Optimal. Leaf size=49 \[ \frac{x^2 \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]
[Out]
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Rubi [A] time = 0.0702721, antiderivative size = 83, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (b x+c x^2\right )^{p+1} \left (-\frac{c x}{b}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{b+c x}{b}\right )}{2 c (p+1)}+\frac{\left (b x+c x^2\right )^{p+1}}{2 c (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x*(b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 10.7041, size = 48, normalized size = 0.98 \[ \frac{x x^{- p - 1} x^{p + 2} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 2 \\ p + 3 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{p + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**2+b*x)**p,x)
[Out]
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Mathematica [A] time = 0.0351655, size = 47, normalized size = 0.96 \[ \frac{x^2 (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+2;p+3;-\frac{c x}{b}\right )}{p+2} \]
Antiderivative was successfully verified.
[In] Integrate[x*(b*x + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int x \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2+b*x)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \left (x \left (b + c x\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2+b*x)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*x,x, algorithm="giac")
[Out]