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

Optimal. Leaf size=236 \[ \frac{9009 a^5 \sqrt{a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac{3003 a^4 \sqrt{a x+b x^{2/3}}}{256 b^6 x}+\frac{3003 a^3 \sqrt{a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac{1287 a^2 \sqrt{a x+b x^{2/3}}}{160 b^4 x^{5/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac{143 a \sqrt{a x+b x^{2/3}}}{20 b^3 x^2}-\frac{13 \sqrt{a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac{6}{b x^{5/3} \sqrt{a x+b x^{2/3}}} \]

[Out]

6/(b*x^(5/3)*Sqrt[b*x^(2/3) + a*x]) - (13*Sqrt[b*x^(2/3) + a*x])/(2*b^2*x^(7/3)) + (143*a*Sqrt[b*x^(2/3) + a*x
])/(20*b^3*x^2) - (1287*a^2*Sqrt[b*x^(2/3) + a*x])/(160*b^4*x^(5/3)) + (3003*a^3*Sqrt[b*x^(2/3) + a*x])/(320*b
^5*x^(4/3)) - (3003*a^4*Sqrt[b*x^(2/3) + a*x])/(256*b^6*x) + (9009*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^7*x^(2/3)
) - (9009*a^6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(512*b^(15/2))

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Rubi [A]  time = 0.410442, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2029, 206} \[ \frac{9009 a^5 \sqrt{a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac{3003 a^4 \sqrt{a x+b x^{2/3}}}{256 b^6 x}+\frac{3003 a^3 \sqrt{a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac{1287 a^2 \sqrt{a x+b x^{2/3}}}{160 b^4 x^{5/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac{143 a \sqrt{a x+b x^{2/3}}}{20 b^3 x^2}-\frac{13 \sqrt{a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac{6}{b x^{5/3} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

6/(b*x^(5/3)*Sqrt[b*x^(2/3) + a*x]) - (13*Sqrt[b*x^(2/3) + a*x])/(2*b^2*x^(7/3)) + (143*a*Sqrt[b*x^(2/3) + a*x
])/(20*b^3*x^2) - (1287*a^2*Sqrt[b*x^(2/3) + a*x])/(160*b^4*x^(5/3)) + (3003*a^3*Sqrt[b*x^(2/3) + a*x])/(320*b
^5*x^(4/3)) - (3003*a^4*Sqrt[b*x^(2/3) + a*x])/(256*b^6*x) + (9009*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^7*x^(2/3)
) - (9009*a^6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(512*b^(15/2))

Rule 2023

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*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] &
& (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]

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 \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}+\frac{13 \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{b}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}-\frac{(143 a) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{12 b^2}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}+\frac{\left (429 a^2\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{40 b^3}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}-\frac{\left (3003 a^3\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{320 b^4}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}+\frac{\left (1001 a^4\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{128 b^5}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}-\frac{\left (3003 a^5\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{512 b^6}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}+\frac{\left (3003 a^6\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{1024 b^7}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac{\left (9009 a^6\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 )}{512 b^7}\\ &=\frac{6}{b x^{5/3} \sqrt{b x^{2/3}+a x}}-\frac{13 \sqrt{b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac{143 a \sqrt{b x^{2/3}+a x}}{20 b^3 x^2}-\frac{1287 a^2 \sqrt{b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac{3003 a^3 \sqrt{b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac{3003 a^4 \sqrt{b x^{2/3}+a x}}{256 b^6 x}+\frac{9009 a^5 \sqrt{b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{512 b^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0578095, size = 48, normalized size = 0.2 \[ \frac{6 a^6 \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},7;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^7 \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 126, normalized size = 0.5 \begin{align*} -{\frac{1}{2560\,x} \left ( b+a\sqrt [3]{x} \right ) \left ( 45045\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{2}{a}^{6}-1664\,{b}^{11/2}\sqrt [3]{x}a+2288\,{b}^{9/2}{x}^{2/3}{a}^{2}-3432\,{b}^{7/2}x{a}^{3}+6006\,{b}^{5/2}{x}^{4/3}{a}^{4}-15015\,{b}^{3/2}{x}^{5/3}{a}^{5}-45045\,{x}^{2}{a}^{6}\sqrt{b}+1280\,{b}^{13/2} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2560*(b+a*x^(1/3))*(45045*(b+a*x^(1/3))^(1/2)*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*x^2*a^6-1664*b^(11/2)*x^
(1/3)*a+2288*b^(9/2)*x^(2/3)*a^2-3432*b^(7/2)*x*a^3+6006*b^(5/2)*x^(4/3)*a^4-15015*b^(3/2)*x^(5/3)*a^5-45045*x
^2*a^6*b^(1/2)+1280*b^(13/2))/x/(b*x^(2/3)+a*x)^(3/2)/b^(15/2)

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

[Out]

integrate(1/((a*x + b*x^(2/3))^(3/2)*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)^(3/2),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} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29799, size = 211, normalized size = 0.89 \begin{align*} \frac{9009 \, a^{6} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{512 \, \sqrt{-b} b^{7}} + \frac{6 \, a^{6}}{\sqrt{a x^{\frac{1}{3}} + b} b^{7}} + \frac{29685 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{6} - 163095 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{6} b + 364194 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{6} b^{2} - 416094 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{6} b^{3} + 246505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{6} b^{4} - 62475 \, \sqrt{a x^{\frac{1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9009/512*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + 6*a^6/(sqrt(a*x^(1/3) + b)*b^7) + 1/2560*(2
9685*(a*x^(1/3) + b)^(11/2)*a^6 - 163095*(a*x^(1/3) + b)^(9/2)*a^6*b + 364194*(a*x^(1/3) + b)^(7/2)*a^6*b^2 -
416094*(a*x^(1/3) + b)^(5/2)*a^6*b^3 + 246505*(a*x^(1/3) + b)^(3/2)*a^6*b^4 - 62475*sqrt(a*x^(1/3) + b)*a^6*b^
5)/(a^6*b^7*x^2)