Optimal. Leaf size=112 \[ \frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6287, 1823,
847, 794, 223, 209} \begin {gather*} \frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {7 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rule 6287
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx &=c \int \frac {x^2 (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x^2 \left (-7 a^2 c+8 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^2}\\ &=\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (-16 a^3 c^2+21 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{12 a^4 c}\\ &=\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {(7 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a^2}\\ &=\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {(7 c) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}\\ &=\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(32-21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 88, normalized size = 0.79 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (32-21 a x+16 a^2 x^2-6 a^3 x^3\right )-21 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{24 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 176, normalized size = 1.57
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+21 a x -32\right ) \left (a^{2} x^{2}-1\right ) c}{24 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {7 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{8 a^{2} \sqrt {c \,a^{2}}}\) | \(89\) |
default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,a^{2}}-\frac {9 \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4 a^{2}}-\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}+\frac {2 \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}+\frac {2 a c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {c \,a^{2}}}}{a^{3}}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 93, normalized size = 0.83 \begin {gather*} -\frac {9 \, \sqrt {-a^{2} c x^{2} + c} x}{8 \, a^{2}} + \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{4 \, a^{2} c} + \frac {7 \, \sqrt {c} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} - \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 168, normalized size = 1.50 \begin {gather*} \left [-\frac {2 \, {\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} - 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, -\frac {{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\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 \left (- \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x^{3} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs.
\(2 (92) = 184\).
time = 0.45, size = 221, normalized size = 1.97 \begin {gather*} -\frac {{\left (336 \, a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + \frac {{\left (75 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 83 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 21 \, a^{5} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - 77 \, a^{5} c^{3} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{4}}{c^{4}}\right )} {\left | a \right |}}{192 \, a^{9}} \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-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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