3.2090 \(\int \frac{x^3}{(a+\frac{b}{x^4})^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{3 b}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{x^4}{4 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

(3*b)/(4*a^2*Sqrt[a + b/x^4]) + x^4/(4*a*Sqrt[a + b/x^4]) - (3*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*a^(5/2))

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Rubi [A]  time = 0.0355107, antiderivative size = 71, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{3 x^4 \sqrt{a+\frac{b}{x^4}}}{4 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^4)^(3/2),x]

[Out]

-x^4/(2*a*Sqrt[a + b/x^4]) + (3*Sqrt[a + b/x^4]*x^4)/(4*a^2) - (3*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*a^(5/
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{x^3}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )}{4 a}\\ &=-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x^4}{4 a^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )}{8 a^2}\\ &=-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x^4}{4 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{4 a^2}\\ &=-\frac{x^4}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt{a+\frac{b}{x^4}} x^4}{4 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0431593, size = 78, normalized size = 1.13 \[ \frac{\sqrt{a} x^2 \left (a x^4+3 b\right )-3 b^{3/2} \sqrt{\frac{a x^4}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )}{4 a^{5/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^4)^(3/2),x]

[Out]

(Sqrt[a]*x^2*(3*b + a*x^4) - 3*b^(3/2)*Sqrt[1 + (a*x^4)/b]*ArcSinh[(Sqrt[a]*x^2)/Sqrt[b]])/(4*a^(5/2)*Sqrt[a +
b/x^4]*x^2)

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Maple [A]  time = 0.013, size = 79, normalized size = 1.1 \begin{align*}{\frac{a{x}^{4}+b}{4\,{x}^{6}} \left ({x}^{6}{a}^{{\frac{7}{2}}}+3\,{x}^{2}b{a}^{5/2}-3\,b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){a}^{2}\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a*x^4+b)*(x^6*a^(7/2)+3*x^2*b*a^(5/2)-3*b*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*a^2*(a*x^4+b)^(1/2))/((a*x^4+b)
/x^4)^(3/2)/x^6/a^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60517, size = 421, normalized size = 6.1 \begin{align*} \left [\frac{3 \,{\left (a b x^{4} + b^{2}\right )} \sqrt{a} \log \left (-2 \, a x^{4} + 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) + 2 \,{\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{8 \,{\left (a^{4} x^{4} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{4} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) +{\left (a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{4 \,{\left (a^{4} x^{4} + a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 4.73916, size = 75, normalized size = 1.09 \begin{align*} \frac{x^{6}}{4 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} + \frac{3 \sqrt{b} x^{2}}{4 a^{2} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6/(4*a*sqrt(b)*sqrt(a*x**4/b + 1)) + 3*sqrt(b)*x**2/(4*a**2*sqrt(a*x**4/b + 1)) - 3*b*asinh(sqrt(a)*x**2/sq
rt(b))/(4*a**(5/2))

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Giac [A]  time = 1.18831, size = 131, normalized size = 1.9 \begin{align*} \frac{1}{4} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{4} + b\right )}}{x^{4}}}{{\left (a \sqrt{\frac{a x^{4} + b}{x^{4}}} - \frac{{\left (a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{4}}\right )} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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