Optimal. Leaf size=163 \[ \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+2}{2},1-p;\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]
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Rubi [A] time = 0.14, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {892, 82, 126, 365, 364} \[ \frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+2}{2},1-p;\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 82
Rule 126
Rule 364
Rule 365
Rule 892
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{d+e x} \, dx &=\left ((d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d-e x)^p (d+e x)^{-1+p} \, dx\\ &=\left (d (d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d-e x)^{-1+p} (d+e x)^{-1+p} \, dx-\frac {\left (e (d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^{1+m} (d-e x)^{-1+p} (d+e x)^{-1+p} \, dx}{g}\\ &=d \int (g x)^m \left (d^2-e^2 x^2\right )^{-1+p} \, dx-\frac {e \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p} \, dx}{g}\\ &=\frac {\left (\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d}-\frac {\left (e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2 g}\\ &=\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},1-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d g (1+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},1-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 124, normalized size = 0.76 \[ \frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d (m+2) \, _2F_1\left (\frac {m+1}{2},1-p;\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )-e (m+1) x \, _2F_1\left (\frac {m}{2}+1,1-p;\frac {m}{2}+2;\frac {e^2 x^2}{d^2}\right )\right )}{d^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{e x + d}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.15, size = 337, normalized size = 2.07 \[ - \frac {0^{p} d d^{2 p} g^{m} m x^{m} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 e^{2} x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} d d^{2 p} g^{m} x^{m} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 e^{2} x \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} + \frac {0^{p} d^{2 p} g^{m} m x^{m} \Phi \left (\frac {d^{2}}{e^{2} x^{2}}, 1, \frac {m e^{i \pi }}{2}\right ) \Gamma \left (- \frac {m}{2}\right )}{4 e \Gamma \left (1 - \frac {m}{2}\right )} + \frac {d e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p + \frac {1}{2} \\ - \frac {m}{2} - p + \frac {3}{2} \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {3}{2}\right )} - \frac {e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \relax (p) \Gamma \left (- \frac {m}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - \frac {m}{2} - p \\ - \frac {m}{2} - p + 1 \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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