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

Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]

[Out]

-Sqrt[a + b*x^3]/(3*a*x^3) + (b*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2))

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Rubi [A]  time = 0.0264228, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a + b*x^3]),x]

[Out]

-Sqrt[a + b*x^3]/(3*a*x^3) + (b*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2))

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{6 a}\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a}\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.048069, size = 64, normalized size = 1.28 \[ \frac{2 b \sqrt{a+b x^3} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{a}{2 b x^3}\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a + b*x^3]),x]

[Out]

(2*b*Sqrt[a + b*x^3]*(-a/(2*b*x^3) + ArcTanh[Sqrt[1 + (b*x^3)/a]]/(2*Sqrt[1 + (b*x^3)/a])))/(3*a^2)

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Maple [A]  time = 0.014, size = 39, normalized size = 0.8 \begin{align*}{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56232, size = 265, normalized size = 5.3 \begin{align*} \left [\frac{\sqrt{a} b x^{3} \log \left (\frac{b x^{3} + 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \, \sqrt{b x^{3} + a} a}{6 \, a^{2} x^{3}}, -\frac{\sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{3} + a} a}{3 \, a^{2} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.48261, size = 49, normalized size = 0.98 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1062, size = 65, normalized size = 1.3 \begin{align*} -\frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{3} + a}}{a b x^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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