Optimal. Leaf size=89 \[ \frac {2 c (1+a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (1+a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6278, 6275, 45}
\begin {gather*} \frac {2 c (a x+1)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (a x+1)^4 \sqrt {c-a^2 c x^2}}{4 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 )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (1-a x) (1+a x)^2 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int \left (2 (1+a x)^2-(1+a x)^3\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 c (1+a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (1+a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.64 \begin {gather*} -\frac {c x \sqrt {c-a^2 c x^2} \left (-12-6 a x+4 a^2 x^2+3 a^3 x^3\right )}{12 \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 64, normalized size = 0.72
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c x \left (3 a^{3} x^{3}+4 a^{2} x^{2}-6 a x -12\right )}{12 a^{2} x^{2}-12}\) | \(64\) |
gosper | \(\frac {x \left (3 a^{3} x^{3}+4 a^{2} x^{2}-6 a x -12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 66, normalized size = 0.74 \begin {gather*} -\frac {1}{3} \, a^{2} c^{\frac {3}{2}} x^{3} + c^{\frac {3}{2}} x + \frac {1}{4} \, {\left (a^{2} c^{\frac {3}{2}} x^{4} + 2 \, c^{\frac {3}{2}} x^{2} - \frac {4 \, \sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} c}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 68, normalized size = 0.76 \begin {gather*} \frac {{\left (3 \, a^{3} c x^{4} + 4 \, a^{2} c x^{3} - 6 \, a c x^{2} - 12 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{12 \, {\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 {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 [B]
time = 1.06, size = 55, normalized size = 0.62 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (-\frac {c\,a^3\,x^4}{4}-\frac {c\,a^2\,x^3}{3}+\frac {c\,a\,x^2}{2}+c\,x\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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