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

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

[Out]

(b*x^2 + c*x^4)^(5/2)/(5*c*x^5)

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Rubi [A]  time = 0.0475792, 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} \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(3/2)/x^2,x]

[Out]

(b*x^2 + c*x^4)^(5/2)/(5*c*x^5)

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

Mathematica [A]  time = 0.0090662, size = 25, normalized size = 1. \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 c x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(b + c*x^2))^(5/2)/(5*c*x^5)

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Maple [A]  time = 0.044, size = 29, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+b}{5\,c{x}^{3}} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.02987, size = 43, normalized size = 1.72 \begin{align*} \frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{2} + b}}{5 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3784, size = 80, normalized size = 3.2 \begin{align*} \frac{{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{5 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13411, size = 78, normalized size = 3.12 \begin{align*} -\frac{b^{\frac{5}{2}} \mathrm{sgn}\left (x\right )}{5 \, c} + \frac{5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) +{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} \mathrm{sgn}\left (x\right )}{15 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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