Optimal. Leaf size=61 \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{3}{2},-p;p-\frac{1}{2};-\frac{c x}{b}\right )}{(3-2 p) (d x)^{5/2}} \]
[Out]
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Rubi [A] time = 0.0745717, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{3}{2},-p;p-\frac{1}{2};-\frac{c x}{b}\right )}{(3-2 p) (d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^p/(d*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.0486, size = 63, normalized size = 1.03 \[ - \frac{2 x^{- p + \frac{5}{2}} x^{p - \frac{3}{2}} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p - \frac{3}{2} \\ p - \frac{1}{2} \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{\left (d x\right )^{\frac{5}{2}} \left (- 2 p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**p/(d*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0673107, size = 59, normalized size = 0.97 \[ \frac{2 x (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (p-\frac{3}{2},-p;p-\frac{1}{2};-\frac{c x}{b}\right )}{(2 p-3) (d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^p/(d*x)^(5/2),x]
[Out]
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Maple [F] time = 0.032, size = 0, normalized size = 0. \[ \int{ \left ( c{x}^{2}+bx \right ) ^{p} \left ( dx \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^p/(d*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{p}}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p/(d*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x\right )}^{p}}{\sqrt{d x} d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p/(d*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{p}}{\left (d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**p/(d*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{p}}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p/(d*x)^(5/2),x, algorithm="giac")
[Out]