3.1310 \(\int \frac{x^{3/2}}{\sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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Rubi [A]  time = 0.0289824, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {329, 275, 217, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[a + b*x^5],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.005264, size = 32, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[a + b*x^5],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.66927, size = 244, normalized size = 7.62 \begin{align*} \left [\frac{\log \left (-8 \, b^{2} x^{10} - 8 \, a b x^{5} - 4 \,{\left (2 \, b x^{7} + a x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{b} \sqrt{x} - a^{2}\right )}{10 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/10*log(-8*b^2*x^10 - 8*a*b*x^5 - 4*(2*b*x^7 + a*x^2)*sqrt(b*x^5 + a)*sqrt(b)*sqrt(x) - a^2)/sqrt(b), -1/5*s
qrt(-b)*arctan(2*sqrt(b*x^5 + a)*sqrt(-b)*x^(5/2)/(2*b*x^5 + a))/b]

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Sympy [A]  time = 2.44544, size = 24, normalized size = 0.75 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a}} \right )}}{5 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*asinh(sqrt(b)*x**(5/2)/sqrt(a))/(5*sqrt(b))

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Giac [A]  time = 1.24313, size = 55, normalized size = 1.72 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b + \frac{a}{x^{5}}}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} + \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*arctan(sqrt(b + a/x^5)/sqrt(-b))/sqrt(-b) + 2/5*arctan(sqrt(b)/sqrt(-b))/sqrt(-b)