Optimal. Leaf size=157 \[ \frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2};-p,1;\frac {m+3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d (m+1)}-\frac {e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {m+2}{2};-p,1;\frac {m+4}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (m+2)} \]
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Rubi [A] time = 0.14, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 511, 510} \[ \frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2};-p,1;\frac {m+3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d (m+1)}-\frac {e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {m+2}{2};-p,1;\frac {m+4}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 959
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (a+c x^2\right )^p}{d+e x} \, dx &=\left (d x^{-m} (g x)^m\right ) \int \frac {x^m \left (a+c x^2\right )^p}{d^2-e^2 x^2} \, dx-\left (e x^{-m} (g x)^m\right ) \int \frac {x^{1+m} \left (a+c x^2\right )^p}{d^2-e^2 x^2} \, dx\\ &=\left (d x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^m \left (1+\frac {c x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx-\left (e x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^{1+m} \left (1+\frac {c x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx\\ &=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1+m}{2};-p,1;\frac {3+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d (1+m)}-\frac {e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {2+m}{2};-p,1;\frac {4+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (2+m)}\\ \end {align*}
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Mathematica [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{d+e x} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{e x + d}\,{d x} \]
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 \[ \int \frac {{\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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