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

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

[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 = 0.687773, 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 \[ -\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac{9009 a^5 \sqrt{a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac{3003 a^4 \sqrt{a x+b x^{2/3}}}{256 b^6 x}+\frac{3003 a^3 \sqrt{a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac{1287 a^2 \sqrt{a x+b x^{2/3}}}{160 b^4 x^{5/3}}+\frac{143 a \sqrt{a x+b x^{2/3}}}{20 b^3 x^2}-\frac{13 \sqrt{a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac{6}{b x^{5/3} \sqrt{a x+b x^{2/3}}} \]

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))

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Rubi in Sympy [A]  time = 62.4927, size = 221, normalized size = 0.94 \[ - \frac{9009 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{512 b^{\frac{15}{2}}} + \frac{9009 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{512 b^{7} x^{\frac{2}{3}}} - \frac{3003 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{256 b^{6} x} + \frac{3003 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{320 b^{5} x^{\frac{4}{3}}} - \frac{1287 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{160 b^{4} x^{\frac{5}{3}}} + \frac{143 a \sqrt{a x + b x^{\frac{2}{3}}}}{20 b^{3} x^{2}} + \frac{6}{b x^{\frac{5}{3}} \sqrt{a x + b x^{\frac{2}{3}}}} - \frac{13 \sqrt{a x + b x^{\frac{2}{3}}}}{2 b^{2} x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

-9009*a**6*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(512*b**(15/2)) + 9009
*a**5*sqrt(a*x + b*x**(2/3))/(512*b**7*x**(2/3)) - 3003*a**4*sqrt(a*x + b*x**(2/
3))/(256*b**6*x) + 3003*a**3*sqrt(a*x + b*x**(2/3))/(320*b**5*x**(4/3)) - 1287*a
**2*sqrt(a*x + b*x**(2/3))/(160*b**4*x**(5/3)) + 143*a*sqrt(a*x + b*x**(2/3))/(2
0*b**3*x**2) + 6/(b*x**(5/3)*sqrt(a*x + b*x**(2/3))) - 13*sqrt(a*x + b*x**(2/3))
/(2*b**2*x**(7/3))

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Mathematica [A]  time = 0.302245, size = 149, normalized size = 0.63 \[ \frac{\sqrt{a x+b x^{2/3}} \left (45045 a^6 x^2+15015 a^5 b x^{5/3}-6006 a^4 b^2 x^{4/3}+3432 a^3 b^3 x-2288 a^2 b^4 x^{2/3}+1664 a b^5 \sqrt [3]{x}-1280 b^6\right )}{2560 b^7 x^{7/3} \left (a \sqrt [3]{x}+b\right )}-\frac{9009 a^6 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{512 b^{15/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-1280*b^6 + 1664*a*b^5*x^(1/3) - 2288*a^2*b^4*x^(2/3) +
3432*a^3*b^3*x - 6006*a^4*b^2*x^(4/3) + 15015*a^5*b*x^(5/3) + 45045*a^6*x^2))/(2
560*b^7*(b + a*x^(1/3))*x^(7/3)) - (9009*a^6*ArcTanh[Sqrt[b*x^(2/3) + a*x]/(Sqrt
[b]*x^(1/3))])/(512*b^(15/2))

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Maple [A]  time = 0.026, size = 126, normalized size = 0.5 \[ -{\frac{1}{2560\,x} \left ( b+a\sqrt [3]{x} \right ) \left ( -15015\,{x}^{5/3}{b}^{3/2}{a}^{5}+6006\,{x}^{4/3}{b}^{5/2}{a}^{4}-3432\,x{b}^{7/2}{a}^{3}+2288\,{x}^{2/3}{b}^{9/2}{a}^{2}-1664\,\sqrt [3]{x}{b}^{11/2}a+1280\,{b}^{13/2}-45045\,{x}^{2}{a}^{6}\sqrt{b}+45045\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{2}{a}^{6} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{15}{2}}}} \]

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))*(-15015*x^(5/3)*b^(3/2)*a^5+6006*x^(4/3)*b^(5/2)*a^4-3432*
x*b^(7/2)*a^3+2288*x^(2/3)*b^(9/2)*a^2-1664*x^(1/3)*b^(11/2)*a+1280*b^(13/2)-450
45*x^2*a^6*b^(1/2)+45045*(b+a*x^(1/3))^(1/2)*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2)
)*x^2*a^6)/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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \]

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/XCAS [A]  time = 0.369492, size = 235, normalized size = 1. \[ \frac{9009 \, a^{6} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{512 \, \sqrt{-b} b^{7}{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{6 \, a^{6}}{\sqrt{a x^{\frac{1}{3}} + b} b^{7}{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{29685 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{6} - 163095 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{6} b + 364194 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{6} b^{2} - 416094 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{6} b^{3} + 246505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{6} b^{4} - 62475 \, \sqrt{a x^{\frac{1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9009/512*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7*sign(x^(1/3))) +
6*a^6/(sqrt(a*x^(1/3) + b)*b^7*sign(x^(1/3))) + 1/2560*(29685*(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*sign(x^(1/3)))