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

Optimal. Leaf size=21 \[ -\frac{x}{c \sqrt{b x^2+c x^4}} \]

[Out]

-(x/(c*Sqrt[b*x^2 + c*x^4]))

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Rubi [A]  time = 0.0194372, antiderivative size = 21, 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 = {1588} \[ -\frac{x}{c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(x/(c*Sqrt[b*x^2 + c*x^4]))

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

Mathematica [A]  time = 0.0047655, size = 21, normalized size = 1. \[ -\frac{x}{c \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/(c*Sqrt[x^2*(b + c*x^2)]))

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Maple [A]  time = 0.044, size = 29, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3} \left ( c{x}^{2}+b \right ) }{c} \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^4/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.00746, size = 19, normalized size = 0.9 \begin{align*} -\frac{1}{\sqrt{c x^{2} + b} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(c*x^2 + b)*c)

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Fricas [A]  time = 1.28105, size = 54, normalized size = 2.57 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}}{c^{2} x^{3} + b c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.16458, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{\sqrt{c + \frac{b}{x^{2}}} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(c + b/x^2)*c*x)