Optimal. Leaf size=113 \[ -\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1+p)} \]
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Rubi [A]
time = 0.01, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {638}
\begin {gather*} -\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 638
Rubi steps
\begin {align*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 96, normalized size = 0.85 \begin {gather*} \frac {(a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \, _2F_1\left (-p,1+p;2+p;\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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