Optimal. Leaf size=220 \[ -\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6295, 6285, 90}
\begin {gather*} \frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6295
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2}}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^3 (1+a x)^2}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (-a^5+\frac {1}{x^5}-\frac {a}{x^4}-\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.37 \begin {gather*} -\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (3-4 a x-12 a^2 x^2+24 a^3 x^3+12 a^5 x^5-12 a^4 x^4 \log (x)\right )}{12 a^4 x^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 86, normalized size = 0.39
method | result | size |
default | \(-\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-12 a^{5} x^{5}+12 \ln \left (x \right ) a^{4} x^{4}-24 a^{3} x^{3}+12 a^{2} x^{2}+4 a x -3\right )}{12 \left (a^{2} x^{2}-1\right )^{3}}\) | \(86\) |
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 = 0.47, size = 478, normalized size = 2.17 \begin {gather*} \left [\frac {6 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) + {\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left (12 \, a^{5} + 24 \, a^{3} - 12 \, a^{2} - 4 \, a + 3\right )} c^{2} x^{4} - 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, -\frac {12 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - {\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} - {\left (12 \, a^{5} + 24 \, a^{3} - 12 \, a^{2} - 4 \, a + 3\right )} c^{2} x^{4} - 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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