Optimal. Leaf size=140 \[ -\frac {c^2 (1-a x)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {4 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 140, 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 = {6278, 6275, 45}
\begin {gather*} -\frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}+\frac {4 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (1-a x)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^3 (1+a x)^2 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (4 (1-a x)^3-4 (1-a x)^4+(1-a x)^5\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {c^2 (1-a x)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {4 c^2 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (1-a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.43 \begin {gather*} -\frac {c^2 (-1+a x)^4 \left (11+14 a x+5 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{30 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 82, normalized size = 0.59
method | result | size |
gosper | \(\frac {x \left (5 x^{5} a^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {-a^{2} x^{2}+1}}{30 \left (a x -1\right )^{3} \left (a x +1\right )^{3}}\) | \(81\) |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, c^{2} x \left (5 x^{5} a^{5}-6 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}+15 a x -30\right )}{30 a^{2} x^{2}-30}\) | \(82\) |
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 = 0.37, size = 98, normalized size = 0.70 \begin {gather*} \frac {{\left (5 \, a^{5} c^{2} x^{6} - 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} - 30 \, c^{2} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{2} x^{2} - 1\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 (a x - 1\right ) \left (a x + 1\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 = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,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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