3.2.40 \(\int x^2 (b \sqrt [3]{x}+a x)^{3/2} \, dx\) [140]

Optimal. Leaf size=298 \[ \frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {884 b^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{100947 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

2/9*x^3*(b*x^(1/3)+a*x)^(3/2)+1768/100947*b^6*(b*x^(1/3)+a*x)^(1/2)/a^5-1768/168245*b^5*x^(2/3)*(b*x^(1/3)+a*x
)^(1/2)/a^4+1768/216315*b^4*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^3-136/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^2+8/13
11*b^2*x^(8/3)*(b*x^(1/3)+a*x)^(1/2)/a+4/69*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)-884/100947*b^(27/4)*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^(21/4)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.35, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2046, 2049, 2036, 335, 226} \begin {gather*} -\frac {884 b^{27/4} \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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{100947 a^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {1768 b^6 \sqrt {a x+b \sqrt [3]{x}}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {a x+b \sqrt [3]{x}}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}+\frac {2}{9} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(b*x^(1/3) + a*x)^(3/2),x]

[Out]

(1768*b^6*Sqrt[b*x^(1/3) + a*x])/(100947*a^5) - (1768*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(168245*a^4) + (1768*
b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^3) - (136*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^2) + (8*b^2*x^(
8/3)*Sqrt[b*x^(1/3) + a*x])/(1311*a) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/69 + (2*x^3*(b*x^(1/3) + a*x)^(3/2
))/9 - (884*b^(27/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*E
llipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(100947*a^(21/4)*Sqrt[b*x^(1/3) + a*x])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

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 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2043

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 2046

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 2049

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]

Rubi steps

\begin {align*} \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=3 \text {Subst}\left (\int x^8 \left (b x+a x^3\right )^{3/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{3} (2 b) \text {Subst}\left (\int x^9 \sqrt {b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{69} \left (4 b^2\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (68 b^3\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a}\\ &=-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (884 b^4\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 a^2}\\ &=\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (884 b^5\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 a^3}\\ &=-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (884 b^6\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 a^4}\\ &=\frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (884 b^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{100947 a^5}\\ &=\frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (884 b^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{100947 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (1768 b^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{100947 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {884 b^{27/4} \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 )}{100947 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.13, size = 142, normalized size = 0.48 \begin {gather*} \frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (\left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}} \left (3315 b^4-7293 a b^3 x^{2/3}+12155 a^2 b^2 x^{4/3}-17765 a^3 b x^2+24035 a^4 x^{8/3}\right )-3315 b^6 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )\right )}{216315 a^5 \sqrt {1+\frac {a x^{2/3}}{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(1/3) + a*x]*((b + a*x^(2/3))^2*Sqrt[1 + (a*x^(2/3))/b]*(3315*b^4 - 7293*a*b^3*x^(2/3) + 12155*a^2
*b^2*x^(4/3) - 17765*a^3*b*x^2 + 24035*a^4*x^(8/3)) - 3315*b^6*Hypergeometric2F1[-3/2, 1/4, 5/4, -((a*x^(2/3))
/b)]))/(216315*a^5*Sqrt[1 + (a*x^(2/3))/b])

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Maple [A]
time = 0.38, size = 196, normalized size = 0.66

method	result	size
default	\(-\frac {2 \left (-216755 x^{\frac {11}{3}} a^{6} b^{2}-380380 x^{\frac {13}{3}} a^{7} b +616 a^{5} b^{3} x^{3}+1768 x^{\frac {5}{3}} a^{3} b^{5}-952 x^{\frac {7}{3}} a^{4} b^{4}-168245 a^{8} x^{5}+6630 b^{7} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-5304 a^{2} b^{6} x -13260 x^{\frac {1}{3}} a \,b^{7}\right )}{1514205 a^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\)	\(196\)
derivativedivides	\(\frac {2 a \,x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9}+\frac {58 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207}+\frac {8 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a}-\frac {136 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{19665 a^{2}}+\frac {1768 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{216315 a^{3}}-\frac {1768 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{168245 a^{4}}+\frac {1768 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{100947 a^{5}}-\frac {884 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{100947 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1514205*(-216755*x^(11/3)*a^6*b^2-380380*x^(13/3)*a^7*b+616*a^5*b^3*x^3+1768*x^(5/3)*a^3*b^5-952*x^(7/3)*a^
4*b^4-168245*a^8*x^5+6630*b^7*(-a*b)^(1/2)*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(a*x^(1/3)-(-a*b)
^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)*a/(-a*b)^(1/2))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^
(1/2),1/2*2^(1/2))-5304*a^2*b^6*x-13260*x^(1/3)*a*b^7)/a^6/(x^(1/3)*(b+a*x^(2/3)))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*x^3 + b*x^(7/3))*sqrt(a*x + b*x^(1/3)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x + b*x^(1/3))^(3/2),x)

[Out]

int(x^2*(a*x + b*x^(1/3))^(3/2), x)

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