Optimal. Leaf size=40 \[ \frac{2 \sqrt{x} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{4};\frac{5}{4};-\frac{b x^2}{a}\right )}{a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0359386, antiderivative size = 49, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ 2 \sqrt{x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^p/Sqrt[x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.93054, size = 39, normalized size = 0.98 \[ 2 \sqrt{x} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**p/x**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0211848, size = 49, normalized size = 1.22 \[ 2 \sqrt{x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^p/Sqrt[x],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.028, size = 0, normalized size = 0. \[ \int{ \left ( b{x}^{2}+a \right ) ^{p}{\frac{1}{\sqrt{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^p/x^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**p/x**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}}{\sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^p/sqrt(x),x, algorithm="giac")
[Out]