Optimal. Leaf size=39 \[ \frac{x \left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac{5}{2}+\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0329826, antiderivative size = 49, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a x \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.53029, size = 42, normalized size = 1.08 \[ \frac{a x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{\sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.121871, size = 94, normalized size = 2.41 \[ \frac{x \left (3 a^2 n^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+2 \left (a+b x^n\right ) \left (a (4 n+2)+b (n+2) x^n\right )\right )}{(n+2) (3 n+2) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2),x, algorithm="giac")
[Out]