Optimal. Leaf size=72 \[ -\frac {\left (a^2-b^2 x^{2 n}\right ) \left (b x^n-a\right )^p \left (a+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {267, 126, 266, 65} \[ -\frac {\left (a^2-b^2 x^{2 n}\right ) \left (b x^n-a\right )^p \left (a+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 126
Rule 266
Rule 267
Rubi steps
\begin {align*} \int \frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+b x)^p (a+b x)^p}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\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 ) \operatorname {Subst}\left (\int \frac {\left (-a^2+b^2 x^2\right )^p}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\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 ) \operatorname {Subst}\left (\int \frac {\left (-a^2+b^2 x\right )^p}{x} \, dx,x,x^{2 n}\right )}{2 n}\\ &=-\frac {\left (-a+b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right ) \, _2F_1\left (1,1+p;2+p;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 73, normalized size = 1.01 \[ \frac {\left (b^2 x^{2 n}-a^2\right ) \left (b x^n-a\right )^p \left (a+b x^n\right )^p \, _2F_1\left (1,p+1;p+2;1-\frac {b^2 x^{2 n}}{a^2}\right )}{2 a^2 n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{p} \left (b \,x^{n}-a \right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{p} {\left (b x^{n} - a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^p\,{\left (b\,x^n-a\right )}^p}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- a + b x^{n}\right )^{p} \left (a + b x^{n}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________