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

Optimal. Leaf size=437 \[ -\frac{4807 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{442 a^{27/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{4807 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{13/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{6555 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}}{1547 a^4}+\frac{4807 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{663 a^6}-\frac{24035 b^3 x \sqrt{a x+b \sqrt [3]{x}}}{4641 a^5}-\frac{437 b x^{7/3} \sqrt{a x+b \sqrt [3]{x}}}{119 a^3}+\frac{23 x^3 \sqrt{a x+b \sqrt [3]{x}}}{7 a^2}-\frac{3 x^4}{a \sqrt{a x+b \sqrt [3]{x}}} \]

[Out]

(-4807*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(13/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (3*x^4)
/(a*Sqrt[b*x^(1/3) + a*x]) + (4807*b^4*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(663*a^6) - (24035*b^3*x*Sqrt[b*x^(1/3)
+ a*x])/(4641*a^5) + (6555*b^2*x^(5/3)*Sqrt[b*x^(1/3) + a*x])/(1547*a^4) - (437*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x
])/(119*a^3) + (23*x^3*Sqrt[b*x^(1/3) + a*x])/(7*a^2) + (4807*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^(
27/4)*Sqrt[b*x^(1/3) + a*x]) - (4807*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])/(442*a^(27/4)*Sqrt[b*x^(1/3) + a*
x])

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Rubi [A]  time = 0.673587, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2022, 2024, 2032, 329, 305, 220, 1196} \[ -\frac{4807 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{13/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{6555 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}}{1547 a^4}-\frac{4807 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{442 a^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{27/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{663 a^6}-\frac{24035 b^3 x \sqrt{a x+b \sqrt [3]{x}}}{4641 a^5}-\frac{437 b x^{7/3} \sqrt{a x+b \sqrt [3]{x}}}{119 a^3}+\frac{23 x^3 \sqrt{a x+b \sqrt [3]{x}}}{7 a^2}-\frac{3 x^4}{a \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(b*x^(1/3) + a*x)^(3/2),x]

[Out]

(-4807*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(13/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) - (3*x^4)
/(a*Sqrt[b*x^(1/3) + a*x]) + (4807*b^4*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(663*a^6) - (24035*b^3*x*Sqrt[b*x^(1/3)
+ a*x])/(4641*a^5) + (6555*b^2*x^(5/3)*Sqrt[b*x^(1/3) + a*x])/(1547*a^4) - (437*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x
])/(119*a^3) + (23*x^3*Sqrt[b*x^(1/3) + a*x])/(7*a^2) + (4807*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^(
27/4)*Sqrt[b*x^(1/3) + a*x]) - (4807*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])/(442*a^(27/4)*Sqrt[b*x^(1/3) + a*
x])

Rule 2018

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 2022

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*(n - j)*(p + 1)), x] - Dist[(c^n*(m + j*p - n + j + 1))/(b*(n - j)*(p + 1)), I
nt[(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] && (I
ntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1] && GtQ[m + j*p + 1, n - j]

Rule 2024

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 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[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]

Rule 329

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 305

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 220

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

Rule 1196

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

Rubi steps

\begin{align*} \int \frac{x^4}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{14}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{69 \operatorname{Subst}\left (\int \frac{x^{11}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{(437 b) \operatorname{Subst}\left (\int \frac{x^9}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14 a^2}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}+\frac{\left (6555 b^2\right ) \operatorname{Subst}\left (\int \frac{x^7}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 a^3}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{\left (72105 b^3\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3094 a^4}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}+\frac{\left (24035 b^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1326 a^5}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^6}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{\left (4807 b^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 a^6}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^6}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{\left (4807 b^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{442 a^6 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^6}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{\left (4807 b^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^6 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^6}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}-\frac{\left (4807 b^{11/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^{13/2} \sqrt{b \sqrt [3]{x}+a x}}+\frac{\left (4807 b^{11/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{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^{13/2} \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{4807 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{13/2} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{3 x^4}{a \sqrt{b \sqrt [3]{x}+a x}}+\frac{4807 b^4 \sqrt [3]{x} \sqrt{b \sqrt [3]{x}+a x}}{663 a^6}-\frac{24035 b^3 x \sqrt{b \sqrt [3]{x}+a x}}{4641 a^5}+\frac{6555 b^2 x^{5/3} \sqrt{b \sqrt [3]{x}+a x}}{1547 a^4}-\frac{437 b x^{7/3} \sqrt{b \sqrt [3]{x}+a x}}{119 a^3}+\frac{23 x^3 \sqrt{b \sqrt [3]{x}+a x}}{7 a^2}+\frac{4807 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^{27/4} \sqrt{b \sqrt [3]{x}+a x}}-\frac{4807 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 )}{442 a^{27/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.0933211, size = 131, normalized size = 0.3 \[ \frac{2 x^{2/3} \left (1311 a^3 b^2 x^2-2185 a^2 b^3 x^{4/3}-897 a^4 b x^{8/3}+663 a^5 x^{10/3}+33649 b^5 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{a x^{2/3}}{b}\right )+4807 a b^4 x^{2/3}-33649 b^5\right )}{4641 a^6 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(2/3)*(-33649*b^5 + 4807*a*b^4*x^(2/3) - 2185*a^2*b^3*x^(4/3) + 1311*a^3*b^2*x^2 - 897*a^4*b*x^(8/3) + 66
3*a^5*x^(10/3) + 33649*b^5*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[3/4, 3/2, 7/4, -((a*x^(2/3))/b)]))/(4641*
a^6*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.026, size = 384, normalized size = 0.9 \begin{align*} -{\frac{1}{9282\,{a}^{7}} \left ( -5244\,{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{4}{b}^{2}+3588\,{x}^{10/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{5}b+8740\,{x}^{2}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{3}{b}^{3}+201894\,{b}^{6}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -100947\,{b}^{6}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -2652\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4}{a}^{6}-39452\,{x}^{2/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }a{b}^{5}-27846\,{x}^{2/3}\sqrt{b\sqrt [3]{x}+ax}a{b}^{5}-19228\,{x}^{4/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{a}^{2}{b}^{4} \right ){\frac{1}{\sqrt [3]{x}}} \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9282/a^7*(-5244*x^(8/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^4*b^2+3588*x^(10/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a
^5*b+8740*x^2*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^3*b^3+201894*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)*(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*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)*(x^(1/3)*(b+a
*x^(2/3)))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-2652*(x^(1/3)*(b+a*x^(2/
3)))^(1/2)*x^4*a^6-39452*x^(2/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a*b^5-27846*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)*a*b^5
-19228*x^(4/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^2*b^4)/x^(1/3)/(b+a*x^(2/3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} x^{6} + 3 \, a^{2} b^{2} x^{\frac{14}{3}} - 2 \, a b^{3} x^{4} -{\left (2 \, a^{3} b x^{5} - b^{4} x^{3}\right )} x^{\frac{1}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + b^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*x^6 + 3*a^2*b^2*x^(14/3) - 2*a*b^3*x^4 - (2*a^3*b*x^5 - b^4*x^3)*x^(1/3))*sqrt(a*x + b*x^(1/3))/
(a^6*x^4 + 2*a^3*b^3*x^2 + b^6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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