3.367 \(\int \frac{\sqrt{a+b x^2}}{x^8} \, dx\)

Optimal. Leaf size=68 \[ -\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}+\frac{4 b \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac{\left (a+b x^2\right )^{3/2}}{7 a x^7} \]

[Out]

-(a + b*x^2)^(3/2)/(7*a*x^7) + (4*b*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*x^3)

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Rubi [A]  time = 0.0179749, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}+\frac{4 b \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac{\left (a+b x^2\right )^{3/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]/x^8,x]

[Out]

-(a + b*x^2)^(3/2)/(7*a*x^7) + (4*b*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*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^8} \, dx &=-\frac{\left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac{(4 b) \int \frac{\sqrt{a+b x^2}}{x^6} \, dx}{7 a}\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac{4 b \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{a+b x^2}}{x^4} \, dx}{35 a^2}\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac{4 b \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac{8 b^2 \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0101219, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (15 a^2-12 a b x^2+8 b^2 x^4\right )}{105 a^3 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^2]/x^8,x]

[Out]

-((a + b*x^2)^(3/2)*(15*a^2 - 12*a*b*x^2 + 8*b^2*x^4))/(105*a^3*x^7)

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Maple [A]  time = 0.003, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{4}-12\,ab{x}^{2}+15\,{a}^{2}}{105\,{a}^{3}{x}^{7}} \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^8,x)

[Out]

-1/105*(b*x^2+a)^(3/2)*(8*b^2*x^4-12*a*b*x^2+15*a^2)/a^3/x^7

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58286, size = 112, normalized size = 1.65 \begin{align*} -\frac{{\left (8 \, b^{3} x^{6} - 4 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + 15 \, a^{3}\right )} \sqrt{b x^{2} + a}}{105 \, a^{3} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)/x^8,x, algorithm="fricas")

[Out]

-1/105*(8*b^3*x^6 - 4*a*b^2*x^4 + 3*a^2*b*x^2 + 15*a^3)*sqrt(b*x^2 + a)/(a^3*x^7)

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Sympy [B]  time = 1.31844, size = 359, normalized size = 5.28 \begin{align*} - \frac{15 a^{5} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{33 a^{4} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{17 a^{3} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{3 a^{2} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{12 a b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{8 b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/2)/x**8,x)

[Out]

-15*a**5*b**(9/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 33*a*
*4*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 17*a*
*3*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 3*a**
2*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 12*a*b
**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 8*b**(19/
2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10)

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Giac [B]  time = 2.13103, size = 186, normalized size = 2.74 \begin{align*} \frac{16 \,{\left (70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} b^{\frac{7}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b^{\frac{7}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{7}{2}} - 7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{7}{2}} + a^{4} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)/x^8,x, algorithm="giac")

[Out]

16/105*(70*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(7/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(7/2) + 21*(sqrt(b
)*x - sqrt(b*x^2 + a))^4*a^2*b^(7/2) - 7*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(7/2) + a^4*b^(7/2))/((sqrt(b)*
x - sqrt(b*x^2 + a))^2 - a)^7