Optimal. Leaf size=89 \[ \frac {(g x)^{m+1} \, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};a^2 x^2\right )}{g (m+1)}-\frac {a (g x)^{m+2} \, _2F_1\left (\frac {m+2}{2},1-p;\frac {m+4}{2};a^2 x^2\right )}{g^2 (m+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {890, 82, 125, 364} \[ \frac {(g x)^{m+1} \, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};a^2 x^2\right )}{g (m+1)}-\frac {a (g x)^{m+2} \, _2F_1\left (\frac {m+2}{2},1-p;\frac {m+4}{2};a^2 x^2\right )}{g^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 82
Rule 125
Rule 364
Rule 890
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (1-a^2 x^2\right )^p}{1+a x} \, dx &=\int (g x)^m (1-a x)^p (1+a x)^{-1+p} \, dx\\ &=-\frac {a \int (g x)^{1+m} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx}{g}+\int (g x)^m (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx\\ &=-\frac {a \int (g x)^{1+m} \left (1-a^2 x^2\right )^{-1+p} \, dx}{g}+\int (g x)^m \left (1-a^2 x^2\right )^{-1+p} \, dx\\ &=\frac {(g x)^{1+m} \, _2F_1\left (\frac {1+m}{2},1-p;\frac {3+m}{2};a^2 x^2\right )}{g (1+m)}-\frac {a (g x)^{2+m} \, _2F_1\left (\frac {2+m}{2},1-p;\frac {4+m}{2};a^2 x^2\right )}{g^2 (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.87 \[ x (g x)^m \left (\frac {\, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};a^2 x^2\right )}{m+1}-\frac {a x \, _2F_1\left (\frac {m}{2}+1,1-p;\frac {m}{2}+2;a^2 x^2\right )}{m+2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (g x\right )^{m}}{a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (g x\right )^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x \right )^{m} \left (-a^{2} x^{2}+1\right )^{p}}{a x +1}\, dx \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{p} \left (g x\right )^{m}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (g\,x\right )}^m\,{\left (1-a^2\,x^2\right )}^p}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.81, size = 308, normalized size = 3.46 \[ \frac {0^{p} g^{m} m x^{m} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {m e^{i \pi }}{2}\right ) \Gamma \left (- \frac {m}{2}\right )}{4 a \Gamma \left (1 - \frac {m}{2}\right )} - \frac {0^{p} g^{m} m x^{m} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 a^{2} x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} g^{m} x^{m} \Phi \left (\frac {1}{a^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 a^{2} x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} - \frac {a^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- \frac {m}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p \\ - \frac {m}{2} - p + 1 \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + 1\right )} + \frac {a^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p + \frac {1}{2} \\ - \frac {m}{2} - p + \frac {3}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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