Optimal. Leaf size=198 \[ \frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]
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Rubi [A]
time = 0.15, 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 = {866, 1821,
849, 821, 272, 43, 65, 214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 866
Rule 1821
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 \text {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 \text {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 \text {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 = 0.50, size = 134, normalized size = 0.68 \begin {gather*} \frac {\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 )+210 e^7 x^7 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{840 d^4 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1574\) vs.
\(2(170)=340\).
time = 0.08, size = 1575, normalized size = 7.95
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-176 e^{6} x^{6}+105 d \,e^{5} x^{5}-88 d^{2} e^{4} x^{4}+70 d^{3} e^{3} x^{3}+144 d^{4} e^{2} x^{2}-280 e \,d^{5} x +120 d^{6}\right )}{840 x^{7} d^{4}}-\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{3} \sqrt {d^{2}}}\) | \(132\) |
default | \(\text {Expression too large to display}\) | \(1575\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 192, normalized size = 0.97 \begin {gather*} -\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{4}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}} e^{7}}{8 \, d^{5}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{8 \, d^{5} x^{2}} - \frac {22 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{105 \, d^{4} x^{3}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{4}} - \frac {11 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d^{2} x^{5}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e}{3 \, d x^{6}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.23, size = 112, normalized size = 0.57 \begin {gather*} \frac {105 \, x^{7} e^{7} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (176 \, x^{6} e^{6} - 105 \, d x^{5} e^{5} + 88 \, d^{2} x^{4} e^{4} - 70 \, d^{3} x^{3} e^{3} - 144 \, d^{4} x^{2} e^{2} + 280 \, d^{5} x e - 120 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{840 \, d^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 9.39, size = 835, normalized size = 4.22 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.10, size = 321, normalized size = 1.62 \begin {gather*} -\frac {1}{53760} \, {\left (\frac {6720 \, e^{6} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} - \frac {6720 \, e^{6} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} + \frac {32 \, {\left (105 \, e^{6} \log \left (2\right ) - 210 \, e^{6} \log \left (i + 1\right ) + 352 i \, e^{6}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}} - \frac {105 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3780 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 189 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4992 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1981 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 700 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 105 \, \sqrt {\frac {2 \, d}{x e + d} - 1} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4} {\left (\frac {d}{x e + d} - 1\right )}^{7}}\right )} e \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 = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^8\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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