3.89 \(\int \frac{x^4}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=240 \[ \frac{x^2 \left (a+b x^3\right )}{2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.123902, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1355, 321, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (a+b x^3\right )}{2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

Rule 1355

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]

Rule 321

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

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

Mathematica [A]  time = 0.0464957, size = 131, normalized size = 0.55 \[ \frac{\left (a+b x^3\right ) \left (-a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+3 b^{2/3} x^2\right )}{6 b^{5/3} \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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Maple [A]  time = 0.009, size = 113, normalized size = 0.5 \begin{align*}{\frac{b{x}^{3}+a}{6\,{b}^{2}} \left ( 3\,{x}^{2}b\sqrt [3]{{\frac{a}{b}}}+2\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}a+2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) a-\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((b*x^3+a)^2)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73455, size = 298, normalized size = 1.24 \begin{align*} \frac{3 \, x^{2} - 2 \, \sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) - \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/((b*x^3+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.359269, size = 32, normalized size = 0.13 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} + \frac{x^{2}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/((b*x**3+a)**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.15128, size = 197, normalized size = 0.82 \begin{align*} \frac{x^{2} \mathrm{sgn}\left (b x^{3} + a\right )}{2 \, b} + \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right ) \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) \mathrm{sgn}\left (b x^{3} + a\right )}{3 \, b^{3}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) \mathrm{sgn}\left (b x^{3} + a\right )}{6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

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