\(\int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx\) [138]

Optimal result

Integrand size = 19, antiderivative size = 413 \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=-\frac {308 a^{9/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}+\frac {308 a^{17/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {154 a^{17/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

-308/1105*a^(9/2)*(b+a*x^(2/3))*x^(1/3)/b^4/(x^(1/3)*a^(1/2)+b^(1/2))/(b*x^(1/3)+a*x)^(1/2)-6/17*(b*x^(1/3)+a*
x)^(1/2)/x^3-12/221*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(7/3)+44/663*a^2*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(5/3)-308/3315*a^
3*(b*x^(1/3)+a*x)^(1/2)/b^3/x+308/1105*a^4*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(1/3)+308/1105*a^(17/4)*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)/b
^(15/4)/(b*x^(1/3)+a*x)^(1/2)-154/1105*a^(17/4)*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)/b^(15/4)/(b*x^(1/3)+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2043, 2045, 2050, 2057, 335, 311, 226, 1210} \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=-\frac {154 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^4 \sqrt {a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac {308 a^3 \sqrt {a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac {44 a^2 \sqrt {a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{17 x^3} \]

[In]

Int[Sqrt[b*x^(1/3) + a*x]/x^4,x]

[Out]

(-308*a^(9/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (6*Sqrt[
b*x^(1/3) + a*x])/(17*x^3) - (12*a*Sqrt[b*x^(1/3) + a*x])/(221*b*x^(7/3)) + (44*a^2*Sqrt[b*x^(1/3) + a*x])/(66
3*b^2*x^(5/3)) - (308*a^3*Sqrt[b*x^(1/3) + a*x])/(3315*b^3*x) + (308*a^4*Sqrt[b*x^(1/3) + a*x])/(1105*b^4*x^(1
/3)) + (308*a^(17/4)*(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
llipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a*x]) - (154*a^(17/4)*(Sqr
t[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])/(1105*b^(15/4)*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 2045

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 + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2050

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(j - 1)*(c*x)^(m - j +
1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*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]) && LtQ[m + j*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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{10}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}+\frac {1}{17} (6 a) \text {Subst}\left (\int \frac {1}{x^7 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}-\frac {\left (66 a^2\right ) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 b} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}+\frac {\left (154 a^3\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 b^2} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}-\frac {\left (154 a^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^3} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (154 a^5\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^4} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (154 a^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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (308 a^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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (308 a^{9/2} \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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (308 a^{9/2} \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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {308 a^{9/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}+\frac {308 a^{17/4} \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 )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {154 a^{17/4} \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 )}{1105 b^{15/4} \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 = 10.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=-\frac {6 \sqrt {b \sqrt [3]{x}+a x} \operatorname {Hypergeometric2F1}\left (-\frac {17}{4},-\frac {1}{2},-\frac {13}{4},-\frac {a x^{2/3}}{b}\right )}{17 \sqrt {1+\frac {a x^{2/3}}{b}} x^3} \]

[In]

Integrate[Sqrt[b*x^(1/3) + a*x]/x^4,x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-17/4, -1/2, -13/4, -((a*x^(2/3))/b)])/(17*Sqrt[1 + (a*x^(2/3))/b]
*x^3)

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {6 \sqrt {b \,x^{\frac {1}{3}}+a x}}{17 x^{3}}-\frac {12 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{221 b \,x^{\frac {7}{3}}}+\frac {44 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 b^{2} x^{\frac {5}{3}}}-\frac {308 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3315 b^{3} x}+\frac {308 \left (b +a \,x^{\frac {2}{3}}\right ) a^{4}}{1105 b^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}-\frac {154 a^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{1105 b^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(281\)
default	\(-\frac {6 \sqrt {b \,x^{\frac {1}{3}}+a x}}{17 x^{3}}-\frac {12 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{221 b \,x^{\frac {7}{3}}}+\frac {44 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 b^{2} x^{\frac {5}{3}}}-\frac {308 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3315 b^{3} x}+\frac {308 \left (b +a \,x^{\frac {2}{3}}\right ) a^{4}}{1105 b^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}-\frac {154 a^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{1105 b^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(281\)

[In]

int((b*x^(1/3)+a*x)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-6/17*(b*x^(1/3)+a*x)^(1/2)/x^3-12/221*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(7/3)+44/663*a^2*(b*x^(1/3)+a*x)^(1/2)/b^2/
x^(5/3)-308/3315*a^3*(b*x^(1/3)+a*x)^(1/2)/b^3/x+308/1105*(b+a*x^(2/3))*a^4/b^4/(x^(1/3)*(b+a*x^(2/3)))^(1/2)-
154/1105*a^4/b^4*(-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)))

Fricas [F]

\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{4}} \,d x } \]

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(a*x + b*x^(1/3))/x^4, x)

Sympy [F]

\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=\int \frac {\sqrt {a x + b \sqrt [3]{x}}}{x^{4}}\, dx \]

[In]

integrate((b*x**(1/3)+a*x)**(1/2)/x**4,x)

[Out]

Integral(sqrt(a*x + b*x**(1/3))/x**4, x)

Maxima [F]

\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{4}} \,d x } \]

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(1/3))/x^4, x)

Giac [F]

\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{4}} \,d x } \]

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(a*x + b*x^(1/3))/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx=\int \frac {\sqrt {a\,x+b\,x^{1/3}}}{x^4} \,d x \]

[In]

int((a*x + b*x^(1/3))^(1/2)/x^4,x)

[Out]

int((a*x + b*x^(1/3))^(1/2)/x^4, x)