\(\int \frac {1}{(b+a x^2) \sqrt [3]{x+x^3}} \, dx\) [2383]

Optimal result

Integrand size = 19, antiderivative size = 191 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x}{\sqrt [3]{a-b} x-2 \sqrt [3]{b} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{a-b} b^{2/3}}+\frac {\log \left (\sqrt [3]{a-b} x+\sqrt [3]{b} \sqrt [3]{x+x^3}\right )}{2 \sqrt [3]{a-b} b^{2/3}}-\frac {\log \left ((a-b)^{2/3} x^2-\sqrt [3]{a-b} \sqrt [3]{b} x \sqrt [3]{x+x^3}+b^{2/3} \left (x+x^3\right )^{2/3}\right )}{4 \sqrt [3]{a-b} b^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2081, 477, 476, 384} \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=-\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{b} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (a x^2+b\right )}{4 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}}+\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3} \sqrt [3]{a-b}+\sqrt [3]{b} \sqrt [3]{x^2+1}\right )}{4 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}} \]

[In]

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

[Out]

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

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^2\right )} \, dx}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^6} \left (b+a x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (b+a x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{b} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (b+a x^2\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}+\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{a-b} x^{2/3}+\sqrt [3]{b} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.82 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{a-b} x^{2/3}-2 \sqrt [3]{b} \sqrt [3]{1+x^2}}\right )-2 \log \left (\sqrt [3]{a-b} x^{2/3}+\sqrt [3]{b} \sqrt [3]{1+x^2}\right )+\log \left ((a-b)^{2/3} x^{4/3}-\sqrt [3]{a-b} \sqrt [3]{b} x^{2/3} \sqrt [3]{1+x^2}+b^{2/3} \left (1+x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x -2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}\right )}{3 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a -b}{b}\right )^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}}{2 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} b}\)	\(150\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{2} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int { \frac {1}{{\left (a x^{2} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

-3/4*(x^3 + x)/((a*x^(7/3) + b*x^(1/3))*(x^2 + 1)^(1/3)) + integrate(3/2*(b*x^2 + b)/((a^2*x^(13/3) + 2*a*b*x^
(7/3) + b^2*x^(1/3))*(x^2 + 1)^(1/3)), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {3 \, {\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a - b}{b}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} a b^{2} - \sqrt {3} b^{3}\right )}} - \frac {{\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \log \left (\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} + \left (-\frac {a - b}{b}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )}{4 \, {\left (a b^{2} - b^{3}\right )}} + \frac {\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} \log \left ({\left | -\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a - b\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int \frac {1}{\left (a\,x^2+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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