Optimal. Leaf size=206 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]
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Rubi [A] time = 0.31, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1807, 811, 813, 844, 217, 203, 266, 63, 208} \[ -\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 811
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^8} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-21 d^4 e-21 d^3 e^2 x-7 d^2 e^3 x^2\right )}{x^7} \, dx}{7 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {\int \frac {\left (126 d^5 e^2+21 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx}{42 d^4}\\ &=-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {\int \frac {\left (1008 d^7 e^4+210 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx}{336 d^6}\\ &=\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {\int \frac {\left (4032 d^9 e^6+1260 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{1344 d^8}\\ &=-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac {\int \frac {-2520 d^{10} e^7+8064 d^9 e^8 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{2688 d^8}\\ &=-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {1}{16} \left (15 d^2 e^7\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\left (3 d e^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}+\frac {1}{32} \left (15 d^2 e^7\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\left (3 d e^8\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 {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{16} \left (15 d^2 e^5\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 {3 e^6 (16 d-5 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3}-\frac {e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-3 d e^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {15}{16} d e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.15, size = 247, normalized size = 1.20 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^7 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^6}-\frac {3 d^5 e^2 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{5 x^5 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {-8 d^8 e+34 d^6 e^3 x^2-59 d^4 e^5 x^4+33 d^2 e^7 x^6+15 d^2 e^7 x^6 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{16 x^6 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 173, normalized size = 0.84 \[ \frac {3360 \, d e^{7} x^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 525 \, d e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 560 \, d e^{7} x^{7} + {\left (560 \, e^{7} x^{7} - 2496 \, d e^{6} x^{6} - 525 \, d^{2} e^{5} x^{5} + 992 \, d^{3} e^{4} x^{4} + 770 \, d^{4} e^{3} x^{3} - 96 \, d^{5} e^{2} x^{2} - 280 \, d^{6} e x - 80 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{560 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 510, normalized size = 2.48 \[ -3 \, d \arcsin \left (\frac {x e}{d}\right ) e^{7} \mathrm {sgn}\relax (d) - \frac {15}{16} \, d e^{7} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (5 \, d e^{16} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{14}}{x} + \frac {49 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{12}}{x^{2}} - \frac {245 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{10}}{x^{3}} - \frac {875 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d e^{8}}{x^{4}} + \frac {455 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d e^{6}}{x^{5}} + \frac {9065 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d e^{4}}{x^{6}}\right )} x^{7} e^{5}}{4480 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7}} - \frac {1}{4480} \, {\left (\frac {9065 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{68}}{x} + \frac {455 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{66}}{x^{2}} - \frac {875 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{64}}{x^{3}} - \frac {245 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d e^{62}}{x^{4}} + \frac {49 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d e^{60}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d e^{58}}{x^{6}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d e^{56}}{x^{7}}\right )} e^{\left (-63\right )} + \sqrt {-x^{2} e^{2} + d^{2}} e^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 377, normalized size = 1.83 \[ -\frac {15 d^{2} e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}}-\frac {3 d \,e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8} x}{d}+\frac {15 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}{16}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8} x}{d^{3}}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}{16 d^{2}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8} x}{5 d^{5}}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7}}{16 d^{4}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{5 d^{5} x}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{16 d^{4} x^{2}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{5 d^{3} x^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{8 d^{2} x^{4}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d \,x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{2 x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 326, normalized size = 1.58 \[ -3 \, d e^{7} \arcsin \left (\frac {e x}{d}\right ) - \frac {15}{16} \, d e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8} x}{d} + \frac {15}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8} x}{d^{3}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{16 \, d^{2}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{16 \, d^{4}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{5 \, d^{3} x} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{16 \, d^{4} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{5 \, d^{3} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{5 \, d x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, x^{7}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.29, size = 1513, normalized size = 7.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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