3.281 \(\int \frac{x^6}{(b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{2 \sqrt{b x^2+c x^4}}{c^2 x}-\frac{x^3}{c \sqrt{b x^2+c x^4}} \]

[Out]

-(x^3/(c*Sqrt[b*x^2 + c*x^4])) + (2*Sqrt[b*x^2 + c*x^4])/(c^2*x)

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Rubi [A]  time = 0.067865, antiderivative size = 47, 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 = {2015, 1588} \[ \frac{2 \sqrt{b x^2+c x^4}}{c^2 x}-\frac{x^3}{c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(x^3/(c*Sqrt[b*x^2 + c*x^4])) + (2*Sqrt[b*x^2 + c*x^4])/(c^2*x)

Rule 2015

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] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0152588, size = 29, normalized size = 0.62 \[ \frac{x \left (2 b+c x^2\right )}{c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(2*b + c*x^2))/(c^2*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.046, size = 37, normalized size = 0.8 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( c{x}^{2}+2\,b \right ){x}^{3}}{{c}^{2}} \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(x^6/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.99026, size = 30, normalized size = 0.64 \begin{align*} \frac{c x^{2} + 2 \, b}{\sqrt{c x^{2} + b} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

(c*x^2 + 2*b)/(sqrt(c*x^2 + b)*c^2)

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Fricas [A]  time = 1.24328, size = 74, normalized size = 1.57 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 2 \, b\right )}}{c^{3} x^{3} + b c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**6/(x**2*(b + c*x**2))**(3/2), x)

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Giac [A]  time = 1.16965, size = 70, normalized size = 1.49 \begin{align*} -\frac{2 \, \sqrt{b}}{{\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c\right )} c} + \frac{b}{\sqrt{c + \frac{b}{x^{2}}} c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(b)/(((sqrt(c + b/x^2) - sqrt(b)/x)^2 - c)*c) + b/(sqrt(c + b/x^2)*c^2*x)