3.1063 \(\int x^{-5-2 p} (a+b x^2)^p \, dx\)

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]

(b*(a + b*x^2)^(1 + p))/(2*a^2*(1 + p)*(2 + p)*x^(2*(1 + p))) - (a + b*x^2)^(1 + p)/(2*a*(2 + p)*x^(2*(2 + p))
)

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Rubi [A]  time = 0.0202722, antiderivative size = 67, normalized size of antiderivative = 1., 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]

Int[x^(-5 - 2*p)*(a + b*x^2)^p,x]

[Out]

(b*(a + b*x^2)^(1 + p))/(2*a^2*(1 + p)*(2 + p)*x^(2*(1 + p))) - (a + b*x^2)^(1 + p)/(2*a*(2 + p)*x^(2*(2 + p))
)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 = 0.013967, 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]

Integrate[x^(-5 - 2*p)*(a + b*x^2)^p,x]

[Out]

-((a + b*x^2)^p*Hypergeometric2F1[-2 - p, -p, -1 - p, -((b*x^2)/a)])/(2*(2 + p)*x^(2*(2 + p))*(1 + (b*x^2)/a)^
p)

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Maple [A]  time = 0.003, size = 45, normalized size = 0.7 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) ^{1+p}{x}^{-4-2\,p} \left ( -b{x}^{2}+ap+a \right ) }{ \left ( 4+2\,p \right ) \left ( 1+p \right ){a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-5-2*p)*(b*x^2+a)^p,x)

[Out]

-1/2*(b*x^2+a)^(1+p)*x^(-4-2*p)*(-b*x^2+a*p+a)/(2+p)/(1+p)/a^2

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Maxima [A]  time = 1.49775, size = 80, normalized size = 1.19 \begin{align*} \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 \left (x\right )\right )}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-5-2*p)*(b*x^2+a)^p,x, algorithm="maxima")

[Out]

1/2*(b^2*x^4 - a*b*p*x^2 - a^2*(p + 1))*e^(p*log(b*x^2 + a) - 2*p*log(x))/((p^2 + 3*p + 2)*a^2*x^4)

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Fricas [A]  time = 1.61193, size = 135, normalized size = 2.01 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-5-2*p)*(b*x^2+a)^p,x, algorithm="fricas")

[Out]

1/2*(b^2*x^5 - a*b*p*x^3 - (a^2*p + a^2)*x)*(b*x^2 + a)^p*x^(-2*p - 5)/(a^2*p^2 + 3*a^2*p + 2*a^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-5-2*p)*(b*x**2+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-5-2*p)*(b*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*x^(-2*p - 5), x)