Optimal. Leaf size=49 \[ \frac {x^{2-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,2;2-p;-\frac {b x^2}{a}\right )}{2 a (1-p)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac {x^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {b x^2}{a}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rubi steps
\begin {align*} \int x^{1-2 p} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^{1-2 p} \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {x^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {b x^2}{a}\right )}{2 (1-p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 61, normalized size = 1.24 \[ \frac {x^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {b x^2}{a}\right )}{2-2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int x^{-2 p +1} \left (b \,x^{2}+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{1-2\,p}\,{\left (b\,x^2+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________