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

Optimal. Leaf size=241 \[ -\frac{1287 a^6 \sqrt{a x+b x^{2/3}}}{1024 b^7 x^{2/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{512 b^6 x}-\frac{429 a^4 \sqrt{a x+b x^{2/3}}}{640 b^5 x^{4/3}}+\frac{1287 a^3 \sqrt{a x+b x^{2/3}}}{2240 b^4 x^{5/3}}-\frac{143 a^2 \sqrt{a x+b x^{2/3}}}{280 b^3 x^2}+\frac{1287 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{1024 b^{15/2}}+\frac{13 a \sqrt{a x+b x^{2/3}}}{28 b^2 x^{7/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{7 b x^{8/3}} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(7*b*x^(8/3)) + (13*a*Sqrt[b*x^(2/3) + a*x])/(28*b^2*x^(7/3)) - (143*a^2*Sqrt[b*x^(
2/3) + a*x])/(280*b^3*x^2) + (1287*a^3*Sqrt[b*x^(2/3) + a*x])/(2240*b^4*x^(5/3)) - (429*a^4*Sqrt[b*x^(2/3) + a
*x])/(640*b^5*x^(4/3)) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^6*x) - (1287*a^6*Sqrt[b*x^(2/3) + a*x])/(1024*
b^7*x^(2/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

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Rubi [A]  time = 0.407532, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \[ -\frac{1287 a^6 \sqrt{a x+b x^{2/3}}}{1024 b^7 x^{2/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{512 b^6 x}-\frac{429 a^4 \sqrt{a x+b x^{2/3}}}{640 b^5 x^{4/3}}+\frac{1287 a^3 \sqrt{a x+b x^{2/3}}}{2240 b^4 x^{5/3}}-\frac{143 a^2 \sqrt{a x+b x^{2/3}}}{280 b^3 x^2}+\frac{1287 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{1024 b^{15/2}}+\frac{13 a \sqrt{a x+b x^{2/3}}}{28 b^2 x^{7/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{7 b x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(7*b*x^(8/3)) + (13*a*Sqrt[b*x^(2/3) + a*x])/(28*b^2*x^(7/3)) - (143*a^2*Sqrt[b*x^(
2/3) + a*x])/(280*b^3*x^2) + (1287*a^3*Sqrt[b*x^(2/3) + a*x])/(2240*b^4*x^(5/3)) - (429*a^4*Sqrt[b*x^(2/3) + a
*x])/(640*b^5*x^(4/3)) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^6*x) - (1287*a^6*Sqrt[b*x^(2/3) + a*x])/(1024*
b^7*x^(2/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

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^3 \sqrt{b x^{2/3}+a x}} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}-\frac{(13 a) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{14 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}+\frac{\left (143 a^2\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{168 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}-\frac{\left (429 a^3\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{560 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}+\frac{\left (429 a^4\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{640 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac{429 a^4 \sqrt{b x^{2/3}+a x}}{640 b^5 x^{4/3}}-\frac{\left (143 a^5\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{256 b^5}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac{429 a^4 \sqrt{b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{512 b^6 x}+\frac{\left (429 a^6\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{1024 b^6}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac{429 a^4 \sqrt{b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{512 b^6 x}-\frac{1287 a^6 \sqrt{b x^{2/3}+a x}}{1024 b^7 x^{2/3}}-\frac{\left (429 a^7\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{2048 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac{429 a^4 \sqrt{b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{512 b^6 x}-\frac{1287 a^6 \sqrt{b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac{\left (1287 a^7\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 )}{1024 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{7 b x^{8/3}}+\frac{13 a \sqrt{b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac{143 a^2 \sqrt{b x^{2/3}+a x}}{280 b^3 x^2}+\frac{1287 a^3 \sqrt{b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac{429 a^4 \sqrt{b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{512 b^6 x}-\frac{1287 a^6 \sqrt{b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac{1287 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{1024 b^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0558736, size = 48, normalized size = 0.2 \[ \frac{6 a^7 \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{1}{2},8;\frac{3}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^8 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a^7*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[1/2, 8, 3/2, 1 + (a*x^(1/3))/b])/(b^8*x^(1/3))

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Maple [A]  time = 0.006, size = 188, normalized size = 0.8 \begin{align*} -{\frac{1}{35840\,{x}^{4}}\sqrt{b+a\sqrt [3]{x}} \left ( 24024\,{b}^{7/2}{x}^{10/3}\sqrt{b+a\sqrt [3]{x}}{a}^{4}+45045\,{b}^{3/2}{x}^{4}\sqrt{b+a\sqrt [3]{x}}{a}^{6}-45045\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{13/3}{a}^{7}b-16640\,{b}^{13/2}{x}^{7/3}\sqrt{b+a\sqrt [3]{x}}a-30030\,{b}^{5/2}{x}^{11/3}\sqrt{b+a\sqrt [3]{x}}{a}^{5}-20592\,{b}^{9/2}{x}^{3}\sqrt{b+a\sqrt [3]{x}}{a}^{3}+18304\,{b}^{11/2}{x}^{8/3}\sqrt{b+a\sqrt [3]{x}}{a}^{2}+15360\,\sqrt{b+a\sqrt [3]{x}}{b}^{15/2}{x}^{2} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}{b}^{-{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35840/x^4*(b+a*x^(1/3))^(1/2)*(24024*b^(7/2)*x^(10/3)*(b+a*x^(1/3))^(1/2)*a^4+45045*b^(3/2)*x^4*(b+a*x^(1/3
))^(1/2)*a^6-45045*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*x^(13/3)*a^7*b-16640*b^(13/2)*x^(7/3)*(b+a*x^(1/3))^(1
/2)*a-30030*b^(5/2)*x^(11/3)*(b+a*x^(1/3))^(1/2)*a^5-20592*b^(9/2)*x^3*(b+a*x^(1/3))^(1/2)*a^3+18304*b^(11/2)*
x^(8/3)*(b+a*x^(1/3))^(1/2)*a^2+15360*(b+a*x^(1/3))^(1/2)*b^(15/2)*x^2)/(b*x^(2/3)+a*x)^(1/2)/b^(17/2)

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Giac [A]  time = 1.26854, size = 216, normalized size = 0.9 \begin{align*} -\frac{\frac{45045 \, a^{8} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{7}} + \frac{45045 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{8} - 300300 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{8} b + 849849 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{8} b^{2} - 1317888 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{8} b^{3} + 1200199 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{8} b^{4} - 631540 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{8} b^{5} + 169995 \, \sqrt{a x^{\frac{1}{3}} + b} a^{8} b^{6}}{a^{7} b^{7} x^{\frac{7}{3}}}}{35840 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35840*(45045*a^8*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(13/2)*a^8 -
300300*(a*x^(1/3) + b)^(11/2)*a^8*b + 849849*(a*x^(1/3) + b)^(9/2)*a^8*b^2 - 1317888*(a*x^(1/3) + b)^(7/2)*a^8
*b^3 + 1200199*(a*x^(1/3) + b)^(5/2)*a^8*b^4 - 631540*(a*x^(1/3) + b)^(3/2)*a^8*b^5 + 169995*sqrt(a*x^(1/3) +
b)*a^8*b^6)/(a^7*b^7*x^(7/3)))/a