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

Optimal. Leaf size=411 \[ \frac{22 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{221 a^{19/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{44 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{9/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{60 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}}{1547 a^2}-\frac{44 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{19/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{44 b^4 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{663 a^4}+\frac{220 b^3 x \sqrt{a x+b \sqrt [3]{x}}}{4641 a^3}+\frac{4 b x^{7/3} \sqrt{a x+b \sqrt [3]{x}}}{119 a}+\frac{2}{7} x^3 \sqrt{a x+b \sqrt [3]{x}} \]

[Out]

(44*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(9/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (44*b^4*x^(
1/3)*Sqrt[b*x^(1/3) + a*x])/(663*a^4) + (220*b^3*x*Sqrt[b*x^(1/3) + a*x])/(4641*a^3) - (60*b^2*x^(5/3)*Sqrt[b*
x^(1/3) + a*x])/(1547*a^2) + (4*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x])/(119*a) + (2*x^3*Sqrt[b*x^(1/3) + a*x])/7 - (
44*b^(21/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE[
2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(221*a^(19/4)*Sqrt[b*x^(1/3) + a*x]) + (22*b^(21/4)*(Sqrt[b] + Sqrt
[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/
b^(1/4)], 1/2])/(221*a^(19/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.577651, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2021, 2024, 2032, 329, 305, 220, 1196} \[ \frac{44 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{9/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{60 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}}{1547 a^2}+\frac{22 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{19/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{44 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{19/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{44 b^4 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{663 a^4}+\frac{220 b^3 x \sqrt{a x+b \sqrt [3]{x}}}{4641 a^3}+\frac{4 b x^{7/3} \sqrt{a x+b \sqrt [3]{x}}}{119 a}+\frac{2}{7} x^3 \sqrt{a x+b \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[b*x^(1/3) + a*x],x]

[Out]

(44*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(9/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (44*b^4*x^(
1/3)*Sqrt[b*x^(1/3) + a*x])/(663*a^4) + (220*b^3*x*Sqrt[b*x^(1/3) + a*x])/(4641*a^3) - (60*b^2*x^(5/3)*Sqrt[b*
x^(1/3) + a*x])/(1547*a^2) + (4*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x])/(119*a) + (2*x^3*Sqrt[b*x^(1/3) + a*x])/7 - (
44*b^(21/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE[
2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(221*a^(19/4)*Sqrt[b*x^(1/3) + a*x]) + (22*b^(21/4)*(Sqrt[b] + Sqrt
[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/
b^(1/4)], 1/2])/(221*a^(19/4)*Sqrt[b*x^(1/3) + a*x])

Rule 2018

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

Rule 2021

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

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 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

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 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

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

Rubi steps

\begin{align*} \int x^2 \sqrt{b \sqrt [3]{x}+a x} \, dx &=3 \operatorname{Subst}\left (\int x^8 \sqrt{b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{1}{7} (2 b) \operatorname{Subst}\left (\int \frac{x^9}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}-\frac{\left (30 b^2\right ) \operatorname{Subst}\left (\int \frac{x^7}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{119 a}\\ &=-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (330 b^3\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 a^2}\\ &=\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}-\frac{\left (110 b^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 a^3}\\ &=-\frac{44 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^4}+\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (22 b^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 a^4}\\ &=-\frac{44 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^4}+\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (22 b^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{221 a^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{44 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^4}+\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (44 b^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{44 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^4}+\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (44 b^{11/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^{9/2} \sqrt{b \sqrt [3]{x}+a x}}-\frac{\left (44 b^{11/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{a} x^2}{\sqrt{b}}}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^{9/2} \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{44 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{9/2} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{44 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^4}+\frac{220 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^3}-\frac{60 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^2}+\frac{4 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a}+\frac{2}{7} x^3 \sqrt{b \sqrt [3]{x}+a x}-\frac{44 b^{21/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{19/4} \sqrt{b \sqrt [3]{x}+a x}}+\frac{22 b^{21/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{19/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.113741, size = 136, normalized size = 0.33 \[ \frac{2 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}} \left (\sqrt{\frac{a x^{2/3}}{b}+1} \left (-90 a^2 b^2 x^{4/3}+78 a^3 b x^2+663 a^4 x^{8/3}+110 a b^3 x^{2/3}-385 b^4\right )+385 b^4 \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{a x^{2/3}}{b}\right )\right )}{4641 a^4 \sqrt{\frac{a x^{2/3}}{b}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1/3)*Sqrt[b*x^(1/3) + a*x]*(Sqrt[1 + (a*x^(2/3))/b]*(-385*b^4 + 110*a*b^3*x^(2/3) - 90*a^2*b^2*x^(4/3) +
78*a^3*b*x^2 + 663*a^4*x^(8/3)) + 385*b^4*Hypergeometric2F1[-1/2, 3/4, 7/4, -((a*x^(2/3))/b)]))/(4641*a^4*Sqr
t[1 + (a*x^(2/3))/b])

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Maple [A]  time = 0.011, size = 273, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{3}}{7}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{4\,b}{119\,a}{x}^{{\frac{7}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{60\,{b}^{2}}{1547\,{a}^{2}}{x}^{{\frac{5}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{220\,{b}^{3}x}{4641\,{a}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{44\,{b}^{4}}{663\,{a}^{4}}\sqrt [3]{x}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{22\,{b}^{5}}{221\,{a}^{5}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{a}{\it EllipticE} \left ( \sqrt{{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}+{\frac{\sqrt{-ab}}{a}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{a}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*x^3*(b*x^(1/3)+a*x)^(1/2)+4/119*b*x^(7/3)*(b*x^(1/3)+a*x)^(1/2)/a-60/1547*b^2*x^(5/3)*(b*x^(1/3)+a*x)^(1/2
)/a^2+220/4641*b^3*x*(b*x^(1/3)+a*x)^(1/2)/a^3-44/663*b^4*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^4+22/221*b^5/a^5*(-a
*b)^(1/2)*((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)-1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/
2)*(-x^(1/3)*a/(-a*b)^(1/2))^(1/2)/(b*x^(1/3)+a*x)^(1/2)*(-2/a*(-a*b)^(1/2)*EllipticE(((x^(1/3)+1/a*(-a*b)^(1/
2))*a/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/a*(-a*b)^(1/2)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^
(1/2),1/2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x + b x^{\frac{1}{3}}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a x + b x^{\frac{1}{3}}} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a x + b \sqrt [3]{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x + b x^{\frac{1}{3}}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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