3.385 \(\int \frac{\sqrt{a x^4}}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

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Rubi [A]  time = 0.0040289, antiderivative size = 25, 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 = {15, 261} \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^4]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a x^4}}{\sqrt{1+x^3}} \, dx &=\frac{\sqrt{a x^4} \int \frac{x^2}{\sqrt{1+x^3}} \, dx}{x^2}\\ &=\frac{2 \sqrt{a x^4} \sqrt{1+x^3}}{3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.004558, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^4]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

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Maple [A]  time = 0.003, size = 31, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) }{3\,{x}^{2}}\sqrt{a{x}^{4}}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4)^(1/2)/(x^3+1)^(1/2),x)

[Out]

2/3*(1+x)*(x^2-x+1)/x^2*(a*x^4)^(1/2)/(x^3+1)^(1/2)

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Maxima [A]  time = 1.82366, size = 38, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (\sqrt{a} x^{3} + \sqrt{a}\right )}}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/3*(sqrt(a)*x^3 + sqrt(a))/(sqrt(x^2 - x + 1)*sqrt(x + 1))

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Fricas [A]  time = 0.992094, size = 47, normalized size = 1.88 \begin{align*} \frac{2 \, \sqrt{a x^{4}} \sqrt{x^{3} + 1}}{3 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(a*x^4)*sqrt(x^3 + 1)/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{4}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4)**(1/2)/(x**3+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**4)/sqrt((x + 1)*(x**2 - x + 1)), x)

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Giac [A]  time = 1.15782, size = 16, normalized size = 0.64 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(x^3 + 1)*sqrt(a)