3.1190 \(\int x^2 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=232 \[ \frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a-b x^4} \]

[Out]

(x^3*(a - b*x^4)^(1/4))/4 - (a*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(8*Sqrt[2]*b^(3/4)) + (a*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(8*Sqrt[2]*b^(3/4)) + (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4]
- (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(3/4)) - (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (
Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(3/4))

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Rubi [A]  time = 0.110704, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {279, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a-b x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a - b*x^4)^(1/4),x]

[Out]

(x^3*(a - b*x^4)^(1/4))/4 - (a*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(8*Sqrt[2]*b^(3/4)) + (a*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(8*Sqrt[2]*b^(3/4)) + (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4]
- (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(3/4)) - (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (
Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(3/4))

Rule 279

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 x^2 \sqrt [4]{a-b x^4} \, dx &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}+\frac{1}{4} a \int \frac{x^2}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )\\ &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}-\frac{a \operatorname{Subst}\left (\int \frac{1-\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{b}}+\frac{a \operatorname{Subst}\left (\int \frac{1+\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{b}}\\ &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{16 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{16 b}+\frac{a \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}+2 x}{-\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}+\frac{a \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}-2 x}{-\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}\\ &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}+\frac{a \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}\\ &=\frac{1}{4} x^3 \sqrt [4]{a-b x^4}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{16 \sqrt{2} b^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0084273, size = 52, normalized size = 0.22 \[ \frac{x^3 \sqrt [4]{a-b x^4} \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{3 \sqrt [4]{1-\frac{b x^4}{a}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a - b*x^4)^(1/4),x]

[Out]

(x^3*(a - b*x^4)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, (b*x^4)/a])/(3*(1 - (b*x^4)/a)^(1/4))

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Maple [F]  time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt [4]{-b{x}^{4}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2*(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.92115, size = 471, normalized size = 2.03 \begin{align*} \frac{1}{4} \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3} - \frac{1}{4} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{3}{4}} b^{2} x \sqrt{\frac{\sqrt{-\frac{a^{4}}{b^{3}}} b^{2} x^{2} + \sqrt{-b x^{4} + a} a^{2}}{x^{2}}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{3}{4}} b^{2}}{a^{4} x}\right ) - \frac{1}{16} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) + \frac{1}{16} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-b*x^4 + a)^(1/4)*x^3 - 1/4*(-a^4/b^3)^(1/4)*arctan(((-a^4/b^3)^(3/4)*b^2*x*sqrt((sqrt(-a^4/b^3)*b^2*x^2
+ sqrt(-b*x^4 + a)*a^2)/x^2) - (-b*x^4 + a)^(1/4)*a*(-a^4/b^3)^(3/4)*b^2)/(a^4*x)) - 1/16*(-a^4/b^3)^(1/4)*log
(((-a^4/b^3)^(1/4)*b*x + (-b*x^4 + a)^(1/4)*a)/x) + 1/16*(-a^4/b^3)^(1/4)*log(-((-a^4/b^3)^(1/4)*b*x - (-b*x^4
+ a)^(1/4)*a)/x)

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Sympy [C]  time = 1.77758, size = 41, normalized size = 0.18 \begin{align*} \frac{\sqrt [4]{a} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(1/4)*x**3*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4))

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Giac [A]  time = 1.23266, size = 265, normalized size = 1.14 \begin{align*} \frac{1}{32} \,{\left (\frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}{a} - \frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{3}{4}}} - \frac{2 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{3}{4}}} + \frac{\sqrt{2} \log \left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{3}{4}}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*(-b*x^4 + a)^(1/4)*x^3/a - 2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4)/x)/b^(
1/4))/b^(3/4) - 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(3/4) - sq
rt(2)*log(sqrt(b) + sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4) + sqrt(2)*log(sqrt(b)
- sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4))*a