3.2.66 \(\int \frac {1}{x^4 (b \sqrt [3]{x}+a x)^{3/2}} \, dx\) [166]

Optimal. Leaf size=471 \[ \frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 b^7 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}+\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

3/b/x^(10/3)/(b*x^(1/3)+a*x)^(1/2)-4807/221*a^(11/2)*(b+a*x^(2/3))*x^(1/3)/b^7/(x^(1/3)*a^(1/2)+b^(1/2))/(b*x^
(1/3)+a*x)^(1/2)-23/7*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(11/3)+437/119*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x^3-6555/1547*a^2
*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(7/3)+24035/4641*a^3*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(5/3)-4807/663*a^4*(b*x^(1/3)+a*
x)^(1/2)/b^6/x+4807/221*a^5*(b*x^(1/3)+a*x)^(1/2)/b^7/x^(1/3)+4807/221*a^(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)/b^(27/4)/(b*x^(1
/3)+a*x)^(1/2)-4807/442*a^(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)/b^(27/4)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.49, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2043, 2048, 2050, 2057, 335, 311, 226, 1210} \begin {gather*} -\frac {4807 a^{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 )}{442 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^{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 b^{27/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {4807 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 b^7 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 a^5 \sqrt {a x+b \sqrt [3]{x}}}{221 b^7 \sqrt [3]{x}}-\frac {4807 a^4 \sqrt {a x+b \sqrt [3]{x}}}{663 b^6 x}+\frac {24035 a^3 \sqrt {a x+b \sqrt [3]{x}}}{4641 b^5 x^{5/3}}-\frac {6555 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1547 b^4 x^{7/3}}+\frac {437 a \sqrt {a x+b \sqrt [3]{x}}}{119 b^3 x^3}-\frac {23 \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{11/3}}+\frac {3}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(b*x^(1/3) + a*x)^(3/2)),x]

[Out]

3/(b*x^(10/3)*Sqrt[b*x^(1/3) + a*x]) - (4807*a^(11/2)*(b + a*x^(2/3))*x^(1/3))/(221*b^7*(Sqrt[b] + Sqrt[a]*x^(
1/3))*Sqrt[b*x^(1/3) + a*x]) - (23*Sqrt[b*x^(1/3) + a*x])/(7*b^2*x^(11/3)) + (437*a*Sqrt[b*x^(1/3) + a*x])/(11
9*b^3*x^3) - (6555*a^2*Sqrt[b*x^(1/3) + a*x])/(1547*b^4*x^(7/3)) + (24035*a^3*Sqrt[b*x^(1/3) + a*x])/(4641*b^5
*x^(5/3)) - (4807*a^4*Sqrt[b*x^(1/3) + a*x])/(663*b^6*x) + (4807*a^5*Sqrt[b*x^(1/3) + a*x])/(221*b^7*x^(1/3))
+ (4807*a^(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)*Ellip
ticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(221*b^(27/4)*Sqrt[b*x^(1/3) + a*x]) - (4807*a^(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])/(442*b^(27/4)*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 2048

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*(n - j)*(p + 1))), x] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(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]) && LtQ[p, -1]

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*} \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {Subst}\left (\int \frac {1}{x^{10} \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {69 \text {Subst}\left (\int \frac {1}{x^{11} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}-\frac {(437 a) \text {Subst}\left (\int \frac {1}{x^9 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14 b^2}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}+\frac {\left (6555 a^2\right ) \text {Subst}\left (\int \frac {1}{x^7 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 b^3}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}-\frac {\left (72105 a^3\right ) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3094 b^4}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}+\frac {\left (24035 a^4\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1326 b^5}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}-\frac {\left (4807 a^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^6}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 b^7}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6 \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 )}{442 b^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^6 \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 b^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}-\frac {\left (4807 a^{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 b^{13/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (4807 a^{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 b^{13/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 b^7 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}+\frac {4807 a^{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 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{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 )}{442 b^{27/4} \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 = 10.04, size = 64, normalized size = 0.14 \begin {gather*} -\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (-\frac {21}{4},\frac {3}{2};-\frac {17}{4};-\frac {a x^{2/3}}{b}\right )}{7 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(b*x^(1/3) + a*x)^(3/2)),x]

[Out]

(-2*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-21/4, 3/2, -17/4, -((a*x^(2/3))/b)])/(7*b*x^(10/3)*Sqrt[b*x^(1/
3) + a*x])

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Maple [A]
time = 0.35, size = 411, normalized size = 0.87

method	result	size
derivativedivides	\(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 b^{2} x^{\frac {11}{3}}}+\frac {80 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 b^{3} x^{3}}-\frac {1914 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 b^{4} x^{\frac {7}{3}}}+\frac {10112 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 b^{5} x^{\frac {5}{3}}}-\frac {2818 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 b^{6} x}+\frac {4144 \left (b +a \,x^{\frac {2}{3}}\right ) a^{5}}{221 b^{7} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}+\frac {3 x^{\frac {2}{3}} a^{6}}{b^{7} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}-\frac {4807 a^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{442 b^{7} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(333\)
default	\(-\frac {201894 a^{5} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-100947 a^{5} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-201894 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {22}{3}} a^{6}-174048 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {20}{3}} a^{5} b +19228 x^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} b^{2}+39452 x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} b +5244 x^{\frac {14}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{4}-8740 x^{\frac {16}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b^{3}-3588 x^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{5}+2652 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{6}}{9282 x^{7} \left (b +a \,x^{\frac {2}{3}}\right ) b^{7}}\)	\(411\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9282*(201894*a^5*b*((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)*x^(20/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*EllipticE(((a*x^(1/3)+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-100947*a^5*b*((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)*x^(20/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*Elli
pticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-201894*(b*x^(1/3)+a*x)^(1/2)*x^(22/3)*a^6-174
048*(b*x^(1/3)+a*x)^(1/2)*x^(20/3)*a^5*b+19228*x^6*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^4*b^2+39452*x^(20/3)*(x^(1/
3)*(b+a*x^(2/3)))^(1/2)*a^5*b+5244*x^(14/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^2*b^4-8740*x^(16/3)*(x^(1/3)*(b+a*
x^(2/3)))^(1/2)*a^3*b^3-3588*x^4*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a*b^5+2652*x^(10/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/
2)*b^6)/x^7/(b+a*x^(2/3))/b^7

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

[Out]

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

[Out]

integral((a^4*x^3 + 3*a^2*b^2*x^(5/3) - 2*a*b^3*x - (2*a^3*b*x^2 - b^4)*x^(1/3))*sqrt(a*x + b*x^(1/3))/(a^6*x^
9 + 2*a^3*b^3*x^7 + b^6*x^5), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \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(1/x**4/(b*x**(1/3)+a*x)**(3/2),x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\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(1/(x^4*(a*x + b*x^(1/3))^(3/2)),x)

[Out]

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

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