3.5.86 \(\int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \, dx\) [486]

Optimal. Leaf size=190 \[ \frac {3 b^2 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 b \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {3 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \begin {gather*} -\frac {3 b x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {x \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {3 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {3 b^2 \sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^3 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 269

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

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 86, normalized size = 0.45 \begin {gather*} \frac {\left (b+a \sqrt [3]{x}\right ) \left (6 a b^2 \sqrt [3]{x}-3 a^2 b x^{2/3}+2 a^3 x-6 b^3 \log \left (b+a \sqrt [3]{x}\right )\right )}{2 a^4 \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 78, normalized size = 0.41

method	result	size
derivativedivides	\(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-2 a^{3} x +3 a^{2} b \,x^{\frac {2}{3}}+6 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )-6 a \,b^{2} x^{\frac {1}{3}}\right )}{2 \sqrt {\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x^{\frac {1}{3}} a^{4}}\)	\(78\)
default	\(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-2 a^{3} x +3 a^{2} b \,x^{\frac {2}{3}}+6 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )-6 a \,b^{2} x^{\frac {1}{3}}\right )}{2 \sqrt {\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, x^{\frac {1}{3}} a^{4}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 44, normalized size = 0.23 \begin {gather*} -\frac {3 \, b^{3} \log \left (a x^{\frac {1}{3}} + b\right )}{a^{4}} + \frac {2 \, a^{2} x - 3 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{2 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 3.48, size = 77, normalized size = 0.41 \begin {gather*} -\frac {3 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{4} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} + \frac {2 \, a^{2} x - 3 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{2 \, a^{3} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________