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

Optimal. Leaf size=52 \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0540039, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2002, 2014} \[ \frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 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 \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac{(2 b) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{7 c}\\ &=-\frac{2 b \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5}+\frac{\left (b x^2+c x^4\right )^{5/2}}{7 c x^3}\\ \end{align*}

Mathematica [A]  time = 0.0191722, size = 42, normalized size = 0.81 \[ \frac{x \left (b+c x^2\right )^3 \left (5 c x^2-2 b\right )}{35 c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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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 ( -5\,c{x}^{2}+2\,b \right ) }{35\,{c}^{2}{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)

[Out]

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

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Maxima [A]  time = 1.04154, size = 61, normalized size = 1.17 \begin{align*} \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{2} + b}}{35 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.31199, size = 108, normalized size = 2.08 \begin{align*} \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x^{2} + c x^{4}\right )^{\frac{3}{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)

[Out]

Integral((b*x**2 + c*x**4)**(3/2), x)

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Giac [B]  time = 1.15115, size = 126, normalized size = 2.42 \begin{align*} \frac{2 \, b^{\frac{7}{2}} \mathrm{sgn}\left (x\right )}{35 \, c^{2}} + \frac{\frac{7 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} b \mathrm{sgn}\left (x\right )}{c} + \frac{{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} \mathrm{sgn}\left (x\right )}{c}}{105 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*b^(7/2)*sgn(x)/c^2 + 1/105*(7*(3*(c*x^2 + b)^(5/2) - 5*(c*x^2 + b)^(3/2)*b)*b*sgn(x)/c + (15*(c*x^2 + b)^
(7/2) - 42*(c*x^2 + b)^(5/2)*b + 35*(c*x^2 + b)^(3/2)*b^2)*sgn(x)/c)/c