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

Optimal. Leaf size=304 \[ -\frac{5525 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{14421 a^{29/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{2210 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^3}+\frac{11050 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^7}-\frac{50 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a^2}+\frac{2 x^4 \sqrt{a x+b \sqrt [3]{x}}}{9 a} \]

[Out]

(11050*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^7) - (2210*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(4807*a^6) + (15470*b
^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(43263*a^5) - (1190*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(3933*a^4) + (350*b^2*x^(
8/3)*Sqrt[b*x^(1/3) + a*x])/(1311*a^3) - (50*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a^2) + (2*x^4*Sqrt[b*x^(1/
3) + a*x])/(9*a) - (5525*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^(29/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.50605, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2018, 2024, 2011, 329, 220} \[ -\frac{2210 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^3}-\frac{5525 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{29/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{11050 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^7}-\frac{50 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a^2}+\frac{2 x^4 \sqrt{a x+b \sqrt [3]{x}}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[b*x^(1/3) + a*x],x]

[Out]

(11050*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^7) - (2210*b^5*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(4807*a^6) + (15470*b
^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(43263*a^5) - (1190*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(3933*a^4) + (350*b^2*x^(
8/3)*Sqrt[b*x^(1/3) + a*x])/(1311*a^3) - (50*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a^2) + (2*x^4*Sqrt[b*x^(1/
3) + a*x])/(9*a) - (5525*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^(29/4)*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 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 \frac{x^4}{\sqrt{b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{14}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{(25 b) \operatorname{Subst}\left (\int \frac{x^{12}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{9 a}\\ &=-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}+\frac{\left (175 b^2\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{69 a^2}\\ &=\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{\left (2975 b^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a^3}\\ &=-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}+\frac{\left (7735 b^4\right ) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3933 a^4}\\ &=\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{\left (7735 b^5\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^5}\\ &=-\frac{2210 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}+\frac{\left (5525 b^6\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^6}\\ &=\frac{11050 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{\left (5525 b^7\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^7}\\ &=\frac{11050 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{\left (5525 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^7 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{11050 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{\left (11050 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^7 \sqrt{b \sqrt [3]{x}+a x}}\\ &=\frac{11050 b^6 \sqrt{b \sqrt [3]{x}+a x}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{b \sqrt [3]{x}+a x}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{b \sqrt [3]{x}+a x}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{b \sqrt [3]{x}+a x}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{b \sqrt [3]{x}+a x}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{b \sqrt [3]{x}+a x}}{207 a^2}+\frac{2 x^4 \sqrt{b \sqrt [3]{x}+a x}}{9 a}-\frac{5525 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^{29/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.0922186, size = 161, normalized size = 0.53 \[ \frac{2 \sqrt{a x+b \sqrt [3]{x}} \left (550 a^5 b^2 x^{10/3}-770 a^4 b^3 x^{8/3}+1190 a^3 b^4 x^2-2210 a^2 b^5 x^{4/3}-418 a^6 b x^4+4807 a^7 x^{14/3}-16575 b^7 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{a x^{2/3}}{b}\right )+6630 a b^6 x^{2/3}+16575 b^7\right )}{43263 a^7 \left (a x^{2/3}+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(1/3) + a*x]*(16575*b^7 + 6630*a*b^6*x^(2/3) - 2210*a^2*b^5*x^(4/3) + 1190*a^3*b^4*x^2 - 770*a^4*b
^3*x^(8/3) + 550*a^5*b^2*x^(10/3) - 418*a^6*b*x^4 + 4807*a^7*x^(14/3) - 16575*b^7*Sqrt[1 + (a*x^(2/3))/b]*Hype
rgeometric2F1[1/4, 1/2, 5/4, -((a*x^(2/3))/b)]))/(43263*a^7*(b + a*x^(2/3)))

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Maple [A]  time = 0.032, size = 196, normalized size = 0.6 \begin{align*} -{\frac{1}{43263\,{a}^{8}} \left ( -1100\,{x}^{11/3}{a}^{6}{b}^{2}+836\,{x}^{13/3}{a}^{7}b+1540\,{x}^{3}{a}^{5}{b}^{3}+4420\,{x}^{5/3}{a}^{3}{b}^{5}-2380\,{x}^{7/3}{a}^{4}{b}^{4}-9614\,{x}^{5}{a}^{8}+16575\,{b}^{7}\sqrt{-ab}\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 ) -13260\,x{a}^{2}{b}^{6}-33150\,\sqrt [3]{x}a{b}^{7} \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^4/(b*x^(1/3)+a*x)^(1/2),x)

[Out]

-1/43263*(-1100*x^(11/3)*a^6*b^2+836*x^(13/3)*a^7*b+1540*x^3*a^5*b^3+4420*x^(5/3)*a^3*b^5-2380*x^(7/3)*a^4*b^4
-9614*x^5*a^8+16575*b^7*(-a*b)^(1/2)*((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))-13260*x*a^2*b^6-33150*x^(1/3)*a*b^7)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)/a^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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^4/(b*x^(1/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/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^{5} - a b x^{\frac{13}{3}} + b^{2} x^{\frac{11}{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^4/(b*x^(1/3)+a*x)^(1/2),x, algorithm="fricas")

[Out]

integral((a^2*x^5 - a*b*x^(13/3) + b^2*x^(11/3))*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^{4}}{\sqrt{a x + b \sqrt [3]{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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