Optimal. Leaf size=92 \[ x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} (d+e x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} (d f-e f x)^m F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {519, 430, 429} \[ x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} (d+e x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} (d f-e f x)^m F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 519
Rubi steps
\begin {align*} \int (d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \, dx &=\left ((d+e x)^m (d f-e f x)^m \left (d^2 f-e^2 f x^2\right )^{-m}\right ) \int \left (a+c x^2\right )^p \left (d^2 f-e^2 f x^2\right )^m \, dx\\ &=\left ((d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d^2 f-e^2 f x^2\right )^{-m}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (d^2 f-e^2 f x^2\right )^m \, dx\\ &=\left ((d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^m \, dx\\ &=x (d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )\\ \end {align*}
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Mathematica [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int (d+e x)^m (d f-e f x)^m \left (a+c x^2\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-e f x + d f\right )}^{m} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int {\left (-e f x + d f\right )}^{m} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{m} \left (-e f x +d f \right )^{m}\, dx \]
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 \[ \int {\left (-e f x + d f\right )}^{m} {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
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 \[ \int {\left (d\,f-e\,f\,x\right )}^m\,{\left (c\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^m \,d x \]
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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