3.96 \(\int \frac{1}{x^8 \sqrt{a x+b x^4}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 b^2 \sqrt{a x+b x^4}}{45 a^3 x^2}+\frac{8 b \sqrt{a x+b x^4}}{45 a^2 x^5}-\frac{2 \sqrt{a x+b x^4}}{15 a x^8} \]

[Out]

(-2*Sqrt[a*x + b*x^4])/(15*a*x^8) + (8*b*Sqrt[a*x + b*x^4])/(45*a^2*x^5) - (16*b^2*Sqrt[a*x + b*x^4])/(45*a^3*
x^2)

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Rubi [A]  time = 0.101669, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2016, 2014} \[ -\frac{16 b^2 \sqrt{a x+b x^4}}{45 a^3 x^2}+\frac{8 b \sqrt{a x+b x^4}}{45 a^2 x^5}-\frac{2 \sqrt{a x+b x^4}}{15 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[a*x + b*x^4])/(15*a*x^8) + (8*b*Sqrt[a*x + b*x^4])/(45*a^2*x^5) - (16*b^2*Sqrt[a*x + b*x^4])/(45*a^3*
x^2)

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^8 \sqrt{a x+b x^4}} \, dx &=-\frac{2 \sqrt{a x+b x^4}}{15 a x^8}-\frac{(4 b) \int \frac{1}{x^5 \sqrt{a x+b x^4}} \, dx}{5 a}\\ &=-\frac{2 \sqrt{a x+b x^4}}{15 a x^8}+\frac{8 b \sqrt{a x+b x^4}}{45 a^2 x^5}+\frac{\left (8 b^2\right ) \int \frac{1}{x^2 \sqrt{a x+b x^4}} \, dx}{15 a^2}\\ &=-\frac{2 \sqrt{a x+b x^4}}{15 a x^8}+\frac{8 b \sqrt{a x+b x^4}}{45 a^2 x^5}-\frac{16 b^2 \sqrt{a x+b x^4}}{45 a^3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0146873, size = 44, normalized size = 0.59 \[ -\frac{2 \sqrt{x \left (a+b x^3\right )} \left (3 a^2-4 a b x^3+8 b^2 x^6\right )}{45 a^3 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*Sqrt[a*x + b*x^4]),x]

[Out]

(-2*Sqrt[x*(a + b*x^3)]*(3*a^2 - 4*a*b*x^3 + 8*b^2*x^6))/(45*a^3*x^8)

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Maple [A]  time = 0.005, size = 48, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( 8\,{b}^{2}{x}^{6}-4\,ab{x}^{3}+3\,{a}^{2} \right ) }{45\,{x}^{7}{a}^{3}}{\frac{1}{\sqrt{b{x}^{4}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^4+a*x)^(1/2),x)

[Out]

-2/45*(b*x^3+a)*(8*b^2*x^6-4*a*b*x^3+3*a^2)/x^7/a^3/(b*x^4+a*x)^(1/2)

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Maxima [A]  time = 1.08143, size = 68, normalized size = 0.92 \begin{align*} -\frac{2 \,{\left (8 \, b^{3} x^{10} + 4 \, a b^{2} x^{7} - a^{2} b x^{4} + 3 \, a^{3} x\right )}}{45 \, \sqrt{b x^{3} + a} a^{3} x^{\frac{17}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a*x)^(1/2),x, algorithm="maxima")

[Out]

-2/45*(8*b^3*x^10 + 4*a*b^2*x^7 - a^2*b*x^4 + 3*a^3*x)/(sqrt(b*x^3 + a)*a^3*x^(17/2))

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Fricas [A]  time = 1.18574, size = 90, normalized size = 1.22 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} x^{6} - 4 \, a b x^{3} + 3 \, a^{2}\right )} \sqrt{b x^{4} + a x}}{45 \, a^{3} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a*x)^(1/2),x, algorithm="fricas")

[Out]

-2/45*(8*b^2*x^6 - 4*a*b*x^3 + 3*a^2)*sqrt(b*x^4 + a*x)/(a^3*x^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{8} \sqrt{x \left (a + b x^{3}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**4+a*x)**(1/2),x)

[Out]

Integral(1/(x**8*sqrt(x*(a + b*x**3))), x)

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Giac [A]  time = 1.23487, size = 58, normalized size = 0.78 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b + \frac{a}{x^{3}}\right )}^{\frac{5}{2}} - 10 \,{\left (b + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} b + 15 \, \sqrt{b + \frac{a}{x^{3}}} b^{2}\right )}}{45 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a*x)^(1/2),x, algorithm="giac")

[Out]

-2/45*(3*(b + a/x^3)^(5/2) - 10*(b + a/x^3)^(3/2)*b + 15*sqrt(b + a/x^3)*b^2)/a^3