Optimal. Leaf size=137 \[ \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {3 (5 a x+8) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4} \]
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Rubi [A] time = 0.41, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {3 (5 a x+8) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4} \]
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^3 \sqrt {c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {x^3 (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x^3 \left (-9 a^2 c-10 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{5 a^2}\\ &=\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x^2 \left (30 a^3 c^2+36 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{20 a^4 c}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (-72 a^4 c^3-90 a^5 c^3 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{60 a^6 c^2}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {(3 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^3}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {(3 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 )}{4 a^3}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 96, normalized size = 0.70 \[ \frac {15 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\left (4 a^4 x^4+10 a^3 x^3+12 a^2 x^2+15 a x+24\right ) \sqrt {c-a^2 c x^2}}{20 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 184, normalized size = 1.34 \[ \left [\frac {2 \, {\left (4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} + 15 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{40 \, a^{4}}, \frac {{\left (4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} + 15 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{20 \, a^{4}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 210, normalized size = 1.53 \[ -\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 a^{2} c}-\frac {4 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,a^{4}}-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 a^{3} c}+\frac {5 x \sqrt {-a^{2} c \,x^{2}+c}}{4 a^{3}}+\frac {5 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{4 a^{3} \sqrt {a^{2} c}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{4}}-\frac {2 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a^{3} \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 117, normalized size = 0.85 \[ -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{5 \, a^{2} c} + \frac {5 \, \sqrt {-a^{2} c x^{2} + c} x}{4 \, a^{3}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{2 \, a^{3} c} - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{4 \, a^{4}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4}} - \frac {4 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{5 \, a^{4} c} \]
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 \[ \int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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