Optimal. Leaf size=214 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3} \]
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Rubi [A] time = 0.31, antiderivative size = 214, 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, 813, 811, 844, 217, 203, 266, 63, 208} \[ -\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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^7} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-18 d^4 e-17 d^3 e^2 x-6 d^2 e^3 x^2\right )}{x^6} \, dx}{6 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {\int \frac {\left (85 d^5 e^2-6 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx}{30 d^4}\\ &=-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (48 d^6 e^3+340 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx}{96 d^4}\\ &=\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {\int \frac {\left (192 d^8 e^5+2040 d^7 e^6 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{384 d^6}\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {-4080 d^9 e^6+384 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^6}\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {1}{16} \left (85 d^3 e^6\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (d^2 e^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {1}{32} \left (85 d^3 e^6\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (d^2 e^7\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 e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{16} \left (85 d^3 e^4\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 e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.22, size = 286, normalized size = 1.34 \[ -\frac {3 d^6 e \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{5 x^5 \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {3 e^6 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^5}-\frac {d^4 e^3 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {-8 d^9+34 d^7 e^2 x^2-59 d^5 e^4 x^4+33 d^3 e^6 x^6+15 d^3 e^6 x^6 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{48 x^6 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 179, normalized size = 0.84 \[ \frac {240 \, d^{2} e^{6} x^{6} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 1275 \, d^{2} e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 720 \, d^{2} e^{6} x^{6} + {\left (120 \, e^{7} x^{7} + 720 \, d e^{6} x^{6} - 544 \, d^{2} e^{5} x^{5} + 645 \, d^{3} e^{4} x^{4} + 448 \, d^{4} e^{3} x^{3} - 50 \, d^{5} e^{2} x^{2} - 144 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 485, normalized size = 2.27 \[ -\frac {1}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{6} \mathrm {sgn}\relax (d) - \frac {85}{16} \, d^{2} e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (5 \, d^{2} e^{14} + \frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{12}}{x} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{10}}{x^{2}} - \frac {340 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{8}}{x^{3}} - \frac {1215 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{6}}{x^{4}} + \frac {1800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{4}}{x^{5}}\right )} x^{6} e^{4}}{1920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6}} - \frac {1}{1920} \, {\left (\frac {1800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{52}}{x} - \frac {1215 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{50}}{x^{2}} - \frac {340 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{48}}{x^{3}} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{46}}{x^{4}} + \frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{44}}{x^{5}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{2} e^{42}}{x^{6}}\right )} e^{\left (-48\right )} + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e^{7} + 6 \, d e^{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 352, normalized size = 1.64 \[ -\frac {85 d^{3} e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}}-\frac {d^{2} e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{7} x}{2}+\frac {85 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{6}}{16}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7} x}{3 d^{2}}+\frac {85 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}{48 d}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7} x}{15 d^{4}}+\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}{16 d^{3}}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{15 d^{4} x}+\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{16 d^{3} x^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{15 d^{2} x^{3}}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{24 d \,x^{4}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{5 x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 303, normalized size = 1.42 \[ -\frac {1}{2} \, d^{2} e^{6} \arcsin \left (\frac {e x}{d}\right ) - \frac {85}{16} \, d^{2} e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7} x + \frac {85}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{6} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7} x}{3 \, d^{2}} + \frac {85 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{48 \, d} + \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{16 \, d^{3}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{15 \, d^{2} x} + \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{16 \, d^{3} x^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{15 \, d^{2} x^{3}} - \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{24 \, d x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{5 \, x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{6 \, x^{6}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 21.71, size = 1397, normalized size = 6.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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