3.267 \(\int \frac{1}{x^7 \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=108 \[ \frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{\sqrt{b x^2+c x^4}}{7 b x^8} \]

[Out]

-Sqrt[b*x^2 + c*x^4]/(7*b*x^8) + (6*c*Sqrt[b*x^2 + c*x^4])/(35*b^2*x^6) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(35*b^3*
x^4) + (16*c^3*Sqrt[b*x^2 + c*x^4])/(35*b^4*x^2)

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Rubi [A]  time = 0.169381, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ \frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{\sqrt{b x^2+c x^4}}{7 b x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-Sqrt[b*x^2 + c*x^4]/(7*b*x^8) + (6*c*Sqrt[b*x^2 + c*x^4])/(35*b^2*x^6) - (8*c^2*Sqrt[b*x^2 + c*x^4])/(35*b^3*
x^4) + (16*c^3*Sqrt[b*x^2 + c*x^4])/(35*b^4*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^7 \sqrt{b x^2+c x^4}} \, dx &=-\frac{\sqrt{b x^2+c x^4}}{7 b x^8}-\frac{(6 c) \int \frac{1}{x^5 \sqrt{b x^2+c x^4}} \, dx}{7 b}\\ &=-\frac{\sqrt{b x^2+c x^4}}{7 b x^8}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}+\frac{\left (24 c^2\right ) \int \frac{1}{x^3 \sqrt{b x^2+c x^4}} \, dx}{35 b^2}\\ &=-\frac{\sqrt{b x^2+c x^4}}{7 b x^8}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}-\frac{\left (16 c^3\right ) \int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx}{35 b^3}\\ &=-\frac{\sqrt{b x^2+c x^4}}{7 b x^8}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0147405, size = 57, normalized size = 0.53 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (6 b^2 c x^2-5 b^3-8 b c^2 x^4+16 c^3 x^6\right )}{35 b^4 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-5*b^3 + 6*b^2*c*x^2 - 8*b*c^2*x^4 + 16*c^3*x^6))/(35*b^4*x^8)

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Maple [A]  time = 0.045, size = 61, normalized size = 0.6 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -16\,{c}^{3}{x}^{6}+8\,b{c}^{2}{x}^{4}-6\,{b}^{2}c{x}^{2}+5\,{b}^{3} \right ) }{35\,{b}^{4}{x}^{6}}{\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^7/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/35*(c*x^2+b)*(-16*c^3*x^6+8*b*c^2*x^4-6*b^2*c*x^2+5*b^3)/x^6/b^4/(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^7/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32417, size = 115, normalized size = 1.06 \begin{align*} \frac{{\left (16 \, c^{3} x^{6} - 8 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} - 5 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, b^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/35*(16*c^3*x^6 - 8*b*c^2*x^4 + 6*b^2*c*x^2 - 5*b^3)*sqrt(c*x^4 + b*x^2)/(b^4*x^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{7} \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**7/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(1/(x**7*sqrt(x**2*(b + c*x**2))), x)

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Giac [A]  time = 1.17345, size = 77, normalized size = 0.71 \begin{align*} -\frac{5 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} - 21 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{c + \frac{b}{x^{2}}} c^{3}}{35 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/35*(5*(c + b/x^2)^(7/2) - 21*(c + b/x^2)^(5/2)*c + 35*(c + b/x^2)^(3/2)*c^2 - 35*sqrt(c + b/x^2)*c^3)/b^4