Optimal. Leaf size=70 \[ \frac {1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {b^2 x^{2 n}}{a^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {366, 365, 364} \[ \frac {1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {b^2 x^{2 n}}{a^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rule 366
Rubi steps
\begin {align*} \int x \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \, dx &=\left (\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (-a^2+b^2 x^{2 n}\right )^{-p}\right ) \int x \left (-a^2+b^2 x^{2 n}\right )^p \, dx\\ &=\left (\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p}\right ) \int x \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^p \, dx\\ &=\frac {1}{2} x^2 \left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {b^2 x^{2 n}}{a^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 72, normalized size = 1.03 \[ \frac {1}{2} x^2 \left (b x^n-a\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {b^2 x^{2 n}}{a^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int x \left (b \,x^{n}+a \right )^{p} \left (b \,x^{n}-a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,x^n\right )}^p\,{\left (b\,x^n-a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (- a + b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________