Optimal. Leaf size=76 \[ \frac {c x^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}+\frac {a c x^{m+2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2} \]
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Rubi [A] time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6148, 808, 364} \[ \frac {c x^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}+\frac {a c x^{m+2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 6148
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx &=c \int x^m (1+a x) \sqrt {1-a^2 x^2} \, dx\\ &=c \int x^m \sqrt {1-a^2 x^2} \, dx+(a c) \int x^{1+m} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c x^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{1+m}+\frac {a c x^{2+m} \, _2F_1\left (-\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{2+m}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.95 \[ c x^{m+1} \left (\frac {a x \, _2F_1\left (-\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;a^2 x^2\right )}{m+2}+\frac {\, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} x^{2} + 1} {\left (a c x + c\right )} x^{m}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 143, normalized size = 1.88 \[ -\frac {a^{3} c \,x^{4+m} \hypergeom \left (\left [\frac {1}{2}, 2+\frac {m}{2}\right ], \left [3+\frac {m}{2}\right ], a^{2} x^{2}\right )}{4+m}+\frac {a c \,x^{2+m} \hypergeom \left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], a^{2} x^{2}\right )}{2+m}-\frac {c \,a^{2} x^{3+m} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {5}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{3+m}+\frac {c \,x^{1+m} \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{1+m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (c-a^2\,c\,x^2\right )\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.50, size = 104, normalized size = 1.37 \[ \frac {a c x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {c x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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