3.244 \(\int \frac{(b x^2+c x^4)^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]

[Out]

-(b*x^2 + c*x^4)^(5/2)/(5*b*x^10)

________________________________________________________________________________________

Rubi [A]  time = 0.0459559, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ -\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(3/2)/x^9,x]

[Out]

-(b*x^2 + c*x^4)^(5/2)/(5*b*x^10)

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{\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx &=-\frac{\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0141597, size = 25, normalized size = 1. \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^(3/2)/x^9,x]

[Out]

-(x^2*(b + c*x^2))^(5/2)/(5*b*x^10)

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 29, normalized size = 1.2 \begin{align*} -{\frac{c{x}^{2}+b}{5\,{x}^{8}b} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(3/2)/x^9,x)

[Out]

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

________________________________________________________________________________________

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)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.26809, size = 84, normalized size = 3.36 \begin{align*} -\frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \, b x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(3/2)/x^9,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{9}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(3/2)/x**9,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**9, x)

________________________________________________________________________________________

Giac [B]  time = 1.26731, size = 124, normalized size = 4.96 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{2} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + b^{4} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right )\right )}}{5 \,{\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)^(3/2)/x^9,x, algorithm="giac")

[Out]

2/5*(5*(sqrt(c)*x - sqrt(c*x^2 + b))^8*c^(5/2)*sgn(x) + 10*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b^2*c^(5/2)*sgn(x)
+ b^4*c^(5/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^5