Optimal. Leaf size=176 \[ -\frac {x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+\frac {c^2 (7+2 m) x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (6+m) \sqrt {1-a^2 x^2}}+\frac {2 a c^2 x^{2+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{(2+m) \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 176, 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 = {6286, 1823,
822, 372, 371} \begin {gather*} \frac {c^2 (2 m+7) x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+6) \sqrt {1-a^2 x^2}}+\frac {2 a c^2 x^{m+2} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{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 )^{5/2}}{m+6} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 822
Rule 1823
Rule 6286
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x^m (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}-\frac {\int x^m \left (-a^2 c (7+2 m)-2 a^3 c (6+m) x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{a^2 (6+m)}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+(2 a c) \int x^{1+m} \left (c-a^2 c x^2\right )^{3/2} \, dx+\frac {(c (7+2 m)) \int x^m \left (c-a^2 c x^2\right )^{3/2} \, dx}{6+m}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+\frac {\left (2 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x^{1+m} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}+\frac {\left (c^2 (7+2 m) \sqrt {c-a^2 c x^2}\right ) \int x^m \left (1-a^2 x^2\right )^{3/2} \, dx}{(6+m) \sqrt {1-a^2 x^2}}\\ &=-\frac {x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+\frac {c^2 (7+2 m) x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (6+m) \sqrt {1-a^2 x^2}}+\frac {2 a c^2 x^{2+m} \sqrt {c-a^2 c x^2} \, _2F_1\left (-\frac {3}{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 = 0.28, size = 180, normalized size = 1.02 \begin {gather*} \frac {c^2 x^{1+m} \sqrt {c-a^2 c x^2} \left (\frac {2 a x \, _2F_1\left (-\frac {1}{2},1+\frac {m}{2};2+\frac {m}{2};a^2 x^2\right )}{2+m}-\frac {2 a^3 x^3 \, _2F_1\left (-\frac {1}{2},2+\frac {m}{2};3+\frac {m}{2};a^2 x^2\right )}{4+m}+\frac {\, _2F_1\left (-\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{1+m}-\frac {a^4 x^4 \, _2F_1\left (-\frac {1}{2},\frac {5+m}{2};\frac {7+m}{2};a^2 x^2\right )}{5+m}\right )}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right )^{2} x^{m} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{-a^{2} x^{2}+1}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 16.02, size = 226, normalized size = 1.28 \begin {gather*} - \frac {a^{4} c^{\frac {5}{2}} x^{5} x^{m} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} - \frac {a^{3} c^{\frac {5}{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 )}}{\Gamma \left (\frac {m}{2} + 3\right )} + \frac {a c^{\frac {5}{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 {5}{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 )} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int -\frac {x^m\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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