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

Optimal. Leaf size=22 \[ \frac{x^2}{b \sqrt{b x^2+c x^4}} \]

[Out]

x^2/(b*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.0553208, antiderivative size = 22, 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} \[ \frac{x^2}{b \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^2 + c*x^4)^(3/2),x]

[Out]

x^2/(b*Sqrt[b*x^2 + c*x^4])

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

Mathematica [A]  time = 0.0075433, size = 22, normalized size = 1. \[ \frac{x^2}{b \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(b*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.043, size = 28, normalized size = 1.3 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ){x}^{4}}{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(x^3/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.986302, size = 27, normalized size = 1.23 \begin{align*} \frac{x^{2}}{\sqrt{c x^{4} + b x^{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2/(sqrt(c*x^4 + b*x^2)*b)

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Fricas [A]  time = 1.25365, size = 50, normalized size = 2.27 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}}{b c x^{2} + b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^4 + b*x^2)/(b*c*x^2 + b^2)

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

[Out]

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

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Giac [A]  time = 1.20895, size = 47, normalized size = 2.14 \begin{align*} \frac{1}{{\left ({\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} + b\right )} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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