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

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

[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])

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Rubi [A]  time = 0.566179, antiderivative size = 414, normalized size of antiderivative = 1., 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 = {2018, 2024, 2032, 329, 305, 220, 1196} \[ -\frac{418 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{11/2} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{570 b^2 x^{5/3} \sqrt{a x+b \sqrt [3]{x}}}{1547 a^3}-\frac{209 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{23/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{418 b^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{221 a^{23/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{418 b^4 \sqrt [3]{x} \sqrt{a x+b \sqrt [3]{x}}}{663 a^5}-\frac{2090 b^3 x \sqrt{a x+b \sqrt [3]{x}}}{4641 a^4}-\frac{38 b x^{7/3} \sqrt{a x+b \sqrt [3]{x}}}{119 a^2}+\frac{2 x^3 \sqrt{a x+b \sqrt [3]{x}}}{7 a} \]

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 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 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^3}{\sqrt{b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \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^{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*}

Mathematica [C]  time = 0.0707897, size = 143, normalized size = 0.35 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (114 a^3 b^2 x^{7/3}-190 a^2 b^3 x^{5/3}-78 a^4 b x^3+663 a^5 x^{11/3}-1463 b^5 \sqrt [3]{x} \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{a x^{2/3}}{b}\right )+418 a b^4 x+1463 b^5 \sqrt [3]{x}\right )}{4641 a^5 \left (a x^{2/3}+b\right )} \]

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 = 0.024, size = 261, normalized size = 0.6 \begin{align*} -{\frac{1}{4641\,{a}^{6}} \left ( -228\,{x}^{8/3}{a}^{4}{b}^{2}+156\,{x}^{10/3}{a}^{5}b+380\,{x}^{2}{a}^{3}{b}^{3}+8778\,{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}}}}{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -4389\,{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}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -1326\,{x}^{4}{a}^{6}-2926\,{x}^{2/3}a{b}^{5}-836\,{x}^{4/3}{a}^{2}{b}^{4} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4641/a^6*(-228*x^(8/3)*a^4*b^2+156*x^(10/3)*a^5*b+380*x^2*a^3*b^3+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*x^4*a^6-2926*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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} x^{4} - a b x^{\frac{10}{3}} + b^{2} x^{\frac{8}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{3} x^{2} + b^{3}}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a x + b \sqrt [3]{x}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \end{align*}

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)