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

Optimal. Leaf size=301 \[ \frac{442 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{14421 a^{25/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{884 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^2}-\frac{884 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^6}+\frac{4 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a}+\frac{2}{9} x^4 \sqrt{a x+b \sqrt [3]{x}} \]

[Out]

(-884*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^6) + (884*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(24035*a^5) - (6188*b^4
*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^4) + (476*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^3) - (28*b^2*x^(8/
3)*Sqrt[b*x^(1/3) + a*x])/(1311*a^2) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a) + (2*x^4*Sqrt[b*x^(1/3) +
a*x])/9 + (442*b^(27/4)*(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])/(14421*a^(25/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.509048, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 2021, 2024, 2011, 329, 220} \[ \frac{884 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^2}+\frac{442 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{25/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{884 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^6}+\frac{4 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a}+\frac{2}{9} x^4 \sqrt{a x+b \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[b*x^(1/3) + a*x],x]

[Out]

(-884*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^6) + (884*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(24035*a^5) - (6188*b^4
*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^4) + (476*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^3) - (28*b^2*x^(8/
3)*Sqrt[b*x^(1/3) + a*x])/(1311*a^2) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a) + (2*x^4*Sqrt[b*x^(1/3) +
a*x])/9 + (442*b^(27/4)*(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])/(14421*a^(25/4)*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 2021

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 + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*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]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

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 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[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 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]

Rubi steps

\begin{align*} \int x^3 \sqrt{b \sqrt [3]{x}+a x} \, dx &=3 \operatorname{Subst}\left (\int x^{11} \sqrt{b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{1}{9} (2 b) \operatorname{Subst}\left (\int \frac{x^{12}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}-\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{69 a}\\ &=-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (238 b^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a^2}\\ &=\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}-\frac{\left (3094 b^4\right ) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 a^3}\\ &=-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (3094 b^5\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 a^4}\\ &=\frac{884 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}-\frac{\left (442 b^6\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^5}\\ &=-\frac{884 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (442 b^7\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^6}\\ &=-\frac{884 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (442 b^7 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^6 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{884 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{\left (884 b^7 \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 )}{14421 a^6 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{884 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^6}+\frac{884 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{24035 a^5}-\frac{6188 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{216315 a^4}+\frac{476 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{19665 a^3}-\frac{28 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^2}+\frac{4 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a}+\frac{2}{9} x^4 \sqrt{b \sqrt [3]{x}+a x}+\frac{442 b^{27/4} \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 )}{14421 a^{25/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.160714, size = 155, normalized size = 0.51 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (\sqrt{\frac{a x^{2/3}}{b}+1} \left (-2310 a^4 b^2 x^{8/3}+2618 a^3 b^3 x^2-3094 a^2 b^4 x^{4/3}+2090 a^5 b x^{10/3}+24035 a^6 x^4+3978 a b^5 x^{2/3}-9945 b^6\right )+9945 b^6 \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )\right )}{216315 a^6 \sqrt{\frac{a x^{2/3}}{b}+1}} \]

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]*(Sqrt[1 + (a*x^(2/3))/b]*(-9945*b^6 + 3978*a*b^5*x^(2/3) - 3094*a^2*b^4*x^(4/3) + 261
8*a^3*b^3*x^2 - 2310*a^4*b^2*x^(8/3) + 2090*a^5*b*x^(10/3) + 24035*a^6*x^4) + 9945*b^6*Hypergeometric2F1[-1/2,
1/4, 5/4, -((a*x^(2/3))/b)]))/(216315*a^6*Sqrt[1 + (a*x^(2/3))/b])

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Maple [A]  time = 0.033, size = 264, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{4}}{9}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{4\,b}{207\,a}{x}^{{\frac{10}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{28\,{b}^{2}}{1311\,{a}^{2}}{x}^{{\frac{8}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{476\,{b}^{3}{x}^{2}}{19665\,{a}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{6188\,{b}^{4}}{216315\,{a}^{4}}{x}^{{\frac{4}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{884\,{b}^{5}}{24035\,{a}^{5}}{x}^{{\frac{2}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{884\,{b}^{6}}{14421\,{a}^{6}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{442\,{b}^{7}}{14421\,{a}^{7}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \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]

2/9*x^4*(b*x^(1/3)+a*x)^(1/2)+4/207*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)/a-28/1311*b^2*x^(8/3)*(b*x^(1/3)+a*x)^(1/
2)/a^2+476/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^3-6188/216315*b^4*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^4+884/24035
*b^5*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^5-884/14421*b^6*(b*x^(1/3)+a*x)^(1/2)/a^6+442/14421*b^7/a^7*(-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)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2
*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x + b x^{\frac{1}{3}}} x^{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(sqrt(a*x + b*x^(1/3))*x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a x + b x^{\frac{1}{3}}} x^{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(sqrt(a*x + b*x^(1/3))*x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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 \sqrt{a x + b x^{\frac{1}{3}}} x^{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(sqrt(a*x + b*x^(1/3))*x^3, x)