Optimal. Leaf size=140 \[ -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {7 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1821, 849, 821,
272, 65, 214} \begin {gather*} -\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {\int \frac {-8 d^3 e-7 d^2 e^2 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {\int \frac {21 d^4 e^2+16 d^3 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {\int \frac {-32 d^5 e^3-21 d^4 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^6}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (7 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (7 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {\left (7 e^2\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 )}{8 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {7 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 104, normalized size = 0.74 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3-16 d^2 e x-21 d e^2 x^2-32 e^3 x^3\right )}{24 d^3 x^4}+\frac {7 e^4 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 229, normalized size = 1.64
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (32 e^{3} x^{3}+21 d \,e^{2} x^{2}+16 d^{2} e x +6 d^{3}\right )}{24 d^{3} x^{4}}-\frac {7 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{2} \sqrt {d^{2}}}\) | \(99\) |
default | \(d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 d^{2} x^{4}}+\frac {3 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )+e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )+2 d e \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )\) | \(229\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 126, normalized size = 0.90 \begin {gather*} -\frac {7 \, e^{4} \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^{3}} - \frac {4 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{3}}{3 \, d^{3} x} - \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{2}}{8 \, d^{2} x^{2}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} e}{3 \, d x^{3}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 83, normalized size = 0.59 \begin {gather*} \frac {21 \, x^{4} e^{4} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (32 \, x^{3} e^{3} + 21 \, d x^{2} e^{2} + 16 \, d^{2} x e + 6 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{24 \, d^{3} x^{4}} \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 = 4.87, size = 449, normalized size = 3.21 \begin {gather*} d^{2} \left (\begin {cases} - \frac {1}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e}{8 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e^{3}}{8 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {3 e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e}{8 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e^{3}}{8 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {3 i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \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. 299 vs.
\(2 (113) = 226\).
time = 1.48, size = 299, normalized size = 2.14 \begin {gather*} \frac {x^{4} {\left (\frac {16 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{x} + \frac {144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{x^{3}} + \frac {48 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{x^{2}} + 3 \, e^{4}\right )} e^{8}}{192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3}} - \frac {7 \, e^{4} \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 )}{8 \, d^{3}} - \frac {\frac {144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{9} e^{2}}{x} + \frac {16 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{9} e^{\left (-2\right )}}{x^{3}} + \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{9} e^{\left (-4\right )}}{x^{4}} + \frac {48 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{9}}{x^{2}}}{192 \, d^{12}} \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+e\,x\right )}^2}{x^5\,\sqrt {d^2-e^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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