3.139 \(\int \frac{\sqrt{b \sqrt [3]{x}+a x}}{x^5} \, dx\)

Optimal. Leaf size=276 \[ -\frac{1326 a^{23/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{33649 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{2652 a^5 \sqrt{a x+b \sqrt [3]{x}}}{33649 b^5 x^{2/3}}+\frac{7956 a^4 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{884 a^3 \sqrt{a x+b \sqrt [3]{x}}}{24035 b^3 x^2}+\frac{68 a^2 \sqrt{a x+b \sqrt [3]{x}}}{2185 b^2 x^{8/3}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{437 b x^{10/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{23 x^4} \]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(23*x^4) - (12*a*Sqrt[b*x^(1/3) + a*x])/(437*b*x^(10/3)) + (68*a^2*Sqrt[b*x^(1/3) +
a*x])/(2185*b^2*x^(8/3)) - (884*a^3*Sqrt[b*x^(1/3) + a*x])/(24035*b^3*x^2) + (7956*a^4*Sqrt[b*x^(1/3) + a*x])
/(168245*b^4*x^(4/3)) - (2652*a^5*Sqrt[b*x^(1/3) + a*x])/(33649*b^5*x^(2/3)) - (1326*a^(23/4)*(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])/(33649*b^(21/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.412899, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{2652 a^5 \sqrt{a x+b \sqrt [3]{x}}}{33649 b^5 x^{2/3}}+\frac{7956 a^4 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{884 a^3 \sqrt{a x+b \sqrt [3]{x}}}{24035 b^3 x^2}+\frac{68 a^2 \sqrt{a x+b \sqrt [3]{x}}}{2185 b^2 x^{8/3}}-\frac{1326 a^{23/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{437 b x^{10/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{23 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(1/3) + a*x]/x^5,x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(23*x^4) - (12*a*Sqrt[b*x^(1/3) + a*x])/(437*b*x^(10/3)) + (68*a^2*Sqrt[b*x^(1/3) +
a*x])/(2185*b^2*x^(8/3)) - (884*a^3*Sqrt[b*x^(1/3) + a*x])/(24035*b^3*x^2) + (7956*a^4*Sqrt[b*x^(1/3) + a*x])
/(168245*b^4*x^(4/3)) - (2652*a^5*Sqrt[b*x^(1/3) + a*x])/(33649*b^5*x^(2/3)) - (1326*a^(23/4)*(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])/(33649*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{\sqrt{b \sqrt [3]{x}+a x}}{x^5} \, dx &=3 \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}+\frac{1}{23} (6 a) \operatorname{Subst}\left (\int \frac{1}{x^{10} \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}-\frac{\left (102 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{437 b}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}+\frac{\left (442 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2185 b^2}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}-\frac{\left (3978 a^4\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{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac{7956 a^4 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}+\frac{\left (3978 a^5\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{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac{7956 a^4 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{2652 a^5 \sqrt{b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac{\left (1326 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^5}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac{7956 a^4 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{2652 a^5 \sqrt{b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac{\left (1326 a^6 \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 )}{33649 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac{7956 a^4 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{2652 a^5 \sqrt{b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac{\left (2652 a^6 \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 )}{33649 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{23 x^4}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac{68 a^2 \sqrt{b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac{884 a^3 \sqrt{b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac{7956 a^4 \sqrt{b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac{2652 a^5 \sqrt{b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac{1326 a^{23/4} \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 )}{33649 b^{21/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}

Mathematica [C]  time = 0.0546913, size = 59, normalized size = 0.21 \[ -\frac{6 \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{23}{4},-\frac{1}{2};-\frac{19}{4};-\frac{a x^{2/3}}{b}\right )}{23 x^4 \sqrt{\frac{a x^{2/3}}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^(1/3) + a*x]/x^5,x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-23/4, -1/2, -19/4, -((a*x^(2/3))/b)])/(23*Sqrt[1 + (a*x^(2/3))/b]
*x^4)

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Maple [A]  time = 0.019, size = 245, normalized size = 0.9 \begin{align*} -{\frac{6}{23\,{x}^{4}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{12\,a}{437\,b}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{10}{3}}}}+{\frac{68\,{a}^{2}}{2185\,{b}^{2}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{8}{3}}}}-{\frac{884\,{a}^{3}}{24035\,{b}^{3}{x}^{2}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{7956\,{a}^{4}}{168245\,{b}^{4}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{4}{3}}}}-{\frac{2652\,{a}^{5}}{33649\,{b}^{5}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{2}{3}}}}-{\frac{1326\,{a}^{5}}{33649\,{b}^{5}}\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((b*x^(1/3)+a*x)^(1/2)/x^5,x)

[Out]

-6/23*(b*x^(1/3)+a*x)^(1/2)/x^4-12/437*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(10/3)+68/2185*a^2*(b*x^(1/3)+a*x)^(1/2)/b^
2/x^(8/3)-884/24035*a^3*(b*x^(1/3)+a*x)^(1/2)/b^3/x^2+7956/168245*a^4*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)-2652/3
3649*a^5*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)-1326/33649*a^5/b^5*(-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 \frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(1/3))/x^5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(a*x + b*x^(1/3))/x^5, x)