3.286 \(\int \frac{x^3}{\sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

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Rubi [A]  time = 0.016886, antiderivative size = 25, 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} \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

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^3}{\sqrt{a x^2+b x^5}} \, dx &=\frac{2 \sqrt{a x^2+b x^5}}{3 b x}\\ \end{align*}

Mathematica [A]  time = 0.0077292, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^2 \left (a+b x^3\right )}}{3 b x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[x^2*(a + b*x^3)])/(3*b*x)

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Maple [A]  time = 0.004, size = 27, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) x}{3\,b}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^5+a*x^2)^(1/2),x)

[Out]

2/3*(b*x^3+a)*x/b/(b*x^5+a*x^2)^(1/2)

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Maxima [A]  time = 1.1064, size = 19, normalized size = 0.76 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(b*x^3 + a)/b

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Fricas [A]  time = 0.796274, size = 42, normalized size = 1.68 \begin{align*} \frac{2 \, \sqrt{b x^{5} + a x^{2}}}{3 \, b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(b*x^5 + a*x^2)/(b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**3/sqrt(x**2*(a + b*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^3/sqrt(b*x^5 + a*x^2), x)