Optimal. Leaf size=136 \[ \frac {c^2 (a x+1)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}-\frac {4 c^2 (a x+1)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {c^2 (a x+1)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6143, 6140, 43} \[ \frac {c^2 (a x+1)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}-\frac {4 c^2 (a x+1)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {c^2 (a x+1)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^2 (1+a x)^3 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (4 (1+a x)^3-4 (1+a x)^4+(1+a x)^5\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {c^2 (1+a x)^4 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {4 c^2 (1+a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {c^2 (1+a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.44 \[ \frac {c^2 (a x+1)^4 \left (5 a^2 x^2-14 a x+11\right ) \sqrt {c-a^2 c x^2}}{30 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 98, normalized size = 0.72 \[ -\frac {{\left (5 \, a^{5} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} + 30 \, c^{2} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 81, normalized size = 0.60 \[ \frac {x \left (5 x^{5} a^{5}+6 x^{4} a^{4}-15 x^{3} a^{3}-20 a^{2} x^{2}+15 a x +30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{30 \left (a x -1\right )^{2} \left (a x +1\right )^{2} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 111, normalized size = 0.82 \[ \frac {1}{5} \, a^{4} c^{\frac {5}{2}} x^{5} - \frac {2}{3} \, a^{2} c^{\frac {5}{2}} x^{3} + c^{\frac {5}{2}} x - \frac {1}{6} \, {\left (\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2} c^{2} x^{4} - 2 \, a^{2} c^{\frac {5}{2}} x^{4} - 4 \, c^{\frac {5}{2}} x^{2} + \frac {7 \, \sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} c^{2}}{a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 85, normalized size = 0.62 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a^5\,c^2\,x^6}{6}+\frac {a^4\,c^2\,x^5}{5}-\frac {a^3\,c^2\,x^4}{2}-\frac {2\,a^2\,c^2\,x^3}{3}+\frac {a\,c^2\,x^2}{2}+c^2\,x\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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