Optimal. Leaf size=134 \[ \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{c^2 (m+1) \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (3,\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{c^2 (m+2) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6153, 6150, 82, 73, 364} \[ \frac {\sqrt {1-a^2 x^2} x^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{c^2 (m+1) \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} x^{m+2} \, _2F_1\left (3,\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{c^2 (m+2) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 73
Rule 82
Rule 364
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^m}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^m}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^m}{(1-a x)^3 (1+a x)^3} \, dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{(1-a x)^3 (1+a x)^3} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^m}{\left (1-a^2 x^2\right )^3} \, dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x^{1+m}}{\left (1-a^2 x^2\right )^3} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x^{1+m} \sqrt {1-a^2 x^2} \, _2F_1\left (3,\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{c^2 (1+m) \sqrt {c-a^2 c x^2}}+\frac {a x^{2+m} \sqrt {1-a^2 x^2} \, _2F_1\left (3,\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{c^2 (2+m) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 107, normalized size = 0.80 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+1}{2}+1;a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \, _2F_1\left (3,\frac {m+2}{2};\frac {m+2}{2}+1;a^2 x^2\right )}{m+2}\right )}{c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{m}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{m}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{m}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{m} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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