3.148 \(\int \frac{(b \sqrt [3]{x}+a x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=301 \[ -\frac{884 a^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{100947 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{1768 a^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac{1768 a^5 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{1768 a^4 \sqrt{a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac{136 a^3 \sqrt{a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{69 x^4}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5} \]

[Out]

(-4*a*Sqrt[b*x^(1/3) + a*x])/(69*x^4) - (8*a^2*Sqrt[b*x^(1/3) + a*x])/(1311*b*x^(10/3)) + (136*a^3*Sqrt[b*x^(1
/3) + a*x])/(19665*b^2*x^(8/3)) - (1768*a^4*Sqrt[b*x^(1/3) + a*x])/(216315*b^3*x^2) + (1768*a^5*Sqrt[b*x^(1/3)
+ a*x])/(168245*b^4*x^(4/3)) - (1768*a^6*Sqrt[b*x^(1/3) + a*x])/(100947*b^5*x^(2/3)) - (2*(b*x^(1/3) + a*x)^(
3/2))/(9*x^5) - (884*a^(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])/(100947*b^(21/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.478874, 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, 2020, 2025, 2011, 329, 220} \[ -\frac{1768 a^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac{1768 a^5 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{1768 a^4 \sqrt{a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac{136 a^3 \sqrt{a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac{884 a^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{69 x^4}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

(-4*a*Sqrt[b*x^(1/3) + a*x])/(69*x^4) - (8*a^2*Sqrt[b*x^(1/3) + a*x])/(1311*b*x^(10/3)) + (136*a^3*Sqrt[b*x^(1
/3) + a*x])/(19665*b^2*x^(8/3)) - (1768*a^4*Sqrt[b*x^(1/3) + a*x])/(216315*b^3*x^2) + (1768*a^5*Sqrt[b*x^(1/3)
+ a*x])/(168245*b^4*x^(4/3)) - (1768*a^6*Sqrt[b*x^(1/3) + a*x])/(100947*b^5*x^(2/3)) - (2*(b*x^(1/3) + a*x)^(
3/2))/(9*x^5) - (884*a^(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])/(100947*b^(21/4)*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 2020

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

Rule 2025

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 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{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (b x+a x^3\right )^{3/2}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac{1}{3} (2 a) \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac{1}{69} \left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{10} \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{\left (68 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 b}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac{\left (884 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 b^2}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{\left (884 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 b^3}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac{1768 a^5 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac{\left (884 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^4}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac{1768 a^5 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{1768 a^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{\left (884 a^7\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{100947 b^5}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac{1768 a^5 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{1768 a^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{\left (884 a^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 )}{100947 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac{1768 a^5 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{1768 a^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{\left (1768 a^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 )}{100947 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{4 a \sqrt{b \sqrt [3]{x}+a x}}{69 x^4}-\frac{8 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac{136 a^3 \sqrt{b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac{1768 a^4 \sqrt{b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac{1768 a^5 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{1768 a^6 \sqrt{b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac{2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac{884 a^{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 )}{100947 b^{21/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.06984, size = 62, normalized size = 0.21 \[ -\frac{2 b \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{27}{4},-\frac{3}{2};-\frac{23}{4};-\frac{a x^{2/3}}{b}\right )}{9 x^{14/3} \sqrt{\frac{a x^{2/3}}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

(-2*b*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-27/4, -3/2, -23/4, -((a*x^(2/3))/b)])/(9*Sqrt[1 + (a*x^(2/3))/b
]*x^(14/3))

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Maple [A]  time = 0.026, size = 201, normalized size = 0.7 \begin{align*} -{\frac{2}{1514205\,{b}^{5}} \left ( 6630\,{a}^{6}\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 ){x}^{{\frac{26}{3}}}-1768\,{x}^{{\frac{23}{3}}}{a}^{5}{b}^{2}+5304\,{x}^{{\frac{25}{3}}}{a}^{6}b+952\,{x}^{7}{a}^{4}{b}^{3}+216755\,{x}^{{\frac{17}{3}}}{a}^{2}{b}^{5}-616\,{x}^{{\frac{19}{3}}}{a}^{3}{b}^{4}+380380\,{x}^{5}a{b}^{6}+13260\,{x}^{9}{a}^{7}+168245\,{x}^{13/3}{b}^{7} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}{x}^{-{\frac{26}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^(1/3)+a*x)^(3/2)/x^6,x)

[Out]

-2/1514205*(6630*a^6*(-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))*x^(26/3)-1768*x^(23/3)*a^5*b^2+5304*x^(25/3)*a^6*b+952*x^7*a^4*b^3+216755*x^(17/3)*a^2*b^5-616*x^(19
/3)*a^3*b^4+380380*x^5*a*b^6+13260*x^9*a^7+168245*x^(13/3)*b^7)/b^5/(x^(1/3)*(b+a*x^(2/3)))^(1/2)/x^(26/3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)/x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="fricas")

[Out]

integral((a*x + b*x^(1/3))^(3/2)/x^6, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**(1/3)+a*x)**(3/2)/x**6,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="giac")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)/x^6, x)