3.384 \(\int \frac{(a+b x^2)^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac{\left (a+b x^2\right )^{5/2}}{7 a x^7} \]

[Out]

-(a + b*x^2)^(5/2)/(7*a*x^7) + (2*b*(a + b*x^2)^(5/2))/(35*a^2*x^5)

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Rubi [A]  time = 0.0124121, 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} \[ \frac{2 b \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac{\left (a+b x^2\right )^{5/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^8,x]

[Out]

-(a + b*x^2)^(5/2)/(7*a*x^7) + (2*b*(a + b*x^2)^(5/2))/(35*a^2*x^5)

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

Mathematica [A]  time = 0.0095871, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^2\right )^{5/2} \left (2 b x^2-5 a\right )}{35 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^8,x]

[Out]

((a + b*x^2)^(5/2)*(-5*a + 2*b*x^2))/(35*a^2*x^7)

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Maple [A]  time = 0.002, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,b{x}^{2}+5\,a}{35\,{x}^{7}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/x^8,x)

[Out]

-1/35*(b*x^2+a)^(5/2)*(-2*b*x^2+5*a)/x^7/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)^(3/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57507, size = 105, normalized size = 2.39 \begin{align*} \frac{{\left (2 \, b^{3} x^{6} - a b^{2} x^{4} - 8 \, a^{2} b x^{2} - 5 \, a^{3}\right )} \sqrt{b x^{2} + a}}{35 \, a^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^8,x, algorithm="fricas")

[Out]

1/35*(2*b^3*x^6 - a*b^2*x^4 - 8*a^2*b*x^2 - 5*a^3)*sqrt(b*x^2 + a)/(a^2*x^7)

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Sympy [B]  time = 1.37544, size = 94, normalized size = 2.14 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{6}} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 x^{4}} - \frac{b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a x^{2}} + \frac{2 b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/x**8,x)

[Out]

-a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(7*x**6) - 8*b**(3/2)*sqrt(a/(b*x**2) + 1)/(35*x**4) - b**(5/2)*sqrt(a/(b*x**2
) + 1)/(35*a*x**2) + 2*b**(7/2)*sqrt(a/(b*x**2) + 1)/(35*a**2)

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Giac [B]  time = 2.03754, size = 224, normalized size = 5.09 \begin{align*} \frac{4 \,{\left (35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} b^{\frac{7}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{7}{2}} + 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{7}{2}} + 14 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{7}{2}} + 7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{7}{2}} - a^{5} b^{\frac{7}{2}}\right )}}{35 \,{\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)^(3/2)/x^8,x, algorithm="giac")

[Out]

4/35*(35*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2) + 70*(sqrt(b)
*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2) + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(7/2) + 7*(sqrt(b)*x - sqrt(b*x
^2 + a))^2*a^4*b^(7/2) - a^5*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7