3.2.50 \(\int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx\) [150]

Optimal. Leaf size=414 \[ -\frac {418 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{11/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {418 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {209 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

-418/221*b^5*(b+a*x^(2/3))*x^(1/3)/a^(11/2)/(x^(1/3)*a^(1/2)+b^(1/2))/(b*x^(1/3)+a*x)^(1/2)+418/663*b^4*x^(1/3
)*(b*x^(1/3)+a*x)^(1/2)/a^5-2090/4641*b^3*x*(b*x^(1/3)+a*x)^(1/2)/a^4+570/1547*b^2*x^(5/3)*(b*x^(1/3)+a*x)^(1/
2)/a^3-38/119*b*x^(7/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+2/7*x^3*(b*x^(1/3)+a*x)^(1/2)/a+418/221*b^(21/4)*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)))*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^(23/4)/(b*x^(1/3)+a*x)^(1/2)-209/221*b^(21/4)*x^(1/6)*(cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/co
s(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^(23/4)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2043, 2049, 2057, 335, 311, 226, 1210} \begin {gather*} -\frac {209 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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {418 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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {418 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{11/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{663 a^5}-\frac {2090 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{119 a^2}+\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[b*x^(1/3) + a*x],x]

[Out]

(-418*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(11/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) + (418*b^4
*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(663*a^5) - (2090*b^3*x*Sqrt[b*x^(1/3) + a*x])/(4641*a^4) + (570*b^2*x^(5/3)*S
qrt[b*x^(1/3) + a*x])/(1547*a^3) - (38*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x])/(119*a^2) + (2*x^3*Sqrt[b*x^(1/3) + a*
x])/(7*a) + (418*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^(23/4)*Sqrt[b*x^(1/3) + a*x]) - (209*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^(23/4)*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 311

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 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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

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 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]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[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 \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \text {Subst}\left (\int \frac {x^{11}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {(19 b) \text {Subst}\left (\int \frac {x^9}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 a}\\ &=-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {\left (285 b^2\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{119 a^2}\\ &=\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (3135 b^3\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 a^3}\\ &=-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {\left (1045 b^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 a^4}\\ &=\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (209 b^5\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 a^5}\\ &=\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (209 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{221 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (418 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (418 b^{11/2} \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 )}{221 a^{11/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (418 b^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {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^{11/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {418 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{11/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {418 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^{23/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {209 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^{23/4} \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 = 10.06, size = 143, normalized size = 0.35 \begin {gather*} \frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (1463 b^5 \sqrt [3]{x}+418 a b^4 x-190 a^2 b^3 x^{5/3}+114 a^3 b^2 x^{7/3}-78 a^4 b x^3+663 a^5 x^{11/3}-1463 b^5 \sqrt {1+\frac {a x^{2/3}}{b}} \sqrt [3]{x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{4641 a^5 \left (b+a x^{2/3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(1/3) + a*x]*(1463*b^5*x^(1/3) + 418*a*b^4*x - 190*a^2*b^3*x^(5/3) + 114*a^3*b^2*x^(7/3) - 78*a^4*
b*x^3 + 663*a^5*x^(11/3) - 1463*b^5*Sqrt[1 + (a*x^(2/3))/b]*x^(1/3)*Hypergeometric2F1[1/2, 3/4, 7/4, -((a*x^(2
/3))/b)]))/(4641*a^5*(b + a*x^(2/3)))

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Maple [A]
time = 0.35, size = 261, normalized size = 0.63

method	result	size
default	\(-\frac {-228 x^{\frac {8}{3}} a^{4} b^{2}+156 x^{\frac {10}{3}} a^{5} b +380 a^{3} b^{3} x^{2}+8778 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-4389 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-1326 a^{6} x^{4}-2926 x^{\frac {2}{3}} a \,b^{5}-836 x^{\frac {4}{3}} a^{2} b^{4}}{4641 a^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\)	\(261\)
derivativedivides	\(\frac {2 x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 a}-\frac {38 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 a^{2}}+\frac {570 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a^{3}}-\frac {2090 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{4}}+\frac {418 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 a^{5}}-\frac {209 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{221 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4641/a^6*(-228*x^(8/3)*a^4*b^2+156*x^(10/3)*a^5*b+380*a^3*b^3*x^2+8778*b^6*((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)*EllipticE(((a*x
^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-4389*b^6*((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))-1326*a^6*x^4-2926*x^(2/3)*a*b^5-836*x^(4/3)*a^2*b^4)/(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^3/(b*x^(1/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

integral((a^2*x^4 - a*b*x^(10/3) + b^2*x^(8/3))*sqrt(a*x + b*x^(1/3))/(a^3*x^2 + b^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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