3.1323 \(\int \frac{x^3}{a+b x^6} \, dx\)

Optimal. Leaf size=123 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} \sqrt [3]{a} b^{2/3}} \]

[Out]

-ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(2*Sqrt[3]*a^(1/3)*b^(2/3)) - Log[a^(1/3) + b^(1/3)*x^2]/
(6*a^(1/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(12*a^(1/3)*b^(2/3))

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Rubi [A]  time = 0.0921261, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {275, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^6),x]

[Out]

-ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(2*Sqrt[3]*a^(1/3)*b^(2/3)) - Log[a^(1/3) + b^(1/3)*x^2]/
(6*a^(1/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(12*a^(1/3)*b^(2/3))

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 292

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{a+b x^6} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 \sqrt [3]{a} \sqrt [3]{b}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 \sqrt [3]{a} \sqrt [3]{b}}\\ &=-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 \sqrt [3]{a} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{b}}\\ &=-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{a} b^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} \sqrt [3]{a} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0181929, size = 154, normalized size = 1.25 \[ \frac{-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^6),x]

[Out]

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

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Maple [A]  time = 0.001, size = 97, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{12\,b}\ln \left ({x}^{4}-\sqrt [3]{{\frac{a}{b}}}{x}^{2}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{6\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^6+a),x)

[Out]

-1/6/b/(1/b*a)^(1/3)*ln(x^2+(1/b*a)^(1/3))+1/12/b/(1/b*a)^(1/3)*ln(x^4-(1/b*a)^(1/3)*x^2+(1/b*a)^(2/3))+1/6*3^
(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x^2-1))

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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/(b*x^6+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.66282, size = 765, normalized size = 6.22 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{6} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{4} + a b x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b x^{6} + a}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{12 \, a b^{2}}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{12 \, a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^6+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.177251, size = 26, normalized size = 0.21 \begin{align*} \operatorname{RootSum}{\left (216 t^{3} a b^{2} + 1, \left ( t \mapsto t \log{\left (36 t^{2} a b + x^{2} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*a*b**2 + 1, Lambda(_t, _t*log(36*_t**2*a*b + x**2)))

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Giac [A]  time = 1.18902, size = 159, normalized size = 1.29 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{6 \, a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{6 \, a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{12 \, a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^6+a),x, algorithm="giac")

[Out]

-1/6*(-a/b)^(2/3)*log(abs(x^2 - (-a/b)^(1/3)))/a - 1/6*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/
b)^(1/3))/(-a/b)^(1/3))/(a*b^2) + 1/12*(-a*b^2)^(2/3)*log(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^2)