Optimal. Leaf size=67 \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]
[Out]
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Rubi [A] time = 0.0864399, antiderivative size = 67, 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 \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 13.7773, size = 71, normalized size = 1.06 \[ \frac{2 x^{- m - \frac{1}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ) \sqrt{b x + c x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{1}{2}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0421613, size = 59, normalized size = 0.88 \[ \frac{2 x \sqrt{x (b+c x)} (d x)^m \, _2F_1\left (-\frac{1}{2},m+\frac{3}{2};m+\frac{5}{2};-\frac{c x}{b}\right )}{(2 m+3) \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*Sqrt[b*x + c*x^2],x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{c{x}^{2}+bx}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2} + b x} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{x \left (b + c x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(d*x)^m,x, algorithm="giac")
[Out]