Optimal. Leaf size=274 \[ \frac {c^2 x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a c^2 x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}}-\frac {2 a^2 c^2 x^{m+3} \sqrt {c-a^2 c x^2}}{(m+3) \sqrt {1-a^2 x^2}}+\frac {a^5 c^2 x^{m+6} \sqrt {c-a^2 c x^2}}{(m+6) \sqrt {1-a^2 x^2}}+\frac {a^4 c^2 x^{m+5} \sqrt {c-a^2 c x^2}}{(m+5) \sqrt {1-a^2 x^2}}-\frac {2 a^3 c^2 x^{m+4} \sqrt {c-a^2 c x^2}}{(m+4) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 274, 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, 88} \[ \frac {c^2 x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a c^2 x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}}-\frac {2 a^2 c^2 x^{m+3} \sqrt {c-a^2 c x^2}}{(m+3) \sqrt {1-a^2 x^2}}-\frac {2 a^3 c^2 x^{m+4} \sqrt {c-a^2 c x^2}}{(m+4) \sqrt {1-a^2 x^2}}+\frac {a^4 c^2 x^{m+5} \sqrt {c-a^2 c x^2}}{(m+5) \sqrt {1-a^2 x^2}}+\frac {a^5 c^2 x^{m+6} \sqrt {c-a^2 c x^2}}{(m+6) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \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)} x^m \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 x^m (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 (x^m+a x^{1+m}-2 a^2 x^{2+m}-2 a^3 x^{3+m}+a^4 x^{4+m}+a^5 x^{5+m}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {c^2 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}+\frac {a c^2 x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}-\frac {2 a^2 c^2 x^{3+m} \sqrt {c-a^2 c x^2}}{(3+m) \sqrt {1-a^2 x^2}}-\frac {2 a^3 c^2 x^{4+m} \sqrt {c-a^2 c x^2}}{(4+m) \sqrt {1-a^2 x^2}}+\frac {a^4 c^2 x^{5+m} \sqrt {c-a^2 c x^2}}{(5+m) \sqrt {1-a^2 x^2}}+\frac {a^5 c^2 x^{6+m} \sqrt {c-a^2 c x^2}}{(6+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 102, normalized size = 0.37 \[ \frac {c^2 x^{m+1} \sqrt {c-a^2 c x^2} \left (\frac {a^5 x^5}{m+6}+\frac {a^4 x^4}{m+5}-\frac {2 a^3 x^3}{m+4}-\frac {2 a^2 x^2}{m+3}+\frac {a x}{m+2}+\frac {1}{m+1}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 476, normalized size = 1.74 \[ \frac {{\left ({\left (a^{5} c^{2} m^{5} + 15 \, a^{5} c^{2} m^{4} + 85 \, a^{5} c^{2} m^{3} + 225 \, a^{5} c^{2} m^{2} + 274 \, a^{5} c^{2} m + 120 \, a^{5} c^{2}\right )} x^{6} + {\left (a^{4} c^{2} m^{5} + 16 \, a^{4} c^{2} m^{4} + 95 \, a^{4} c^{2} m^{3} + 260 \, a^{4} c^{2} m^{2} + 324 \, a^{4} c^{2} m + 144 \, a^{4} c^{2}\right )} x^{5} - 2 \, {\left (a^{3} c^{2} m^{5} + 17 \, a^{3} c^{2} m^{4} + 107 \, a^{3} c^{2} m^{3} + 307 \, a^{3} c^{2} m^{2} + 396 \, a^{3} c^{2} m + 180 \, a^{3} c^{2}\right )} x^{4} - 2 \, {\left (a^{2} c^{2} m^{5} + 18 \, a^{2} c^{2} m^{4} + 121 \, a^{2} c^{2} m^{3} + 372 \, a^{2} c^{2} m^{2} + 508 \, a^{2} c^{2} m + 240 \, a^{2} c^{2}\right )} x^{3} + {\left (a c^{2} m^{5} + 19 \, a c^{2} m^{4} + 137 \, a c^{2} m^{3} + 461 \, a c^{2} m^{2} + 702 \, a c^{2} m + 360 \, a c^{2}\right )} x^{2} + {\left (c^{2} m^{5} + 20 \, c^{2} m^{4} + 155 \, c^{2} m^{3} + 580 \, c^{2} m^{2} + 1044 \, c^{2} m + 720 \, c^{2}\right )} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} - {\left (a^{2} m^{6} + 21 \, a^{2} m^{5} + 175 \, a^{2} m^{4} + 735 \, a^{2} m^{3} + 1624 \, a^{2} m^{2} + 1764 \, a^{2} m + 720 \, a^{2}\right )} x^{2} + 1624 \, m^{2} + 1764 \, m + 720} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 377, normalized size = 1.38 \[ \frac {x^{1+m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} m \,x^{5}+95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 x^{5} a^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} m \,x^{4}-214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 x^{4} a^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} m \,x^{3}-242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 x^{3} a^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a m x +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 144, normalized size = 0.53 \[ \frac {{\left ({\left (m^{2} + 6 \, m + 8\right )} a^{4} c^{\frac {5}{2}} x^{6} - 2 \, {\left (m^{2} + 8 \, m + 12\right )} a^{2} c^{\frac {5}{2}} x^{4} + {\left (m^{2} + 10 \, m + 24\right )} c^{\frac {5}{2}} x^{2}\right )} a x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} + \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} a^{4} c^{\frac {5}{2}} x^{5} - 2 \, {\left (m^{2} + 6 \, m + 5\right )} a^{2} c^{\frac {5}{2}} x^{3} + {\left (m^{2} + 8 \, m + 15\right )} c^{\frac {5}{2}} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 468, normalized size = 1.71 \[ \frac {x^m\,\left (\frac {c^2\,x\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,c^2\,x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,c^2\,x^6\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,c^2\,x^5\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,c^2\,x^4\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,c^2\,x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right )}{\sqrt {1-a^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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