Optimal. Leaf size=198 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4} \]
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Rubi [A] time = 0.24, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {852, 1807, 835, 807, 266, 47, 63, 208} \[ \frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^8} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac {\int \frac {\left (14 d^3 e-11 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^7} \, dx}{7 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}+\frac {\int \frac {\left (66 d^4 e^2-42 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^6} \, dx}{42 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {\int \frac {\left (210 d^5 e^3-132 d^4 e^4 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{210 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}+\frac {\int \frac {\left (528 d^6 e^4-210 d^5 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{840 d^8}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{4 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{8 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \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 )}{8 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 128, normalized size = 0.65 \[ \frac {-105 e^7 x^7 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-120 d^6+280 d^5 e x-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4-105 d e^5 x^5+176 e^6 x^6\right )+105 e^7 x^7 \log (x)}{840 d^4 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 119, normalized size = 0.60 \[ \frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (176 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 88 \, d^{2} e^{4} x^{4} - 70 \, d^{3} e^{3} x^{3} - 144 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x - 120 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{840 \, d^{4} x^{7}} \]
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 [B] time = 0.02, size = 591, normalized size = 2.98 \[ -\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}\, d^{3}}+\frac {29 e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {29 e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}+\frac {29 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{8} x}{8 d^{6}}-\frac {29 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8} x}{8 d^{6}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}{8 d^{5}}+\frac {29 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{8} x}{12 d^{8}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8} x}{12 d^{8}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}{24 d^{7}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8} x}{15 d^{10}}+\frac {29 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{7}}{15 d^{9}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7}}{40 d^{9}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{5}}{3 \left (x +\frac {d}{e}\right )^{2} d^{9}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{15 d^{10} x}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{8 d^{9} x^{2}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{15 d^{8} x^{3}}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{12 d^{7} x^{4}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d^{6} x^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{3 d^{5} x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 205, normalized size = 1.04 \[ -\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{4}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{8 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{8 \, d^{5} x^{2}} - \frac {22 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{105 \, d^{4} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d^{2} x^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{3 \, d x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{7 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^8\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 18.18, size = 835, normalized size = 4.22 \[ d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{7 x^{6}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{35 d^{2} x^{4}} + \frac {4 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{4} x^{2}} + \frac {8 e^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{7 x^{6}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{35 d^{2} x^{4}} + \frac {4 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{4} x^{2}} + \frac {8 i e^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{6}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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