3.2068 \(\int \frac {(a+\frac {b}{x^4})^{3/2}}{x} \, dx\)

Optimal. Leaf size=59 \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2} \]

[Out]

-1/6*(a+b/x^4)^(3/2)+1/2*a^(3/2)*arctanh((a+b/x^4)^(1/2)/a^(1/2))-1/2*a*(a+b/x^4)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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, 50, 63, 208} \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^4)^(3/2)/x,x]

[Out]

-(a*Sqrt[a + b/x^4])/2 - (a + b/x^4)^(3/2)/6 + (a^(3/2)*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

Rule 50

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

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]]

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 52, normalized size = 0.88 \[ -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {a x^4}{b}\right )}{6 x^4 \sqrt {\frac {a x^4}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^4)^(3/2)/x,x]

[Out]

-1/6*(b*Sqrt[a + b/x^4]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*x^4)/b)])/(x^4*Sqrt[1 + (a*x^4)/b])

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fricas [A]  time = 0.84, size = 141, normalized size = 2.39 \[ \left [\frac {3 \, a^{\frac {3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - 2 \, {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, -\frac {3 \, \sqrt {-a} a x^{4} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) + {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*a^(3/2)*x^4*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 2*(4*a*x^4 + b)*sqrt((a*x^4 + b
)/x^4))/x^4, -1/6*(3*sqrt(-a)*a*x^4*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)) + (4*a*x^4 + b)*sqr
t((a*x^4 + b)/x^4))/x^4]

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giac [B]  time = 0.25, size = 122, normalized size = 2.07 \[ -\frac {1}{4} \, a^{\frac {3}{2}} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{4} a^{\frac {3}{2}} b - 3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} a^{\frac {3}{2}} b^{2} + 2 \, a^{\frac {3}{2}} b^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 78, normalized size = 1.32 \[ -\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (-3 a^{\frac {3}{2}} x^{6} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )+4 \sqrt {a \,x^{4}+b}\, a \,x^{4}+\sqrt {a \,x^{4}+b}\, b \right )}{6 \left (a \,x^{4}+b \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.95, size = 61, normalized size = 1.03 \[ -\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{6} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^(3/2)*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a))) - 1/6*(a + b/x^4)^(3/2) - 1/2*sqrt(a
+ b/x^4)*a

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mupad [B]  time = 1.50, size = 43, normalized size = 0.73 \[ \frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2}-\frac {a\,\sqrt {a+\frac {b}{x^4}}}{2}-\frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^4)^(3/2)/x,x)

[Out]

(a^(3/2)*atanh((a + b/x^4)^(1/2)/a^(1/2)))/2 - (a*(a + b/x^4)^(1/2))/2 - (a + b/x^4)^(3/2)/6

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sympy [A]  time = 2.48, size = 80, normalized size = 1.36 \[ - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{3} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{4}} \right )}}{4} + \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{2} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{4}}}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x**4))/3 - a**(3/2)*log(b/(a*x**4))/4 + a**(3/2)*log(sqrt(1 + b/(a*x**4)) + 1)/2 - s
qrt(a)*b*sqrt(1 + b/(a*x**4))/(6*x**4)

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