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

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

[Out]

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

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Rubi [A]  time = 0.58, antiderivative size = 411, normalized size of antiderivative = 1.00, 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 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 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 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 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]

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]

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*}

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Mathematica [C]  time = 0.13, size = 136, normalized size = 0.33 \[ \frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}} \left (\sqrt {\frac {a x^{2/3}}{b}+1} \left (663 a^4 x^{8/3}+78 a^3 b x^2-90 a^2 b^2 x^{4/3}+110 a b^3 x^{2/3}-385 b^4\right )+385 b^4 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{4641 a^4 \sqrt {\frac {a x^{2/3}}{b}+1}} \]

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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fricas [F]  time = 7.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a x + b x^{\frac {1}{3}}} x^{2}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + b x^{\frac {1}{3}}} x^{2}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 273, normalized size = 0.66 \[ \frac {2 \sqrt {a x +b \,x^{\frac {1}{3}}}\, x^{3}}{7}+\frac {4 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b \,x^{\frac {7}{3}}}{119 a}-\frac {60 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b^{2} x^{\frac {5}{3}}}{1547 a^{2}}+\frac {220 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b^{3} x}{4641 a^{3}}-\frac {44 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b^{4} x^{\frac {1}{3}}}{663 a^{4}}+\frac {22 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right ) b^{5}}{221 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + b x^{\frac {1}{3}}} x^{2}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {a\,x+b\,x^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x + b*x^(1/3))^(1/2),x)

[Out]

int(x^2*(a*x + b*x^(1/3))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a x + b \sqrt [3]{x}}\, dx \]

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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