Optimal. Leaf size=152 \[ \frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{2 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^4}{3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1+a x)}{a^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6295, 6285, 78}
\begin {gather*} \frac {4 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{a^2 \sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-a^2 x^2}}-\frac {3 x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6285
Rule 6295
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-3 \tanh ^{-1}(a x)} x \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x (1-a x)^2}{1+a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (\frac {4}{a}-3 x+a x^2-\frac {4}{a (1+a x)}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{2 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^4}{3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1+a x)}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 64, normalized size = 0.42 \begin {gather*} \frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\frac {4 x}{a}-\frac {3 x^2}{2}+\frac {a x^3}{3}-\frac {4 \log (1+a x)}{a^2}\right )}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 78, normalized size = 0.51
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{3} x^{3}+9 a^{2} x^{2}-24 a x +24 \ln \left (a x +1\right )\right )}{6 \left (a^{2} x^{2}-1\right ) a^{2}}\) | \(78\) |
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 = 0.43, size = 407, normalized size = 2.68 \begin {gather*} \left [\frac {12 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) - {\left (2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left (a^{5} x^{2} - a^{3}\right )}}, \frac {24 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) - {\left (2 \, a^{4} x^{4} - 9 \, a^{3} x^{3} + 24 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left (a^{5} x^{2} - a^{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 {x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{\left (a x + 1\right )^{3}}\, 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 {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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