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

Optimal. Leaf size=23 \[ \frac{2}{3} \sqrt{a} \sinh ^{-1}\left (\frac{(a x)^{3/2}}{a^{3/2}}\right ) \]

[Out]

(2*Sqrt[a]*ArcSinh[(a*x)^(3/2)/a^(3/2)])/3

________________________________________________________________________________________

Rubi [A]  time = 0.015498, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {329, 275, 215} \[ \frac{2}{3} \sqrt{a} \sinh ^{-1}\left (\frac{(a x)^{3/2}}{a^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a]*ArcSinh[(a*x)^(3/2)/a^(3/2)])/3

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{a x}}{\sqrt{1+x^3}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^6}{a^3}}} \, dx,x,\sqrt{a x}\right )}{a}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^3}}} \, dx,x,(a x)^{3/2}\right )}{3 a}\\ &=\frac{2}{3} \sqrt{a} \sinh ^{-1}\left (\frac{(a x)^{3/2}}{a^{3/2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0048701, size = 22, normalized size = 0.96 \[ \frac{2 \sqrt{a x} \sinh ^{-1}\left (x^{3/2}\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a*x]*ArcSinh[x^(3/2)])/(3*Sqrt[x])

________________________________________________________________________________________

Maple [C]  time = 0.063, size = 321, normalized size = 14. \begin{align*} -4\,{\frac{\sqrt{ax}\sqrt{{x}^{3}+1}a \left ( 1+i\sqrt{3} \right ) \left ( 1+x \right ) ^{2}}{\sqrt{x \left ({x}^{3}+1 \right ) a} \left ( 3+i\sqrt{3} \right ) \sqrt{-ax \left ( 1+x \right ) \left ( i\sqrt{3}+2\,x-1 \right ) \left ( i\sqrt{3}-2\,x+1 \right ) }}\sqrt{{\frac{ \left ( 3+i\sqrt{3} \right ) x}{ \left ( 1+i\sqrt{3} \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{ \left ( i\sqrt{3}-1 \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{ \left ( 1+i\sqrt{3} \right ) \left ( 1+x \right ) }}} \left ({\it EllipticF} \left ( \sqrt{{\frac{ \left ( 3+i\sqrt{3} \right ) x}{ \left ( 1+i\sqrt{3} \right ) \left ( 1+x \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-1 \right ) \left ( 3+i\sqrt{3} \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( 3+i\sqrt{3} \right ) x}{ \left ( 1+i\sqrt{3} \right ) \left ( 1+x \right ) }}},{\frac{1+i\sqrt{3}}{3+i\sqrt{3}}},\sqrt{{\frac{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-1 \right ) \left ( 3+i\sqrt{3} \right ) }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(a*x)^(1/2)*(x^3+1)^(1/2)*a*(1+I*3^(1/2))*((3+I*3^(1/2))*x/(1+I*3^(1/2))/(1+x))^(1/2)*(1+x)^2*((I*3^(1/2)+2
*x-1)/(I*3^(1/2)-1)/(1+x))^(1/2)*((I*3^(1/2)-2*x+1)/(1+I*3^(1/2))/(1+x))^(1/2)*(EllipticF(((3+I*3^(1/2))*x/(1+
I*3^(1/2))/(1+x))^(1/2),((I*3^(1/2)-3)*(1+I*3^(1/2))/(I*3^(1/2)-1)/(3+I*3^(1/2)))^(1/2))-EllipticPi(((3+I*3^(1
/2))*x/(1+I*3^(1/2))/(1+x))^(1/2),(1+I*3^(1/2))/(3+I*3^(1/2)),((I*3^(1/2)-3)*(1+I*3^(1/2))/(I*3^(1/2)-1)/(3+I*
3^(1/2)))^(1/2)))/(x*(x^3+1)*a)^(1/2)/(3+I*3^(1/2))/(-a*x*(1+x)*(I*3^(1/2)+2*x-1)*(I*3^(1/2)-2*x+1))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x)/sqrt(x^3 + 1), x)

________________________________________________________________________________________

Fricas [B]  time = 1.25422, size = 225, normalized size = 9.78 \begin{align*} \left [\frac{1}{6} \, \sqrt{a} \log \left (-8 \, a x^{6} - 8 \, a x^{3} - 4 \,{\left (2 \, x^{4} + x\right )} \sqrt{x^{3} + 1} \sqrt{a x} \sqrt{a} - a\right ), -\frac{1}{3} \, \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{x^{3} + 1} \sqrt{a x} \sqrt{-a} x}{2 \, a x^{3} + a}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*sqrt(a)*log(-8*a*x^6 - 8*a*x^3 - 4*(2*x^4 + x)*sqrt(x^3 + 1)*sqrt(a*x)*sqrt(a) - a), -1/3*sqrt(-a)*arctan
(2*sqrt(x^3 + 1)*sqrt(a*x)*sqrt(-a)*x/(2*a*x^3 + a))]

________________________________________________________________________________________

Sympy [A]  time = 1.05586, size = 14, normalized size = 0.61 \begin{align*} \frac{2 \sqrt{a} \operatorname{asinh}{\left (x^{\frac{3}{2}} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)*asinh(x**(3/2))/3

________________________________________________________________________________________

Giac [B]  time = 1.17155, size = 47, normalized size = 2.04 \begin{align*} -\frac{2 \, a^{\frac{5}{2}} \log \left (-\sqrt{a x} a^{\frac{3}{2}} x + \sqrt{a^{4} x^{3} + a^{4}}\right )}{3 \,{\left | a \right |}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*a^(5/2)*log(-sqrt(a*x)*a^(3/2)*x + sqrt(a^4*x^3 + a^4))/abs(a)^2