\(\int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx\) [131]

Optimal result

Integrand size = 19, antiderivative size = 301 \[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=-\frac {884 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {442 b^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{14421 a^{25/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

-884/14421*b^6*(b*x^(1/3)+a*x)^(1/2)/a^6+884/24035*b^5*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^5-6188/216315*b^4*x^(4/
3)*(b*x^(1/3)+a*x)^(1/2)/a^4+476/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^3-28/1311*b^2*x^(8/3)*(b*x^(1/3)+a*x)^(
1/2)/a^2+4/207*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)/a+2/9*x^4*(b*x^(1/3)+a*x)^(1/2)+442/14421*b^(27/4)*x^(1/6)*(co
s(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^(25/4)/(b*x^(1/3)+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2046, 2049, 2036, 335, 226} \[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\frac {442 b^{27/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{14421 a^{25/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {884 b^6 \sqrt {a x+b \sqrt [3]{x}}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {a x+b \sqrt [3]{x}}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{207 a}+\frac {2}{9} x^4 \sqrt {a x+b \sqrt [3]{x}} \]

[In]

Int[x^3*Sqrt[b*x^(1/3) + a*x],x]

[Out]

(-884*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^6) + (884*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(24035*a^5) - (6188*b^4
*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^4) + (476*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^3) - (28*b^2*x^(8/
3)*Sqrt[b*x^(1/3) + a*x])/(1311*a^2) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a) + (2*x^4*Sqrt[b*x^(1/3) +
a*x])/9 + (442*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
)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(14421*a^(25/4)*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*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} \sqrt {b x+a x^3} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {1}{9} (2 b) \text {Subst}\left (\int \frac {x^{12}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{69 a} \\ & = -\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (238 b^3\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a^2} \\ & = \frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (3094 b^4\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 a^3} \\ & = -\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (3094 b^5\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 a^4} \\ & = \frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}-\frac {\left (442 b^6\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^5} \\ & = -\frac {884 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (442 b^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^6} \\ & = -\frac {884 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (442 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 )}{14421 a^6 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {884 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {\left (884 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 )}{14421 a^6 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {884 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^6}+\frac {884 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{24035 a^5}-\frac {6188 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^4}+\frac {476 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^3}-\frac {28 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^2}+\frac {4 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a}+\frac {2}{9} x^4 \sqrt {b \sqrt [3]{x}+a x}+\frac {442 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 )}{14421 a^{25/4} \sqrt {b \sqrt [3]{x}+a x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.51 \[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (\sqrt {1+\frac {a x^{2/3}}{b}} \left (-9945 b^6+3978 a b^5 x^{2/3}-3094 a^2 b^4 x^{4/3}+2618 a^3 b^3 x^2-2310 a^4 b^2 x^{8/3}+2090 a^5 b x^{10/3}+24035 a^6 x^4\right )+9945 b^6 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{216315 a^6 \sqrt {1+\frac {a x^{2/3}}{b}}} \]

[In]

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

[Out]

(2*Sqrt[b*x^(1/3) + a*x]*(Sqrt[1 + (a*x^(2/3))/b]*(-9945*b^6 + 3978*a*b^5*x^(2/3) - 3094*a^2*b^4*x^(4/3) + 261
8*a^3*b^3*x^2 - 2310*a^4*b^2*x^(8/3) + 2090*a^5*b*x^(10/3) + 24035*a^6*x^4) + 9945*b^6*Hypergeometric2F1[-1/2,
1/4, 5/4, -((a*x^(2/3))/b)]))/(216315*a^6*Sqrt[1 + (a*x^(2/3))/b])

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {2 x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9}+\frac {4 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207 a}-\frac {28 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a^{2}}+\frac {476 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{19665 a^{3}}-\frac {6188 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{216315 a^{4}}+\frac {884 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{24035 a^{5}}-\frac {884 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{14421 a^{6}}+\frac {442 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14421 a^{7} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(264\)
default	\(\frac {2 x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9}+\frac {4 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207 a}-\frac {28 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a^{2}}+\frac {476 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{19665 a^{3}}-\frac {6188 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{216315 a^{4}}+\frac {884 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{24035 a^{5}}-\frac {884 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{14421 a^{6}}+\frac {442 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14421 a^{7} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(264\)

[In]

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

[Out]

2/9*x^4*(b*x^(1/3)+a*x)^(1/2)+4/207*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)/a-28/1311*b^2*x^(8/3)*(b*x^(1/3)+a*x)^(1/
2)/a^2+476/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^3-6188/216315*b^4*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^4+884/24035
*b^5*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^5-884/14421*b^6*(b*x^(1/3)+a*x)^(1/2)/a^6+442/14421*b^7/a^7*(-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)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2
*2^(1/2))

Fricas [F]

\[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {1}{3}}} x^{3} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\int x^{3} \sqrt {a x + b \sqrt [3]{x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {1}{3}}} x^{3} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {1}{3}}} x^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^3 \sqrt {b \sqrt [3]{x}+a x} \, dx=\int x^3\,\sqrt {a\,x+b\,x^{1/3}} \,d x \]

[In]

int(x^3*(a*x + b*x^(1/3))^(1/2),x)

[Out]

int(x^3*(a*x + b*x^(1/3))^(1/2), x)