Optimal. Leaf size=234 \[ -\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}} \]
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Rubi [A] time = 0.38, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {852, 1805, 823, 12, 266, 63, 208} \[ -\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 852
Rule 1805
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac {(d-e x)^4}{x \left (d^2-e^2 x^2\right )^{15/2}} \, dx\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {\int \frac {-13 d^4+44 d^3 e x+13 d^2 e^2 x^2}{x \left (d^2-e^2 x^2\right )^{13/2}} \, dx}{13 d^2}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {\int \frac {143 d^4-440 d^3 e x}{x \left (d^2-e^2 x^2\right )^{11/2}} \, dx}{143 d^4}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {\int \frac {1287 d^6 e^2-3520 d^5 e^3 x}{x \left (d^2-e^2 x^2\right )^{9/2}} \, dx}{1287 d^8 e^2}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {\int \frac {9009 d^8 e^4-21120 d^7 e^5 x}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{9009 d^{12} e^4}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {45045 d^{10} e^6-84480 d^9 e^7 x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{45045 d^{16} e^6}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {135135 d^{12} e^8-168960 d^{11} e^9 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{135135 d^{20} e^8}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {135135 d^{14} e^{10}}{x \sqrt {d^2-e^2 x^2}} \, dx}{135135 d^{24} e^{10}}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^{10}}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^{10}}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\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 )}{d^{10} e^2}\\ &=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 161, normalized size = 0.69 \[ \frac {-4095 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (9839 d^9+22976 d^8 e x-4466 d^7 e^2 x^2-56304 d^6 e^3 x^3-34156 d^5 e^4 x^4+40240 d^4 e^5 x^5+45735 d^3 e^6 x^6-1540 d^2 e^7 x^7-16385 d e^8 x^8-5120 e^9 x^9\right )}{(d-e x)^3 (d+e x)^7}+4095 \log (x)}{4095 d^{11}} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.40, size = 432, normalized size = 1.85 \[ \frac {9839 \, e^{10} x^{10} + 39356 \, d e^{9} x^{9} + 29517 \, d^{2} e^{8} x^{8} - 78712 \, d^{3} e^{7} x^{7} - 137746 \, d^{4} e^{6} x^{6} + 137746 \, d^{6} e^{4} x^{4} + 78712 \, d^{7} e^{3} x^{3} - 29517 \, d^{8} e^{2} x^{2} - 39356 \, d^{9} e x - 9839 \, d^{10} + 4095 \, {\left (e^{10} x^{10} + 4 \, d e^{9} x^{9} + 3 \, d^{2} e^{8} x^{8} - 8 \, d^{3} e^{7} x^{7} - 14 \, d^{4} e^{6} x^{6} + 14 \, d^{6} e^{4} x^{4} + 8 \, d^{7} e^{3} x^{3} - 3 \, d^{8} e^{2} x^{2} - 4 \, d^{9} e x - d^{10}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (5120 \, e^{9} x^{9} + 16385 \, d e^{8} x^{8} + 1540 \, d^{2} e^{7} x^{7} - 45735 \, d^{3} e^{6} x^{6} - 40240 \, d^{4} e^{5} x^{5} + 34156 \, d^{5} e^{4} x^{4} + 56304 \, d^{6} e^{3} x^{3} + 4466 \, d^{7} e^{2} x^{2} - 22976 \, d^{8} e x - 9839 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4095 \, {\left (d^{11} e^{10} x^{10} + 4 \, d^{12} e^{9} x^{9} + 3 \, d^{13} e^{8} x^{8} - 8 \, d^{14} e^{7} x^{7} - 14 \, d^{15} e^{6} x^{6} + 14 \, d^{17} e^{4} x^{4} + 8 \, d^{18} e^{3} x^{3} - 3 \, d^{19} e^{2} x^{2} - 4 \, d^{20} e x - d^{21}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 385, normalized size = 1.65 \[ -\frac {128 e x}{273 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{7}}+\frac {1}{13 \left (x +\frac {d}{e}\right )^{4} \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{2} e^{4}}+\frac {2}{13 \left (x +\frac {d}{e}\right )^{3} \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{3} e^{3}}+\frac {29}{117 \left (x +\frac {d}{e}\right )^{2} \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{4} e^{2}}+\frac {320}{819 \left (x +\frac {d}{e}\right ) \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{5} e}+\frac {1}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{6}}-\frac {512 e x}{819 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{9}}+\frac {1}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{8}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{10}}-\frac {1024 e x}{819 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{11}}+\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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