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

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

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Rubi [A]  time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^2}{b \sqrt {b x^2+c x^4}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^2}{b \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.27, size = 28, normalized size = 1.27 \begin {gather*} \frac {\sqrt {b x^2+c x^4}}{b \left (b+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[b*x^2 + c*x^4]/(b*(b + c*x^2))

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fricas [A]  time = 1.91, size = 26, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}}}{b c x^{2} + b^{2}} \end {gather*}

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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giac [A]  time = 0.18, size = 35, normalized size = 1.59 \begin {gather*} \frac {1}{{\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} + b\right )} \sqrt {c}} \end {gather*}

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))

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maple [A]  time = 0.00, size = 28, normalized size = 1.27 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) x^{4}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b} \end {gather*}

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

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maxima [A]  time = 1.45, size = 20, normalized size = 0.91 \begin {gather*} \frac {x^{2}}{\sqrt {c x^{4} + b x^{2}} b} \end {gather*}

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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mupad [B]  time = 4.13, size = 26, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}}{b\,\left (c\,x^2+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2 + c*x^4)^(3/2),x)

[Out]

(b*x^2 + c*x^4)^(1/2)/(b*(b + c*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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