Optimal. Leaf size=24 \[ \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {245} \[ \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2} \, dx &=\frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (n - 1\right )} \int \frac {1}{a b n x^{n} + a^{2} n}\,{d x} + \frac {x}{a b n x^{n} + a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.17, size = 25, normalized size = 1.04 \[ \frac {x\,{{}}_2{\mathrm {F}}_1\left (2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.16, size = 257, normalized size = 10.71 \[ \frac {n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________