Optimal. Leaf size=174 \[ \frac {c x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a c x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}}-\frac {a^2 c x^{m+3} \sqrt {c-a^2 c x^2}}{(m+3) \sqrt {1-a^2 x^2}}-\frac {a^3 c x^{m+4} \sqrt {c-a^2 c x^2}}{(m+4) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 75} \[ \frac {c x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a c x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}}-\frac {a^2 c x^{m+3} \sqrt {c-a^2 c x^2}}{(m+3) \sqrt {1-a^2 x^2}}-\frac {a^3 c x^{m+4} \sqrt {c-a^2 c x^2}}{(m+4) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int x^m (1-a x) (1+a x)^2 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int \left (x^m+a x^{1+m}-a^2 x^{2+m}-a^3 x^{3+m}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {c x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}+\frac {a c x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}-\frac {a^2 c x^{3+m} \sqrt {c-a^2 c x^2}}{(3+m) \sqrt {1-a^2 x^2}}-\frac {a^3 c x^{4+m} \sqrt {c-a^2 c x^2}}{(4+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 84, normalized size = 0.48 \[ \frac {c x^{m+1} \sqrt {c-a^2 c x^2} \left ((2 m+5) \left (\frac {a^2 x^2}{m+3}+\frac {2 a x}{m+2}+\frac {1}{m+1}\right )-(a x+1)^3\right )}{(m+4) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 211, normalized size = 1.21 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} c m^{3} + 6 \, a^{3} c m^{2} + 11 \, a^{3} c m + 6 \, a^{3} c\right )} x^{4} + {\left (a^{2} c m^{3} + 7 \, a^{2} c m^{2} + 14 \, a^{2} c m + 8 \, a^{2} c\right )} x^{3} - {\left (a c m^{3} + 8 \, a c m^{2} + 19 \, a c m + 12 \, a c\right )} x^{2} - {\left (c m^{3} + 9 \, c m^{2} + 26 \, c m + 24 \, c\right )} x\right )} \sqrt {-a^{2} x^{2} + 1} x^{m}}{m^{4} + 10 \, m^{3} - {\left (a^{2} m^{4} + 10 \, a^{2} m^{3} + 35 \, a^{2} m^{2} + 50 \, a^{2} m + 24 \, a^{2}\right )} x^{2} + 35 \, m^{2} + 50 \, m + 24} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 180, normalized size = 1.03 \[ \frac {x^{1+m} \left (a^{3} m^{3} x^{3}+6 a^{3} m^{2} x^{3}+11 a^{3} m \,x^{3}+a^{2} m^{3} x^{2}+6 x^{3} a^{3}+7 a^{2} m^{2} x^{2}+14 a^{2} m \,x^{2}-a \,m^{3} x +8 a^{2} x^{2}-8 a \,m^{2} x -19 a m x -m^{3}-12 a x -9 m^{2}-26 m -24\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \left (a x -1\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 80, normalized size = 0.46 \[ -\frac {{\left (a^{2} c^{\frac {3}{2}} {\left (m + 2\right )} x^{4} - c^{\frac {3}{2}} {\left (m + 4\right )} x^{2}\right )} a x^{m}}{m^{2} + 6 \, m + 8} - \frac {{\left (a^{2} c^{\frac {3}{2}} {\left (m + 1\right )} x^{3} - c^{\frac {3}{2}} {\left (m + 3\right )} x\right )} x^{m}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 228, normalized size = 1.31 \[ \frac {x^m\,\left (\frac {c\,x\,\sqrt {c-a^2\,c\,x^2}\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,c\,x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^3\,c\,x^4\,\sqrt {c-a^2\,c\,x^2}\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^2\,c\,x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {x^{m} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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