Optimal. Leaf size=96 \[ \frac {(g x)^{m+1} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (\frac {e x}{d}+1\right )^{-n-p} F_1\left (m+1;-p,-n-p;m+2;\frac {e x}{d},-\frac {e x}{d}\right )}{g (m+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {892, 135, 133} \[ \frac {(g x)^{m+1} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (\frac {e x}{d}+1\right )^{-n-p} F_1\left (m+1;-p,-n-p;m+2;\frac {e x}{d},-\frac {e x}{d}\right )}{g (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 892
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, 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)^{n+p} \, dx\\ &=\left ((d+e x)^{-p} \left (1-\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d+e x)^{n+p} \left (1-\frac {e x}{d}\right )^p \, dx\\ &=\left ((d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-n-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m \left (1-\frac {e x}{d}\right )^p \left (1+\frac {e x}{d}\right )^{n+p} \, dx\\ &=\frac {(g x)^{1+m} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-n-p} \left (d^2-e^2 x^2\right )^p F_1\left (1+m;-p,-n-p;2+m;\frac {e x}{d},-\frac {e x}{d}\right )}{g (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 90, normalized size = 0.94 \[ \frac {x (g x)^m (d-e x)^p \left (\frac {d-e x}{d}\right )^{-p} (d+e x)^{n+p} \left (\frac {d+e x}{d}\right )^{-n-p} F_1\left (m+1;-p,-n-p;m+2;\frac {e x}{d},-\frac {e x}{d}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m}, x\right ) \]
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 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (g x \right )^{m} \left (e x +d \right )^{n} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx \]
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 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m}\,{d x} \]
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 \[ \int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^n \,d x \]
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 \[ \int \left (g x\right )^{m} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p} \left (d + e x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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