Optimal. Leaf size=74 \[ \frac {x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6288, 6285, 45}
\begin {gather*} \frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{\tanh ^{-1}(a x)} x^2 \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x^2 (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (x^2+a x^3\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.57 \begin {gather*} \frac {x^3 (4+3 a x) \sqrt {c-a^2 c x^2}}{12 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 49, normalized size = 0.66
method | result | size |
gosper | \(\frac {x^{3} \left (3 a x +4\right ) \sqrt {-a^{2} c \,x^{2}+c}}{12 \sqrt {-a^{2} x^{2}+1}}\) | \(37\) |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, x^{3} \left (3 a x +4\right )}{12 \left (a^{2} x^{2}-1\right )}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 50, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x^{4} + 4 \, x^{3}\right )} \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 {x^{2} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 38, normalized size = 0.51 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a\,x^4}{4}+\frac {x^3}{3}\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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