Optimal. Leaf size=201 \[ \frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {849, 821, 272,
43, 65, 214} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 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
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {\int \frac {\left (-8 d^2 e-3 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx}{8 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}+\frac {\int \frac {\left (21 d^3 e^2+16 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {\int \frac {\left (-96 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{336 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{16 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{32 d^3}\\ &=-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {\left (3 e^8\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{256 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\right ) \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 )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 148, normalized size = 0.74 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-560 d^7-640 d^6 e x+840 d^5 e^2 x^2+1024 d^4 e^3 x^3-70 d^3 e^4 x^4-128 d^2 e^5 x^5-105 d e^6 x^6-256 e^7 x^7\right )}{4480 d^4 x^8}+\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{64 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 256, normalized size = 1.27
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (256 e^{7} x^{7}+105 d \,e^{6} x^{6}+128 d^{2} e^{5} x^{5}+70 d^{3} e^{4} x^{4}-1024 d^{4} e^{3} x^{3}-840 d^{5} e^{2} x^{2}+640 d^{6} e x +560 d^{7}\right )}{4480 x^{8} d^{4}}-\frac {3 e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 d^{3} \sqrt {d^{2}}}\) | \(143\) |
default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 d^{2} x^{8}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d^{2} x^{7}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 d^{4} x^{5}}\right )\) | \(256\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 215, normalized size = 1.07 \begin {gather*} -\frac {3 \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d^{4}} + \frac {3 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{8}}{128 \, d^{5}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{128 \, d^{7}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{128 \, d^{7} x^{2}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{64 \, d^{5} x^{4}} - \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{35 \, d^{4} x^{5}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{16 \, d^{3} x^{6}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e}{7 \, d^{2} x^{7}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{8 \, d x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.99, size = 123, normalized size = 0.61 \begin {gather*} \frac {105 \, x^{8} e^{8} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (256 \, x^{7} e^{7} + 105 \, d x^{6} e^{6} + 128 \, d^{2} x^{5} e^{5} + 70 \, d^{3} x^{4} e^{4} - 1024 \, d^{4} x^{3} e^{3} - 840 \, d^{5} x^{2} e^{2} + 640 \, d^{6} x e + 560 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \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 = 26.95, size = 1159, normalized size = 5.77 \begin {gather*} d^{3} \left (\begin {cases} - \frac {d^{2}}{8 e x^{9} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {7 e}{48 x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{192 d^{2} x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e^{5}}{384 d^{4} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {5 e^{7}}{128 d^{6} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e^{8} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{128 d^{7}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{8 e x^{9} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {7 i e}{48 x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{192 d^{2} x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e^{5}}{384 d^{4} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {5 i e^{7}}{128 d^{6} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e^{8} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{128 d^{7}} & \text {otherwise} \end {cases}\right ) + d^{2} e \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 ) - d e^{2} \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^{3} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 425 vs.
\(2 (162) = 324\).
time = 1.02, size = 425, normalized size = 2.11 \begin {gather*} \frac {x^{8} {\left (\frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{2}}{x^{3}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-2\right )}}{x^{5}} + \frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-6\right )}}{x^{7}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}}{x^{4}} + 35 \, e^{8}\right )} e^{16}}{71680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{4}} - \frac {3 \, e^{8} \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 )}{128 \, d^{4}} - \frac {\frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{28} e^{6}}{x} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{28} e^{2}}{x^{3}} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{28} e^{\left (-2\right )}}{x^{5}} + \frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{28} e^{\left (-6\right )}}{x^{7}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{28} e^{\left (-8\right )}}{x^{8}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{28}}{x^{4}}}{71680 \, d^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.04, size = 212, normalized size = 1.05 \begin {gather*} \frac {3\,d^3\,\sqrt {d^2-e^2\,x^2}}{128\,x^8}-\frac {11\,d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{128\,x^8}-\frac {11\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{128\,d\,x^8}+\frac {3\,{\left (d^2-e^2\,x^2\right )}^{7/2}}{128\,d^3\,x^8}+\frac {8\,e^3\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {e^5\,\sqrt {d^2-e^2\,x^2}}{35\,d^2\,x^3}-\frac {2\,e^7\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,x}-\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}+\frac {e^8\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,3{}\mathrm {i}}{128\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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