3.368 \(\int \frac{\sqrt{a+b x^2}}{x^{10}} \, dx\)

Optimal. Leaf size=92 \[ \frac{16 b^3 \left (a+b x^2\right )^{3/2}}{315 a^4 x^3}-\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac{2 b \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9} \]

[Out]

-(a + b*x^2)^(3/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(3/2))/(21*a^2*x^7) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*x^5)
+ (16*b^3*(a + b*x^2)^(3/2))/(315*a^4*x^3)

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Rubi [A]  time = 0.0291518, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{16 b^3 \left (a+b x^2\right )^{3/2}}{315 a^4 x^3}-\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac{2 b \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]/x^10,x]

[Out]

-(a + b*x^2)^(3/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(3/2))/(21*a^2*x^7) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*x^5)
+ (16*b^3*(a + b*x^2)^(3/2))/(315*a^4*x^3)

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{\sqrt{a+b x^2}}{x^{10}} \, dx &=-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9}-\frac{(2 b) \int \frac{\sqrt{a+b x^2}}{x^8} \, dx}{3 a}\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{2 b \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{a+b x^2}}{x^6} \, dx}{21 a^2}\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{2 b \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}-\frac{\left (16 b^3\right ) \int \frac{\sqrt{a+b x^2}}{x^4} \, dx}{105 a^3}\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac{2 b \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac{16 b^3 \left (a+b x^2\right )^{3/2}}{315 a^4 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0120811, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^{3/2} \left (30 a^2 b x^2-35 a^3-24 a b^2 x^4+16 b^3 x^6\right )}{315 a^4 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^2]/x^10,x]

[Out]

((a + b*x^2)^(3/2)*(-35*a^3 + 30*a^2*b*x^2 - 24*a*b^2*x^4 + 16*b^3*x^6))/(315*a^4*x^9)

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Maple [A]  time = 0.003, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{3}{x}^{6}+24\,a{b}^{2}{x}^{4}-30\,{a}^{2}b{x}^{2}+35\,{a}^{3}}{315\,{x}^{9}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/2)/x^10,x)

[Out]

-1/315*(b*x^2+a)^(3/2)*(-16*b^3*x^6+24*a*b^2*x^4-30*a^2*b*x^2+35*a^3)/x^9/a^4

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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)^(1/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59386, size = 134, normalized size = 1.46 \begin{align*} \frac{{\left (16 \, b^{4} x^{8} - 8 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 5 \, a^{3} b x^{2} - 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, a^{4} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)/x^10,x, algorithm="fricas")

[Out]

1/315*(16*b^4*x^8 - 8*a*b^3*x^6 + 6*a^2*b^2*x^4 - 5*a^3*b*x^2 - 35*a^4)*sqrt(b*x^2 + a)/(a^4*x^9)

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Sympy [B]  time = 1.85392, size = 575, normalized size = 6.25 \begin{align*} - \frac{35 a^{7} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac{110 a^{6} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac{114 a^{5} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac{40 a^{4} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac{5 a^{3} b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac{30 a^{2} b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac{40 a b^{\frac{31}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac{16 b^{\frac{33}{2}} x^{14} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/2)/x**10,x)

[Out]

-35*a**7*b**(19/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 31
5*a**4*b**12*x**14) - 110*a**6*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10
+ 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*a**5*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x
**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*a**4*b**(25/2)*x**6*sqrt(a/(b*x
**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) + 5*a**3*b
**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**
4*b**12*x**14) + 30*a**2*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945
*a**5*b**11*x**12 + 315*a**4*b**12*x**14) + 40*a*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 94
5*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) + 16*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(3
15*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14)

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Giac [B]  time = 2.4691, size = 224, normalized size = 2.43 \begin{align*} \frac{32 \,{\left (315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{9}{2}} + 84 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{9}{2}} - 36 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{9}{2}} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{9}{2}} - a^{5} b^{\frac{9}{2}}\right )}}{315 \,{\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)^(1/2)/x^10,x, algorithm="giac")

[Out]

32/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(9/2) + 189*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(9/2) + 84*(sqr
t(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(9/2) - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(9/2) + 9*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*a^4*b^(9/2) - a^5*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9