3.2.55 \(\int \frac {1}{x^2 \sqrt {b \sqrt [3]{x}+a x}} \, dx\) [155]

Optimal. Leaf size=163 \[ -\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {5 a^{7/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 b^{9/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

-6/7*(b*x^(1/3)+a*x)^(1/2)/b/x^(4/3)+10/7*a*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(2/3)+5/7*a^(7/4)*x^(1/6)*(cos(2*arcta
n(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))*EllipticF(sin(2*arctan(a^(1/4)*x^(
1/6)/b^(1/4))),1/2*2^(1/2))*(x^(1/3)*a^(1/2)+b^(1/2))*((b+a*x^(2/3))/(x^(1/3)*a^(1/2)+b^(1/2))^2)^(1/2)/b^(9/4
)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2043, 2050, 2036, 335, 226} \begin {gather*} \frac {5 a^{7/4} \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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 b^{9/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {10 a \sqrt {a x+b \sqrt [3]{x}}}{7 b^2 x^{2/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[b*x^(1/3) + a*x]),x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(7*b*x^(4/3)) + (10*a*Sqrt[b*x^(1/3) + a*x])/(7*b^2*x^(2/3)) + (5*a^(7/4)*(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])/(7*b^(9/4)*Sqrt[b*x^(1/3) + a*x])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

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 2036

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 2043

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 2050

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]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \text {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}-\frac {(15 a) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (5 a^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{7 b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {\left (10 a^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{7 b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{7 b x^{4/3}}+\frac {10 a \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{2/3}}+\frac {5 a^{7/4} \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 )}{7 b^{9/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.04, size = 59, normalized size = 0.36 \begin {gather*} -\frac {6 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};-\frac {a x^{2/3}}{b}\right )}{7 x \sqrt {b \sqrt [3]{x}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((a*x^(2/3))/b)])/(7*x*Sqrt[b*x^(1/3) + a*x])

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Maple [A]
time = 0.35, size = 142, normalized size = 0.87

method	result	size
default	\(\frac {5 a \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) x^{\frac {4}{3}}+4 a b x +10 x^{\frac {5}{3}} a^{2}-6 b^{2} x^{\frac {1}{3}}}{7 b^{2} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{\frac {4}{3}}}\)	\(142\)
derivativedivides	\(-\frac {6 \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 b \,x^{\frac {4}{3}}}+\frac {10 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 b^{2} x^{\frac {2}{3}}}+\frac {5 a \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 b^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(5*a*(-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^
(4/3)+4*a*b*x+10*x^(5/3)*a^2-6*b^2*x^(1/3))/b^2/(x^(1/3)*(b+a*x^(2/3)))^(1/2)/x^(4/3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*x^2 - a*b*x^(4/3) + b^2*x^(2/3))*sqrt(a*x + b*x^(1/3))/(a^3*x^5 + b^3*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {a x + b \sqrt [3]{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*sqrt(a*x + b*x**(1/3))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,\sqrt {a\,x+b\,x^{1/3}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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