3.191 \(\int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx\)

Optimal. Leaf size=153 \[ \frac{105 a^3 \sqrt{a x+b x^{2/3}}}{64 b^4 x^{2/3}}-\frac{35 a^2 \sqrt{a x+b x^{2/3}}}{32 b^3 x}-\frac{105 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{64 b^{9/2}}+\frac{7 a \sqrt{a x+b x^{2/3}}}{8 b^2 x^{4/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{4 b x^{5/3}} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(4*b*x^(5/3)) + (7*a*Sqrt[b*x^(2/3) + a*x])/(8*b^2*x^(4/3)) - (35*a^2*Sqrt[b*x^(2/3
) + a*x])/(32*b^3*x) + (105*a^3*Sqrt[b*x^(2/3) + a*x])/(64*b^4*x^(2/3)) - (105*a^4*ArcTanh[(Sqrt[b]*x^(1/3))/S
qrt[b*x^(2/3) + a*x]])/(64*b^(9/2))

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Rubi [A]  time = 0.239309, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \[ \frac{105 a^3 \sqrt{a x+b x^{2/3}}}{64 b^4 x^{2/3}}-\frac{35 a^2 \sqrt{a x+b x^{2/3}}}{32 b^3 x}-\frac{105 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{64 b^{9/2}}+\frac{7 a \sqrt{a x+b x^{2/3}}}{8 b^2 x^{4/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{4 b x^{5/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(4*b*x^(5/3)) + (7*a*Sqrt[b*x^(2/3) + a*x])/(8*b^2*x^(4/3)) - (35*a^2*Sqrt[b*x^(2/3
) + a*x])/(32*b^3*x) + (105*a^3*Sqrt[b*x^(2/3) + a*x])/(64*b^4*x^(2/3)) - (105*a^4*ArcTanh[(Sqrt[b]*x^(1/3))/S
qrt[b*x^(2/3) + a*x]])/(64*b^(9/2))

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 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}-\frac{(7 a) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{8 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}+\frac{7 a \sqrt{b x^{2/3}+a x}}{8 b^2 x^{4/3}}+\frac{\left (35 a^2\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{48 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}+\frac{7 a \sqrt{b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac{35 a^2 \sqrt{b x^{2/3}+a x}}{32 b^3 x}-\frac{\left (35 a^3\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{64 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}+\frac{7 a \sqrt{b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac{35 a^2 \sqrt{b x^{2/3}+a x}}{32 b^3 x}+\frac{105 a^3 \sqrt{b x^{2/3}+a x}}{64 b^4 x^{2/3}}+\frac{\left (35 a^4\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{128 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}+\frac{7 a \sqrt{b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac{35 a^2 \sqrt{b x^{2/3}+a x}}{32 b^3 x}+\frac{105 a^3 \sqrt{b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac{\left (105 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{64 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{4 b x^{5/3}}+\frac{7 a \sqrt{b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac{35 a^2 \sqrt{b x^{2/3}+a x}}{32 b^3 x}+\frac{105 a^3 \sqrt{b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac{105 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{64 b^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0545172, size = 48, normalized size = 0.31 \[ -\frac{6 a^4 \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^5 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^4*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[1/2, 5, 3/2, 1 + (a*x^(1/3))/b])/(b^5*x^(1/3))

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Maple [A]  time = 0.007, size = 126, normalized size = 0.8 \begin{align*} -{\frac{1}{64\,{x}^{2}}\sqrt{b+a\sqrt [3]{x}} \left ( -56\,{b}^{7/2}{x}^{4/3}\sqrt{b+a\sqrt [3]{x}}a+70\,{b}^{5/2}{x}^{5/3}\sqrt{b+a\sqrt [3]{x}}{a}^{2}+105\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{7/3}{a}^{4}b+48\,\sqrt{b+a\sqrt [3]{x}}{b}^{9/2}x-105\,{b}^{3/2}{x}^{2}\sqrt{b+a\sqrt [3]{x}}{a}^{3} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}{b}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(b+a*x^(1/3))^(1/2)*(-56*b^(7/2)*x^(4/3)*(b+a*x^(1/3))^(1/2)*a+70*b^(5/2)*x^(5/3)*(b+a*x^(1/3))^(1/2)*a^
2+105*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*x^(7/3)*a^4*b+48*(b+a*x^(1/3))^(1/2)*b^(9/2)*x-105*b^(3/2)*x^2*(b+a
*x^(1/3))^(1/2)*a^3)/x^2/(b*x^(2/3)+a*x)^(1/2)/b^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18958, size = 147, normalized size = 0.96 \begin{align*} \frac{\frac{105 \, a^{5} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{105 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{5} - 385 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{5} b + 511 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{5} b^{2} - 279 \, \sqrt{a x^{\frac{1}{3}} + b} a^{5} b^{3}}{a^{4} b^{4} x^{\frac{4}{3}}}}{64 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(105*a^5*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^4) + (105*(a*x^(1/3) + b)^(7/2)*a^5 - 385*(a*x^
(1/3) + b)^(5/2)*a^5*b + 511*(a*x^(1/3) + b)^(3/2)*a^5*b^2 - 279*sqrt(a*x^(1/3) + b)*a^5*b^3)/(a^4*b^4*x^(4/3)
))/a