Optimal. Leaf size=64 \[ -\frac {(d+e x)^{-2 p-3} \left (e x (d g (2 p+3)+e f)+d (d g (p+1)+e f)+e^2 g (p+2) x^2\right )^{p+1}}{e^2 (p+2)} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {786} \begin {gather*} -\frac {(d+e x)^{-2 p-3} \left (e x (d g (2 p+3)+e f)+d (d g (p+1)+e f)+e^2 g (p+2) x^2\right )^{p+1}}{e^2 (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx &=-\frac {(d+e x)^{-3-2 p} \left (d (e f+d g (1+p))+e (e f+d g (3+2 p)) x+e^2 g (2+p) x^2\right )^{1+p}}{e^2 (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 48, normalized size = 0.75 \begin {gather*} -\frac {(d+e x)^{-2 p-3} ((d+e x) (d g (p+1)+e (f+g (p+2) x)))^{p+1}}{e^2 (p+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 132, normalized size = 2.06 \begin {gather*} -\frac {{\left (d^{2} g p + d e f + d^{2} g + {\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} + {\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )} {\left (d^{2} g p + d e f + d^{2} g + {\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} + {\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{e^{2} p + 2 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 444, normalized size = 6.94 \begin {gather*} -\frac {g p x^{2} e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + 2 \, d g p x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + d^{2} g p e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + 2 \, g x^{2} e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + 3 \, d g x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + d^{2} g e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + f x e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 2\right )} + d f e^{\left (p \log \left (g p x e + d g p + 2 \, g x e + d g + f e\right ) - p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )}}{p e^{2} + 2 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 98, normalized size = 1.53 \begin {gather*} -\frac {\left (e g x p +d g p +2 e g x +d g +e f \right ) \left (e x +d \right )^{-2 p -2} \left (e^{2} g \,x^{2} p +2 d e g p x +2 e^{2} g \,x^{2}+d^{2} g p +3 d e g x +e^{2} f x +d^{2} g +d e f \right )^{p}}{\left (p +2\right ) e^{2}} \end {gather*}
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*} \int {\left (g x + f\right )} {\left (e^{2} g {\left (p + 2\right )} x^{2} + {\left (2 \, d g p + e f + 3 \, d g\right )} e x + {\left (d g p + e f + d g\right )} d\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 138, normalized size = 2.16 \begin {gather*} -{\left (d\,\left (d\,g+e\,f+d\,g\,p\right )+e\,x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )+e^2\,g\,x^2\,\left (p+2\right )\right )}^p\,\left (\frac {g\,x^2}{{\left (d+e\,x\right )}^{2\,p+3}}+\frac {d^2\,g+d\,e\,f+d^2\,g\,p}{e^2\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )}{e\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}\right ) \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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