3.266 \(\int \frac{1}{x^5 \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{8 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^2}+\frac{4 c \sqrt{b x^2+c x^4}}{15 b^2 x^4}-\frac{\sqrt{b x^2+c x^4}}{5 b x^6} \]

[Out]

-Sqrt[b*x^2 + c*x^4]/(5*b*x^6) + (4*c*Sqrt[b*x^2 + c*x^4])/(15*b^2*x^4) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(15*b^3*
x^2)

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Rubi [A]  time = 0.125382, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ -\frac{8 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^2}+\frac{4 c \sqrt{b x^2+c x^4}}{15 b^2 x^4}-\frac{\sqrt{b x^2+c x^4}}{5 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-Sqrt[b*x^2 + c*x^4]/(5*b*x^6) + (4*c*Sqrt[b*x^2 + c*x^4])/(15*b^2*x^4) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(15*b^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^5 \sqrt{b x^2+c x^4}} \, dx &=-\frac{\sqrt{b x^2+c x^4}}{5 b x^6}-\frac{(4 c) \int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx}{5 b}\\ &=-\frac{\sqrt{b x^2+c x^4}}{5 b x^6}+\frac{4 c \sqrt{b x^2+c x^4}}{15 b^2 x^4}+\frac{\left (8 c^2\right ) \int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx}{15 b^2}\\ &=-\frac{\sqrt{b x^2+c x^4}}{5 b x^6}+\frac{4 c \sqrt{b x^2+c x^4}}{15 b^2 x^4}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{15 b^3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0140421, size = 46, normalized size = 0.57 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 b^2-4 b c x^2+8 c^2 x^4\right )}{15 b^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4))/(15*b^3*x^6)

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Maple [A]  time = 0.044, size = 50, normalized size = 0.6 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 8\,{c}^{2}{x}^{4}-4\,bc{x}^{2}+3\,{b}^{2} \right ) }{15\,{b}^{3}{x}^{4}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/15*(c*x^2+b)*(8*c^2*x^4-4*b*c*x^2+3*b^2)/x^4/b^3/(c*x^4+b*x^2)^(1/2)

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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(1/x^5/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29769, size = 93, normalized size = 1.16 \begin{align*} -\frac{{\left (8 \, c^{2} x^{4} - 4 \, b c x^{2} + 3 \, b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, b^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(x**2*(b + c*x**2))), x)

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Giac [A]  time = 1.14868, size = 58, normalized size = 0.72 \begin{align*} -\frac{3 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} - 10 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c + 15 \, \sqrt{c + \frac{b}{x^{2}}} c^{2}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/15*(3*(c + b/x^2)^(5/2) - 10*(c + b/x^2)^(3/2)*c + 15*sqrt(c + b/x^2)*c^2)/b^3