3.368 \(\int \frac{\sqrt{a x^3}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]

[Out]

(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))

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Rubi [A]  time = 0.0078643, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {15, 329, 275, 215} \[ \frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^3]/Sqrt[1 + x^5],x]

[Out]

(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))

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 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^3}}{\sqrt{1+x^5}} \, dx &=\frac{\sqrt{a x^3} \int \frac{x^{3/2}}{\sqrt{1+x^5}} \, dx}{x^{3/2}}\\ &=\frac{\left (2 \sqrt{a x^3}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+x^{10}}} \, dx,x,\sqrt{x}\right )}{x^{3/2}}\\ &=\frac{\left (2 \sqrt{a x^3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^{5/2}\right )}{5 x^{3/2}}\\ &=\frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0056179, size = 24, normalized size = 1. \[ \frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^3]/Sqrt[1 + x^5],x]

[Out]

(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))

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Maple [A]  time = 0.03, size = 17, normalized size = 0.7 \begin{align*}{\frac{2}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{3}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*arcsinh(x^(5/2))*(a*x^3)^(1/2)/x^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{3}}}{\sqrt{x^{5} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^3)/sqrt(x^5 + 1), x)

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Fricas [B]  time = 1.69821, size = 252, normalized size = 10.5 \begin{align*} \left [\frac{1}{10} \, \sqrt{a} \log \left (-8 \, a x^{10} - 8 \, a x^{5} - 4 \,{\left (2 \, x^{6} + x\right )} \sqrt{x^{5} + 1} \sqrt{a x^{3}} \sqrt{a} - a\right ), -\frac{1}{5} \, \sqrt{-a} \arctan \left (\frac{{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{a x^{3}} \sqrt{-a}}{2 \,{\left (a x^{9} + a x^{4}\right )}}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/10*sqrt(a)*log(-8*a*x^10 - 8*a*x^5 - 4*(2*x^6 + x)*sqrt(x^5 + 1)*sqrt(a*x^3)*sqrt(a) - a), -1/5*sqrt(-a)*ar
ctan(1/2*(2*x^5 + 1)*sqrt(x^5 + 1)*sqrt(a*x^3)*sqrt(-a)/(a*x^9 + a*x^4))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{3}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*x**3)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError