Optimal. Leaf size=172 \[ \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6286,
1823, 822, 372, 371} \begin {gather*} -\frac {c (2 m+3) \sqrt {1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{(m+1) (m+2) \sqrt {c-a^2 c x^2}}-\frac {2 a c \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{(m+2) \sqrt {c-a^2 c x^2}}+\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{m+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 822
Rule 1823
Rule 6286
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {x^m (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}+\frac {\int \frac {x^m \left (-a^2 c (3+2 m)-2 a^3 c (2+m) x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 (2+m)}\\ &=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-(2 a c) \int \frac {x^{1+m}}{\sqrt {c-a^2 c x^2}} \, dx-\frac {(c (3+2 m)) \int \frac {x^m}{\sqrt {c-a^2 c x^2}} \, dx}{2+m}\\ &=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {\left (2 a c \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}-\frac {\left (c (3+2 m) \sqrt {1-a^2 x^2}\right ) \int \frac {x^m}{\sqrt {1-a^2 x^2}} \, dx}{(2+m) \sqrt {c-a^2 c x^2}}\\ &=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}-\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.15, size = 129, normalized size = 0.75 \begin {gather*} \frac {x^{1+m} \left (\frac {2 \sqrt {1-a x} \sqrt {-c (1+a x)} F_1\left (1+m;\frac {1}{2},-\frac {1}{2};2+m;a x,-a x\right )}{\sqrt {-1+a x} \sqrt {1+a x}}+\frac {\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 )}{\sqrt {1-a^2 x^2}}\right )}{1+m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{a x -1}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \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\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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