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

Optimal. Leaf size=118 \[ \frac {1}{6} (a-3 b) \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {1}{6} \left (3 \sqrt {3} b-\sqrt {3} a\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {1}{12} (3 b-a) \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )+\frac {1}{2} a \left (x^3+x\right )^{2/3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 223, normalized size of antiderivative = 1.89, number of steps used = 11, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2053, 2011, 329, 275, 239, 2024} \begin {gather*} \frac {1}{2} a \left (x^3+x\right )^{2/3}+\frac {a \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}}-\frac {a \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{x^3+x}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + a*x^2)/(x + x^3)^(1/3),x]

[Out]

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 275

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2053

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 136, normalized size = 1.15 \begin {gather*} \frac {\sqrt [3]{x} \left (a x^{2/3} \left (x^2+1\right )-\frac {1}{6} \sqrt [3]{x^2+1} (a-3 b) \left (-2 \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )+\log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )\right )\right )}{2 \sqrt [3]{x^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.34, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{2} a \left (x+x^3\right )^{2/3}+\frac {1}{6} \left (-\sqrt {3} a+3 \sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )+\frac {1}{6} (a-3 b) \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{12} (-a+3 b) \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(x + x^3)^(2/3))/2 + ((-(Sqrt[3]*a) + 3*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/6 + ((a - 3
*b)*Log[-x + (x + x^3)^(1/3)])/6 + ((-a + 3*b)*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/12

________________________________________________________________________________________

fricas [A]  time = 81.81, size = 100, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, {\left (x^{3} + x\right )}^{\frac {2}{3}} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*(a - 3*b)*arctan(-(196*sqrt(3)*(x^3 + x)^(1/3)*x - sqrt(3)*(539*x^2 + 507) - 1274*sqrt(3)*(x^3 +
x)^(2/3))/(2205*x^2 + 2197)) + 1/12*(a - 3*b)*log(3*(x^3 + x)^(1/3)*x - 3*(x^3 + x)^(2/3) + 1) + 1/2*(x^3 + x)
^(2/3)*a

________________________________________________________________________________________

giac [A]  time = 0.24, size = 83, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, a x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + \frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, {\left (a - 3 \, b\right )} \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2*(1/x^2 + 1)^(2/3) + 1/6*sqrt(3)*(a - 3*b)*arctan(1/3*sqrt(3)*(2*(1/x^2 + 1)^(1/3) + 1)) - 1/12*(a -
3*b)*log((1/x^2 + 1)^(2/3) + (1/x^2 + 1)^(1/3) + 1) + 1/6*(a - 3*b)*log(abs((1/x^2 + 1)^(1/3) - 1))

________________________________________________________________________________________

maple [C]  time = 1.49, size = 36, normalized size = 0.31

method	result	size
meijerg	\(\frac {3 a \,x^{\frac {8}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -x^{2}\right )}{8}+\frac {3 b \,x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}\)	\(36\)
risch	\(\frac {a x \left (x^{2}+1\right )}{2 \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}+\frac {3 b \,x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}-\frac {a \,x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}\)	\(54\)
trager	\(\frac {a \left (x^{3}+x \right )^{\frac {2}{3}}}{2}+\frac {\left (a -3 b \right ) \left (6 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}+144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x +174 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}+36 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+8\right )-6 \ln \left (180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -114 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-4 x^{2}-96 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-3\right ) \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-\ln \left (180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{2}-144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -114 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-180 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-4 x^{2}-96 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-3\right )\right )}{6}\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*a*x^(8/3)*hypergeom([1/3,4/3],[7/3],-x^2)+3/2*b*x^(2/3)*hypergeom([1/3,1/3],[4/3],-x^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + b}{{\left (x^{3} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + a*x^2)/(x + x^3)^(1/3),x)

[Out]

int((b + a*x^2)/(x + x^3)^(1/3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________