Optimal. Leaf size=48 \[ \frac {x^2 \, _2F_1\left (1,\frac {2}{n}-\frac {1}{2};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a \sqrt {a+b x^n}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {365, 364} \[ \frac {x^2 \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^n\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {x}{\left (1+\frac {b x^n}{a}\right )^{3/2}} \, dx}{a \sqrt {a+b x^n}}\\ &=\frac {x^2 \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {3}{2},\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 a \sqrt {a+b x^n}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 60, normalized size = 1.25 \[ \frac {x^2 \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {2}{n};1+\frac {2}{n};-\frac {b x^n}{a}\right )}{2 a \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b \,x^{n}+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.41, size = 41, normalized size = 0.85 \[ \frac {x^{2} \Gamma \left (\frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {2}{n} \\ 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac {3}{2}} n \Gamma \left (1 + \frac {2}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________