Optimal. Leaf size=67 \[ \frac {b x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (a+b x^2\right )^{p+1}}{2 a (p+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {271, 264} \[ \frac {b x^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a^2 (p+1) (p+2)}-\frac {x^{-2 (p+2)} \left (a+b x^2\right )^{p+1}}{2 a (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rubi steps
\begin {align*} \int x^{-5-2 p} \left (a+b x^2\right )^p \, dx &=-\frac {x^{-2 (2+p)} \left (a+b x^2\right )^{1+p}}{2 a (2+p)}-\frac {b \int x^{-3-2 p} \left (a+b x^2\right )^p \, dx}{a (2+p)}\\ &=\frac {b x^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a^2 (1+p) (2+p)}-\frac {x^{-2 (2+p)} \left (a+b x^2\right )^{1+p}}{2 a (2+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 62, normalized size = 0.93 \[ -\frac {x^{-2 (p+2)} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p-2,-p;-p-1;-\frac {b x^2}{a}\right )}{2 (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 67, normalized size = 1.00 \[ \frac {{\left (b^{2} x^{5} - a b p x^{3} - {\left (a^{2} p + a^{2}\right )} x\right )} {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 5}}{2 \, {\left (a^{2} p^{2} + 3 \, a^{2} p + 2 \, a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 45, normalized size = 0.67 \[ -\frac {\left (-b \,x^{2}+a p +a \right ) x^{-2 p -4} \left (b \,x^{2}+a \right )^{p +1}}{2 \left (p +2\right ) \left (p +1\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.41, size = 59, normalized size = 0.88 \[ \frac {{\left (b^{2} x^{4} - a b p x^{2} - a^{2} {\left (p + 1\right )}\right )} e^{\left (p \log \left (b x^{2} + a\right ) - 2 \, p \log \relax (x)\right )}}{2 \, {\left (p^{2} + 3 \, p + 2\right )} a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.02, size = 96, normalized size = 1.43 \[ -{\left (b\,x^2+a\right )}^p\,\left (\frac {x\,\left (p+1\right )}{2\,x^{2\,p+5}\,\left (p^2+3\,p+2\right )}-\frac {b^2\,x^5}{2\,a^2\,x^{2\,p+5}\,\left (p^2+3\,p+2\right )}+\frac {b\,p\,x^3}{2\,a\,x^{2\,p+5}\,\left (p^2+3\,p+2\right )}\right ) \]
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]
________________________________________________________________________________________