Optimal. Leaf size=207 \[ \frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2} \]
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Rubi [A] time = 0.31, antiderivative size = 207, 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, 815, 844, 217, 203, 266, 63, 208} \begin {gather*} \frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 815
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^3} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-6 d^4 e-d^3 e^2 x-2 d^2 e^3 x^2\right )}{x^2} \, dx}{2 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (d^5 e^2-34 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{2 d^4}\\ &=\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {\left (-6 d^7 e^4+170 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4 e^2}\\ &=\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (24 d^9 e^6-510 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^4}\\ &=\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {-48 d^{11} e^8+510 d^{10} e^9 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{96 d^4 e^6}\\ &=\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{2} \left (d^7 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{16} \left (85 d^6 e^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{4} \left (d^7 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{16} \left (85 d^6 e^3\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 {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^7 \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 {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.64, size = 259, normalized size = 1.25 \begin {gather*} -\frac {e \left (5040 d^9 \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {e^2 x^2}{d^2}\right )+e x \left (240 \left (d^2-e^2 x^2\right )^4 \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )-7 d \left (1104 d^7+165 d^6 e x-1632 d^5 e^2 x^2-295 d^4 e^3 x^3+672 d^3 e^4 x^4+170 d^2 e^5 x^5+75 d^7 \sqrt {1-\frac {e^2 x^2}{d^2}} \sin ^{-1}\left (\frac {e x}{d}\right )-720 d^6 \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-144 d e^6 x^6-40 e^7 x^7\right )\right )\right )}{1680 d x \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.72, size = 189, normalized size = 0.91 \begin {gather*} -\frac {85}{16} d^6 e \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )+d^6 e^2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (-120 d^7-720 d^6 e x+544 d^5 e^2 x^2-645 d^4 e^3 x^3-448 d^3 e^4 x^4+50 d^2 e^5 x^5+144 d e^6 x^6+40 e^7 x^7\right )}{240 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 179, normalized size = 0.86 \begin {gather*} \frac {2550 \, d^{6} e^{2} x^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 120 \, d^{6} e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 544 \, d^{6} e^{2} x^{2} + {\left (40 \, e^{7} x^{7} + 144 \, d e^{6} x^{6} + 50 \, d^{2} e^{5} x^{5} - 448 \, d^{3} e^{4} x^{4} - 645 \, d^{4} e^{3} x^{3} + 544 \, d^{5} e^{2} x^{2} - 720 \, d^{6} e x - 120 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 262, normalized size = 1.27 \begin {gather*} -\frac {85}{16} \, d^{6} \arcsin \left (\frac {x e}{d}\right ) e^{2} \mathrm {sgn}\relax (d) - \frac {1}{2} \, d^{6} e^{2} \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 {1}{8} \, {\left (\frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{8}}{x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{6}}{x^{2}}\right )} e^{\left (-8\right )} + \frac {1}{240} \, {\left (544 \, d^{5} e^{2} - {\left (645 \, d^{4} e^{3} + 2 \, {\left (224 \, d^{3} e^{4} - {\left (25 \, d^{2} e^{5} + 4 \, {\left (5 \, x e^{7} + 18 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} + \frac {{\left (d^{6} e^{6} + \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{4}}{x}\right )} x^{2}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 252, normalized size = 1.22 \begin {gather*} -\frac {d^{7} e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}-\frac {85 d^{6} e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}-\frac {85 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{3} x}{16}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{2}}{2}-\frac {85 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2} e^{3} x}{24}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} e^{2}}{6}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3} x}{6}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d \,e^{2}}{10}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{x}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 229, normalized size = 1.11 \begin {gather*} -\frac {85}{16} \, d^{6} e^{2} \arcsin \left (\frac {e x}{d}\right ) - \frac {1}{2} \, d^{6} e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {85}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3} x + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} e^{2} - \frac {85}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} x + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2} + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3} x + \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e}{x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{2 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.22, size = 1059, normalized size = 5.12
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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