Optimal. Leaf size=80 \[ \frac {c^2 x^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}+\frac {a c^2 x^{m+2} \, _2F_1\left (-\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2} \]
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Rubi [A] time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6148, 808, 364} \[ \frac {c^2 x^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}+\frac {a c^2 x^{m+2} \, _2F_1\left (-\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2} \]
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 )^2 \, dx &=c^2 \int x^m (1+a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=c^2 \int x^m \left (1-a^2 x^2\right )^{3/2} \, dx+\left (a c^2\right ) \int x^{1+m} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {c^2 x^{1+m} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{1+m}+\frac {a c^2 x^{2+m} \, _2F_1\left (-\frac {3}{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 = 82, normalized size = 1.02 \[ c^2 \left (\frac {x^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+1}{2}+1;a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \, _2F_1\left (-\frac {3}{2},\frac {m+2}{2};\frac {m+2}{2}+1;a^2 x^2\right )}{m+2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} x^{m}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 227, normalized size = 2.84 \[ \frac {a^{5} c^{2} x^{6+m} \hypergeom \left (\left [\frac {1}{2}, 3+\frac {m}{2}\right ], \left [4+\frac {m}{2}\right ], a^{2} x^{2}\right )}{6+m}-\frac {2 a^{3} c^{2} x^{4+m} \hypergeom \left (\left [\frac {1}{2}, 2+\frac {m}{2}\right ], \left [3+\frac {m}{2}\right ], a^{2} x^{2}\right )}{4+m}+\frac {a \,c^{2} x^{2+m} \hypergeom \left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], a^{2} x^{2}\right )}{2+m}+\frac {c^{2} a^{4} x^{5+m} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{2}+\frac {m}{2}\right ], \left [\frac {7}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{5+m}-\frac {2 c^{2} a^{2} x^{3+m} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {5}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{3+m}+\frac {c^{2} x^{1+m} \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{1+m} \]
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 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\left (c-a^2\,c\,x^2\right )}^2\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.39, size = 223, normalized size = 2.79 \[ - \frac {a^{3} c^{2} x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} - \frac {a^{2} c^{2} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {a c^{2} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {c^{2} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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