Optimal. Leaf size=112 \[ \frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {(21 a x+32) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]
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Rubi [A] time = 0.39, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6167, 6151, 1809, 833, 780, 217, 203} \[ \frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}+\frac {(21 a x+32) \sqrt {c-a^2 c x^2}}{24 a^3}-\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rule 6151
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {x^2 (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\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) \operatorname {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 = 0.11, size = 88, normalized size = 0.79 \[ \frac {21 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\left (6 a^3 x^3+16 a^2 x^2+21 a x+32\right ) \sqrt {c-a^2 c x^2}}{24 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 168, normalized size = 1.50 \[ \left [\frac {2 \, {\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, \frac {{\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 84, normalized size = 0.75 \[ \frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, x + \frac {8}{a}\right )} x + \frac {21}{a^{2}}\right )} x + \frac {32}{a^{3}}\right )} + \frac {7 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt {-c} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 186, normalized size = 1.66 \[ -\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}+\frac {9 x \sqrt {-a^{2} c \,x^{2}+c}}{8 a^{2}}+\frac {9 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 a^{2} \sqrt {a^{2} c}}-\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{3}}-\frac {2 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a^{2} \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 93, normalized size = 0.83 \[ \frac {9 \, \sqrt {-a^{2} c x^{2} + c} x}{8 \, a^{2}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{4 \, a^{2} c} - \frac {7 \, \sqrt {c} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} - \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{3} c} \]
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 \[ \int \frac {x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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