Optimal. Leaf size=85 \[ \frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {(3 a x+5) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
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Rubi [A] time = 0.25, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6167, 6151, 1809, 780, 217, 203} \[ \frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {(3 a x+5) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 1809
Rule 6151
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {x (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (-5 a^2 c-6 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{3 a^2}\\ &=\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {(5+3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {c \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a}\\ &=\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {(5+3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {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 )}{a}\\ &=\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {(5+3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 79, normalized size = 0.93 \[ \frac {\left (a^2 x^2+3 a x+5\right ) \sqrt {c-a^2 c x^2}+3 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 150, normalized size = 1.76 \[ \left [\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x + 5\right )} + 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{6 \, a^{2}}, \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x + 5\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 72, normalized size = 0.85 \[ \frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (x + \frac {3}{a}\right )} x + \frac {5}{a^{2}}\right )} + \frac {c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {-c} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 162, normalized size = 1.91 \[ -\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{2} c}+\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{a}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a \sqrt {a^{2} c}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{2}}-\frac {2 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 70, normalized size = 0.82 \[ \frac {\sqrt {-a^{2} c x^{2} + c} x}{a} - \frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{2} c} \]
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 \[ \int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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