Optimal. Leaf size=55 \[ \frac{x (d x)^m \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]
[Out]
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Rubi [A] time = 0.0696622, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x (d x)^m \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*(b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 11.8825, size = 58, normalized size = 1.05 \[ \frac{x^{- m - p} x^{m + p + 1} \left (d x\right )^{m} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, m + p + 1 \\ m + p + 2 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{m + p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2+b*x)**p,x)
[Out]
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Mathematica [A] time = 0.0550313, size = 53, normalized size = 0.96 \[ \frac{x (d x)^m (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,m+p+1;m+p+2;-\frac{c x}{b}\right )}{m+p+1} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(b*x + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.137, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2+b*x)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \left (x \left (b + c x\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(c*x**2+b*x)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(d*x)^m,x, algorithm="giac")
[Out]