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]

-1/7*(b*x^2+a)^(3/2)/a/x^7+4/35*b*(b*x^2+a)^(3/2)/a^2/x^5-8/105*b^2*(b*x^2+a)^(3/2)/a^3/x^3

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Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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]

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

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fricas [A]  time = 0.98, size = 49, normalized size = 0.72 \[ -\frac {{\left (8 \, b^{3} x^{6} - 4 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + 15 \, a^{3}\right )} \sqrt {b x^{2} + a}}{105 \, a^{3} x^{7}} \]

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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giac [B]  time = 0.65, size = 138, normalized size = 2.03 \[ \frac {16 \, {\left (70 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {7}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {7}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {7}{2}} - 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {7}{2}} + a^{4} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

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

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maple [A]  time = 0.01, size = 39, normalized size = 0.57 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{4}-12 a b \,x^{2}+15 a^{2}\right )}{105 a^{3} x^{7}} \]

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 [A]  time = 1.32, size = 56, normalized size = 0.82 \[ -\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}{105 \, a^{3} x^{3}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{7 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/105*(b*x^2 + a)^(3/2)*b^2/(a^3*x^3) + 4/35*(b*x^2 + a)^(3/2)*b/(a^2*x^5) - 1/7*(b*x^2 + a)^(3/2)/(a*x^7)

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mupad [B]  time = 4.71, size = 73, normalized size = 1.07 \[ \frac {4\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^3}-\frac {b\,\sqrt {b\,x^2+a}}{35\,a\,x^5}-\frac {\sqrt {b\,x^2+a}}{7\,x^7}-\frac {8\,b^3\,\sqrt {b\,x^2+a}}{105\,a^3\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(1/2)/x^8,x)

[Out]

(4*b^2*(a + b*x^2)^(1/2))/(105*a^2*x^3) - (b*(a + b*x^2)^(1/2))/(35*a*x^5) - (a + b*x^2)^(1/2)/(7*x^7) - (8*b^
3*(a + b*x^2)^(1/2))/(105*a^3*x)

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sympy [B]  time = 1.30, size = 359, normalized size = 5.28 \[ - \frac {15 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {33 a^{4} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {17 a^{3} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {3 a^{2} b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {12 a b^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {8 b^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} \]

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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