Optimal. Leaf size=106 \[ -\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6286, 1821,
821, 272, 43, 65, 214} \begin {gather*} -\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6286
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^5} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} \int \frac {\left (-8 a c-5 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^4} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (5 a^2 c\right ) \int \frac {\sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^2 c\right ) \text {Subst}\left (\int \frac {\sqrt {c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac {1}{16} \left (5 a^4 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^2 c\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 96, normalized size = 0.91 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-6-16 a x-9 a^2 x^2+16 a^3 x^3\right )}{24 x^4}-\frac {5}{8} a^4 c^{3/2} \log (x)+\frac {5}{8} a^4 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \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. \(562\) vs.
\(2(86)=172\).
time = 0.06, size = 563, normalized size = 5.31
method | result | size |
risch | \(-\frac {\left (16 x^{5} a^{5}-9 a^{4} x^{4}-32 a^{3} x^{3}+3 a^{2} x^{2}+16 a x +6\right ) c^{2}}{24 x^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {5 a^{4} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\) | \(97\) |
default | \(2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )}{3}\right )-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}+\frac {7 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )}{4}+2 a^{4} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a^{4} \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )\) | \(563\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 191, normalized size = 1.80 \begin {gather*} \left [\frac {15 \, a^{4} c^{\frac {3}{2}} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac {15 \, a^{4} \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \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 = 5.57, size = 447, normalized size = 4.22 \begin {gather*} a^{2} c \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \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. 371 vs.
\(2 (86) = 172\).
time = 0.44, size = 371, normalized size = 3.50 \begin {gather*} -\frac {5 \, a^{4} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c^{2} + 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{3} - 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{3} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{4} + 16 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{4} {\left | a \right |} + 9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{5} - 16 \, a^{3} \sqrt {-c} c^{5} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \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 (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^5\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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