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

Optimal. Leaf size=52 \[ \frac{2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}-\frac{\left (b x^2+c x^4\right )^{3/2}}{5 b x^8} \]

[Out]

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

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Rubi [A]  time = 0.0831615, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 2014} \[ \frac{2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}-\frac{\left (b x^2+c x^4\right )^{3/2}}{5 b x^8} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2 + c*x^4]/x^7,x]

[Out]

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

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{\sqrt{b x^2+c x^4}}{x^7} \, dx &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{5 b x^8}-\frac{(2 c) \int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx}{5 b}\\ &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{5 b x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0112408, size = 35, normalized size = 0.67 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (2 c x^2-3 b\right )}{15 b^2 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2 + c*x^4]/x^7,x]

[Out]

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

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Maple [A]  time = 0.046, size = 39, normalized size = 0.8 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -2\,c{x}^{2}+3\,b \right ) }{15\,{b}^{2}{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(1/2)/x^7,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54929, size = 89, normalized size = 1.71 \begin{align*} \frac{{\left (2 \, c^{2} x^{4} - b c x^{2} - 3 \, b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, b^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(1/2)/x**7,x)

[Out]

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

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Giac [B]  time = 1.31997, size = 162, normalized size = 3.12 \begin{align*} \frac{4 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{2} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) - b^{3} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*(15*(sqrt(c)*x - sqrt(c*x^2 + b))^6*c^(5/2)*sgn(x) + 5*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b*c^(5/2)*sgn(x) +
5*(sqrt(c)*x - sqrt(c*x^2 + b))^2*b^2*c^(5/2)*sgn(x) - b^3*c^(5/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 -
b)^5