3.404 \(\int \frac{(a+b x^2)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]

[Out]

-(a + b*x^2)^(7/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi [A]  time = 0.0105772, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^10,x]

[Out]

-(a + b*x^2)^(7/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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 \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9}-\frac{(2 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{9 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0106238, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^2\right )^{7/2} \left (2 b x^2-7 a\right )}{63 a^2 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^10,x]

[Out]

((a + b*x^2)^(7/2)*(-7*a + 2*b*x^2))/(63*a^2*x^9)

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Maple [A]  time = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,b{x}^{2}+7\,a}{63\,{x}^{9}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^10,x)

[Out]

-1/63*(b*x^2+a)^(7/2)*(-2*b*x^2+7*a)/x^9/a^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60237, size = 130, normalized size = 2.95 \begin{align*} \frac{{\left (2 \, b^{4} x^{8} - a b^{3} x^{6} - 15 \, a^{2} b^{2} x^{4} - 19 \, a^{3} b x^{2} - 7 \, a^{4}\right )} \sqrt{b x^{2} + a}}{63 \, a^{2} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^10,x, algorithm="fricas")

[Out]

1/63*(2*b^4*x^8 - a*b^3*x^6 - 15*a^2*b^2*x^4 - 19*a^3*b*x^2 - 7*a^4)*sqrt(b*x^2 + a)/(a^2*x^9)

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Sympy [B]  time = 2.16759, size = 121, normalized size = 2.75 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{9 x^{8}} - \frac{19 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 x^{6}} - \frac{5 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{21 x^{4}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a x^{2}} + \frac{2 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**10,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(9*x**8) - 19*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(63*x**6) - 5*b**(5/2)*sqrt(a
/(b*x**2) + 1)/(21*x**4) - b**(7/2)*sqrt(a/(b*x**2) + 1)/(63*a*x**2) + 2*b**(9/2)*sqrt(a/(b*x**2) + 1)/(63*a**
2)

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Giac [B]  time = 1.69998, size = 297, normalized size = 6.75 \begin{align*} \frac{4 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} b^{\frac{9}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{2} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{3} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{4} b^{\frac{9}{2}} + 27 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{5} b^{\frac{9}{2}} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{6} b^{\frac{9}{2}} - a^{7} b^{\frac{9}{2}}\right )}}{63 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^10,x, algorithm="giac")

[Out]

4/63*(63*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(9/2) + 105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(9/2) + 315*(sqrt
(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(9/2) + 189*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(9/2) + 189*(sqrt(b)*x - s
qrt(b*x^2 + a))^6*a^4*b^(9/2) + 27*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(9/2) + 9*(sqrt(b)*x - sqrt(b*x^2 + a
))^2*a^6*b^(9/2) - a^7*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9