 203} \[ \frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}+\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {27 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 641
Rule 852
Rule 1635
Rule 1815
Rubi steps
\begin {align*} \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac {x^3 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^3}{e^3}+\frac {d^2 x}{e^2}-\frac {d x^2}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {\frac {20 d^6}{e}-65 d^5 x+80 d^4 e x^2-54 d^3 e^2 x^3+20 d^2 e^3 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{5 d e^2}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-80 d^6 e+260 d^5 e^2 x-380 d^4 e^3 x^2+216 d^3 e^4 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{20 d e^4}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {240 d^6 e^3-1212 d^5 e^4 x+1140 d^4 e^5 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{60 d e^6}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-1620 d^6 e^5+2424 d^5 e^6 x}{\sqrt {d^2-e^2 x^2}} \, dx}{120 d e^8}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\left (27 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^3}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\left (27 d^5\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3}\\ &=\frac {d^2 (d-e x)^4}{e^4 \sqrt {d^2-e^2 x^2}}+\frac {101 d^4 \sqrt {d^2-e^2 x^2}}{5 e^4}-\frac {19 d^3 x \sqrt {d^2-e^2 x^2}}{2 e^3}+\frac {18 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {27 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 109, normalized size = 0.57 \[ \frac {135 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (212 d^5+77 d^4 e x-29 d^3 e^2 x^2+16 d^2 e^3 x^3-8 d e^4 x^4+2 e^5 x^5\right )}{d+e x}}{10 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 134, normalized size = 0.70 \[ \frac {212 \, d^{5} e x + 212 \, d^{6} - 270 \, {\left (d^{5} e x + d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (2 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} + 16 \, d^{2} e^{3} x^{3} - 29 \, d^{3} e^{2} x^{2} + 77 \, d^{4} e x + 212 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (e^{5} x + d e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 285, normalized size = 1.48 \[ \frac {27 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{3}}+\frac {27 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3} x}{2 e^{3}}+\frac {9 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d x}{e^{3}}+\frac {36 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}}}{5 e^{4}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{2}}{\left (x +\frac {d}{e}\right )^{4} e^{8}}+\frac {6 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d}{\left (x +\frac {d}{e}\right )^{3} e^{7}}+\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{2} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.03, size = 407, normalized size = 2.12 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}}{e^{5} x + d e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}}{2 \, {\left (e^{5} x + d e^{4}\right )}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{4 \, {\left (e^{5} x + d e^{4}\right )}} + \frac {3 i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{4}} + \frac {15 \, d^{5} \arcsin \left (\frac {e x}{d}\right )}{e^{4}} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{2 \, e^{3}} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{e^{4}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{2 \, e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e^{3}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{4 \, e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________