Optimal. Leaf size=39 \[ \frac {x \, _2F_1\left (1,\frac {1}{n}-\frac {3}{2};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \left (a+b x^n\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {5}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 246
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {1}{\left (1+\frac {b x^n}{a}\right )^{5/2}} \, dx}{a^2 \sqrt {a+b x^n}}\\ &=\frac {x \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {5}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 51, normalized size = 1.31 \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {5}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}} \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.36, size = 43, normalized size = 1.10 \[ \frac {x\,{\left (\frac {b\,x^n}{a}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.37, size = 39, normalized size = 1.00 \[ \frac {x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac {5}{2}} n \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________