3.1966 \(\int \frac{x^4}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=136 \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]

[Out]

-(b*x^2)/(2*a^2) + x^5/(5*a) - (b^(5/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(8/3)) -
(b^(5/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(8/3)) + (b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*
a^(8/3))

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Rubi [A]  time = 0.0954198, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {263, 302, 292, 31, 634, 617, 204, 628} \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x^3),x]

[Out]

-(b*x^2)/(2*a^2) + x^5/(5*a) - (b^(5/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(8/3)) -
(b^(5/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(8/3)) + (b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*
a^(8/3))

Rule 263

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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^4}{a+\frac{b}{x^3}} \, dx &=\int \frac{x^7}{b+a x^3} \, dx\\ &=\int \left (-\frac{b x}{a^2}+\frac{x^4}{a}+\frac{b^2 x}{a^2 \left (b+a x^3\right )}\right ) \, dx\\ &=-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a}+\frac{b^2 \int \frac{x}{b+a x^3} \, dx}{a^2}\\ &=-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a}-\frac{b^{5/3} \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^{7/3}}+\frac{b^{5/3} \int \frac{\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^{7/3}}\\ &=-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a}-\frac{b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{8/3}}+\frac{b^{5/3} \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{8/3}}+\frac{b^2 \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^{7/3}}\\ &=-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a}-\frac{b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{8/3}}+\frac{b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{8/3}}+\frac{b^{5/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{8/3}}\\ &=-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{8/3}}+\frac{b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.0371644, size = 122, normalized size = 0.9 \[ \frac{5 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-15 a^{2/3} b x^2+6 a^{5/3} x^5-10 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-10 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{30 a^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x^3),x]

[Out]

(-15*a^(2/3)*b*x^2 + 6*a^(5/3)*x^5 - 10*Sqrt[3]*b^(5/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 10*b^(5/
3)*Log[b^(1/3) + a^(1/3)*x] + 5*b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(30*a^(8/3))

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Maple [A]  time = 0.003, size = 117, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,a}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5/a-1/2*b*x^2/a^2-1/3/a^3*b^2/(b/a)^(1/3)*ln(x+(b/a)^(1/3))+1/6/a^3*b^2/(b/a)^(1/3)*ln(x^2-(b/a)^(1/3)*x
+(b/a)^(2/3))+1/3/a^3*b^2*3^(1/2)/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*x-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^4/(a+b/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48431, size = 343, normalized size = 2.52 \begin{align*} \frac{6 \, a x^{5} - 15 \, b x^{2} + 10 \, \sqrt{3} b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + \sqrt{3} b}{3 \, b}\right ) - 5 \, b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 10 \, b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right )}{30 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(6*a*x^5 - 15*b*x^2 + 10*sqrt(3)*b*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^(1/3) + sqrt(3)*
b)/b) - 5*b*(-b^2/a^2)^(1/3)*log(b*x^2 - a*x*(-b^2/a^2)^(2/3) - b*(-b^2/a^2)^(1/3)) + 10*b*(-b^2/a^2)^(1/3)*lo
g(b*x + a*(-b^2/a^2)^(2/3)))/a^2

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Sympy [A]  time = 0.424292, size = 44, normalized size = 0.32 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{8} + b^{5}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{b^{3}} + x \right )} \right )\right )} + \frac{x^{5}}{5 a} - \frac{b x^{2}}{2 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**8 + b**5, Lambda(_t, _t*log(9*_t**2*a**5/b**3 + x))) + x**5/(5*a) - b*x**2/(2*a**2)

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Giac [A]  time = 1.19418, size = 178, normalized size = 1.31 \begin{align*} -\frac{b \left (-\frac{b}{a}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{4}} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}} b \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{4}} + \frac{2 \, a^{4} x^{5} - 5 \, a^{3} b x^{2}}{10 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/a^2 - 1/3*sqrt(3)*(-a^2*b)^(2/3)*b*arctan(1/3*sqrt(3)*(2*x + (-
b/a)^(1/3))/(-b/a)^(1/3))/a^4 + 1/6*(-a^2*b)^(2/3)*b*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^4 + 1/10*(2*a^
4*x^5 - 5*a^3*b*x^2)/a^5