Optimal. Leaf size=172 \[ \frac {c (2 m+5) x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+4) \sqrt {1-a^2 x^2}}+\frac {2 a c x^{m+2} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {1-a^2 x^2}}-\frac {x^{m+1} \left (c-a^2 c x^2\right )^{3/2}}{m+4} \]
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Rubi [A] time = 0.30, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6151, 1809, 808, 365, 364} \[ \frac {c (2 m+5) x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+4) \sqrt {1-a^2 x^2}}+\frac {2 a c x^{m+2} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {1-a^2 x^2}}-\frac {x^{m+1} \left (c-a^2 c x^2\right )^{3/2}}{m+4} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 1809
Rule 6151
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x^m (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{3/2}}{4+m}-\frac {\int x^m \left (-a^2 c (5+2 m)-2 a^3 c (4+m) x\right ) \sqrt {c-a^2 c x^2} \, dx}{a^2 (4+m)}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{3/2}}{4+m}+(2 a c) \int x^{1+m} \sqrt {c-a^2 c x^2} \, dx+\frac {(c (5+2 m)) \int x^m \sqrt {c-a^2 c x^2} \, dx}{4+m}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{3/2}}{4+m}+\frac {\left (2 a c \sqrt {c-a^2 c x^2}\right ) \int x^{1+m} \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}+\frac {\left (c (5+2 m) \sqrt {c-a^2 c x^2}\right ) \int x^m \sqrt {1-a^2 x^2} \, dx}{(4+m) \sqrt {1-a^2 x^2}}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{3/2}}{4+m}+\frac {c (5+2 m) x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (4+m) \sqrt {1-a^2 x^2}}+\frac {2 a c x^{2+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{(2+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 158, normalized size = 0.92 \[ \frac {c x^{m+1} \sqrt {c-a^2 c x^2} \left (2 a \left (m^2+4 m+3\right ) x \, _2F_1\left (-\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;a^2 x^2\right )+(m+2) \left (a^2 (m+1) x^2 \, _2F_1\left (-\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};a^2 x^2\right )+(m+3) \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )\right )\right )}{(m+1) (m+2) (m+3) \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {-a^{2} c x^{2} + c} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{2} x^{m} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{-a^{2} x^{2}+1}\, dx \]
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 \[ -\int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \]
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 \[ \int -\frac {x^m\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.26, size = 172, normalized size = 1.00 \[ \frac {a^{2} c^{\frac {3}{2}} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {a c^{\frac {3}{2}} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {c^{\frac {3}{2}} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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