Optimal. Leaf size=216 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1807, 813, 844, 217, 203, 266, 63, 208} \[ \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 813
Rule 844
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-15 d^4 e-13 d^3 e^2 x-5 d^2 e^3 x^2\right )}{x^5} \, dx}{5 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (52 d^5 e^2-25 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx}{20 d^4}\\ &=-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (150 d^6 e^3+312 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx}{72 d^4}\\ &=\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (-1248 d^7 e^4+600 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{192 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {-1200 d^8 e^5-2496 d^7 e^6 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{384 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{8} \left (25 d^4 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{2} \left (13 d^3 e^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{16} \left (25 d^4 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{2} \left (13 d^3 e^6\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{8} \left (25 d^4 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 199, normalized size = 0.92 \[ \frac {\sqrt {d^2-e^2 x^2} \left (5 e^5 \left (e^2 x^2-d^2\right )^3 \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )+15 e^5 \left (e^2 x^2-d^2\right )^3 \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )-\frac {7 d^{11} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{x^5 \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {35 d^9 e^2 \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{35 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 180, normalized size = 0.83 \[ -\frac {1560 \, d^{3} e^{5} x^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 375 \, d^{3} e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 80 \, d^{3} e^{5} x^{5} - {\left (40 \, e^{7} x^{7} + 180 \, d e^{6} x^{6} + 80 \, d^{2} e^{5} x^{5} + 656 \, d^{3} e^{4} x^{4} + 345 \, d^{4} e^{3} x^{3} - 32 \, d^{5} e^{2} x^{2} - 90 \, d^{6} e x - 24 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 430, normalized size = 1.99 \[ \frac {13}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{5} \mathrm {sgn}\relax (d) - \frac {25}{8} \, d^{3} e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (6 \, d^{3} e^{12} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{10}}{x} + \frac {50 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{8}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{6}}{x^{3}} - \frac {2580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{4}}{x^{4}}\right )} x^{5} e^{3}}{960 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5}} + \frac {1}{960} \, {\left (\frac {2580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{38}}{x} + \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{36}}{x^{2}} - \frac {50 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{34}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{32}}{x^{4}} - \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{3} e^{30}}{x^{5}}\right )} e^{\left (-35\right )} + \frac {1}{6} \, {\left (4 \, d^{2} e^{5} + {\left (2 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 327, normalized size = 1.51 \[ -\frac {25 d^{4} e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+\frac {13 d^{3} e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+\frac {13 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{6} x}{2}+\frac {25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{5}}{8}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6} x}{3 d}+\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}{24}+\frac {52 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6} x}{15 d^{3}}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}{8 d^{2}}+\frac {52 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{15 d^{3} x}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{8 d^{2} x^{2}}-\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{15 d \,x^{3}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{4 x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 278, normalized size = 1.29 \[ \frac {13}{2} \, d^{3} e^{5} \arcsin \left (\frac {e x}{d}\right ) - \frac {25}{8} \, d^{3} e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {13}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{6} x + \frac {25}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{5} + \frac {13 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6} x}{3 \, d} + \frac {25}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{8 \, d^{2}} + \frac {52 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{15 \, d x} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{2}} - \frac {13 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{15 \, d x^{3}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{4 \, x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.70, size = 1178, normalized size = 5.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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