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

Optimal. Leaf size=324 \[ \frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{21/2}}-\frac{692835 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac{230945 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^9 x}-\frac{46189 a^6 \sqrt{a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac{138567 a^5 \sqrt{a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac{46189 a^4 \sqrt{a x+b x^{2/3}}}{5376 b^6 x^2}+\frac{20995 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac{1615 a^2 \sqrt{a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac{323 a \sqrt{a x+b x^{2/3}}}{48 b^3 x^3}-\frac{19 \sqrt{a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac{6}{b x^{8/3} \sqrt{a x+b x^{2/3}}} \]

[Out]

6/(b*x^(8/3)*Sqrt[b*x^(2/3) + a*x]) - (19*Sqrt[b*x^(2/3) + a*x])/(3*b^2*x^(10/3)
) + (323*a*Sqrt[b*x^(2/3) + a*x])/(48*b^3*x^3) - (1615*a^2*Sqrt[b*x^(2/3) + a*x]
)/(224*b^4*x^(8/3)) + (20995*a^3*Sqrt[b*x^(2/3) + a*x])/(2688*b^5*x^(7/3)) - (46
189*a^4*Sqrt[b*x^(2/3) + a*x])/(5376*b^6*x^2) + (138567*a^5*Sqrt[b*x^(2/3) + a*x
])/(14336*b^7*x^(5/3)) - (46189*a^6*Sqrt[b*x^(2/3) + a*x])/(4096*b^8*x^(4/3)) +
(230945*a^7*Sqrt[b*x^(2/3) + a*x])/(16384*b^9*x) - (692835*a^8*Sqrt[b*x^(2/3) +
a*x])/(32768*b^10*x^(2/3)) + (692835*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3
) + a*x]])/(32768*b^(21/2))

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Rubi [A]  time = 0.976924, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{21/2}}-\frac{692835 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac{230945 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^9 x}-\frac{46189 a^6 \sqrt{a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac{138567 a^5 \sqrt{a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac{46189 a^4 \sqrt{a x+b x^{2/3}}}{5376 b^6 x^2}+\frac{20995 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac{1615 a^2 \sqrt{a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac{323 a \sqrt{a x+b x^{2/3}}}{48 b^3 x^3}-\frac{19 \sqrt{a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac{6}{b x^{8/3} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

6/(b*x^(8/3)*Sqrt[b*x^(2/3) + a*x]) - (19*Sqrt[b*x^(2/3) + a*x])/(3*b^2*x^(10/3)
) + (323*a*Sqrt[b*x^(2/3) + a*x])/(48*b^3*x^3) - (1615*a^2*Sqrt[b*x^(2/3) + a*x]
)/(224*b^4*x^(8/3)) + (20995*a^3*Sqrt[b*x^(2/3) + a*x])/(2688*b^5*x^(7/3)) - (46
189*a^4*Sqrt[b*x^(2/3) + a*x])/(5376*b^6*x^2) + (138567*a^5*Sqrt[b*x^(2/3) + a*x
])/(14336*b^7*x^(5/3)) - (46189*a^6*Sqrt[b*x^(2/3) + a*x])/(4096*b^8*x^(4/3)) +
(230945*a^7*Sqrt[b*x^(2/3) + a*x])/(16384*b^9*x) - (692835*a^8*Sqrt[b*x^(2/3) +
a*x])/(32768*b^10*x^(2/3)) + (692835*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3
) + a*x]])/(32768*b^(21/2))

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Rubi in Sympy [A]  time = 97.4313, size = 306, normalized size = 0.94 \[ \frac{692835 a^{9} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{32768 b^{\frac{21}{2}}} - \frac{692835 a^{8} \sqrt{a x + b x^{\frac{2}{3}}}}{32768 b^{10} x^{\frac{2}{3}}} + \frac{230945 a^{7} \sqrt{a x + b x^{\frac{2}{3}}}}{16384 b^{9} x} - \frac{46189 a^{6} \sqrt{a x + b x^{\frac{2}{3}}}}{4096 b^{8} x^{\frac{4}{3}}} + \frac{138567 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{14336 b^{7} x^{\frac{5}{3}}} - \frac{46189 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{5376 b^{6} x^{2}} + \frac{20995 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{2688 b^{5} x^{\frac{7}{3}}} - \frac{1615 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{224 b^{4} x^{\frac{8}{3}}} + \frac{323 a \sqrt{a x + b x^{\frac{2}{3}}}}{48 b^{3} x^{3}} + \frac{6}{b x^{\frac{8}{3}} \sqrt{a x + b x^{\frac{2}{3}}}} - \frac{19 \sqrt{a x + b x^{\frac{2}{3}}}}{3 b^{2} x^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

692835*a**9*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(32768*b**(21/2)) - 6
92835*a**8*sqrt(a*x + b*x**(2/3))/(32768*b**10*x**(2/3)) + 230945*a**7*sqrt(a*x
+ b*x**(2/3))/(16384*b**9*x) - 46189*a**6*sqrt(a*x + b*x**(2/3))/(4096*b**8*x**(
4/3)) + 138567*a**5*sqrt(a*x + b*x**(2/3))/(14336*b**7*x**(5/3)) - 46189*a**4*sq
rt(a*x + b*x**(2/3))/(5376*b**6*x**2) + 20995*a**3*sqrt(a*x + b*x**(2/3))/(2688*
b**5*x**(7/3)) - 1615*a**2*sqrt(a*x + b*x**(2/3))/(224*b**4*x**(8/3)) + 323*a*sq
rt(a*x + b*x**(2/3))/(48*b**3*x**3) + 6/(b*x**(8/3)*sqrt(a*x + b*x**(2/3))) - 19
*sqrt(a*x + b*x**(2/3))/(3*b**2*x**(10/3))

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Mathematica [A]  time = 0.38296, size = 186, normalized size = 0.57 \[ \frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{32768 b^{21/2}}-\frac{\sqrt{a x+b x^{2/3}} \left (14549535 a^9 x^3+4849845 a^8 b x^{8/3}-1939938 a^7 b^2 x^{7/3}+1108536 a^6 b^3 x^2-739024 a^5 b^4 x^{5/3}+537472 a^4 b^5 x^{4/3}-413440 a^3 b^6 x+330752 a^2 b^7 x^{2/3}-272384 a b^8 \sqrt [3]{x}+229376 b^9\right )}{688128 b^{10} x^{10/3} \left (a \sqrt [3]{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[b*x^(2/3) + a*x]*(229376*b^9 - 272384*a*b^8*x^(1/3) + 330752*a^2*b^7*x^(2
/3) - 413440*a^3*b^6*x + 537472*a^4*b^5*x^(4/3) - 739024*a^5*b^4*x^(5/3) + 11085
36*a^6*b^3*x^2 - 1939938*a^7*b^2*x^(7/3) + 4849845*a^8*b*x^(8/3) + 14549535*a^9*
x^3))/(688128*b^10*(b + a*x^(1/3))*x^(10/3)) + (692835*a^9*ArcTanh[Sqrt[b*x^(2/3
) + a*x]/(Sqrt[b]*x^(1/3))])/(32768*b^(21/2))

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Maple [A]  time = 0.03, size = 159, normalized size = 0.5 \[{\frac{1}{688128\,{x}^{2}} \left ( b+a\sqrt [3]{x} \right ) \left ( -14549535\,{x}^{3}{a}^{9}\sqrt{b}-4849845\,{x}^{8/3}{b}^{3/2}{a}^{8}+1939938\,{x}^{7/3}{b}^{5/2}{a}^{7}-1108536\,{x}^{2}{b}^{7/2}{a}^{6}+739024\,{x}^{5/3}{b}^{9/2}{a}^{5}-537472\,{x}^{4/3}{b}^{11/2}{a}^{4}+413440\,x{b}^{13/2}{a}^{3}-330752\,{x}^{2/3}{b}^{15/2}{a}^{2}+272384\,\sqrt [3]{x}{b}^{17/2}a-229376\,{b}^{19/2}[phone]\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{3}{a}^{9} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{21}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/688128*(b+a*x^(1/3))*(-14549535*x^3*a^9*b^(1/2)-4849845*x^(8/3)*b^(3/2)*a^8+19
39938*x^(7/3)*b^(5/2)*a^7-1108536*x^2*b^(7/2)*a^6+739024*x^(5/3)*b^(9/2)*a^5-537
472*x^(4/3)*b^(11/2)*a^4+413440*x*b^(13/2)*a^3-330752*x^(2/3)*b^(15/2)*a^2+27238
4*x^(1/3)*b^(17/2)*a-229376*b^(19/2)[phone]*(b+a*x^(1/3))^(1/2)*arctanh((b+a*x
^(1/3))^(1/2)/b^(1/2))*x^3*a^9)/x^2/(b*x^(2/3)+a*x)^(3/2)/b^(21/2)

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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^3),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^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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/x**3/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.517992, size = 304, normalized size = 0.94 \[ -\frac{692835 \, a^{9} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{32768 \, \sqrt{-b} b^{10}{\rm sign}\left (x^{\frac{1}{3}}\right )} - \frac{6 \, a^{9}}{\sqrt{a x^{\frac{1}{3}} + b} b^{10}{\rm sign}\left (x^{\frac{1}{3}}\right )} - \frac{10420767 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{9} - 88937058 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{9} b + 334408914 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{9} b^{2} - 724860666 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{9} b^{3} + 993296384 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{9} b^{4} - 884769030 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{9} b^{5} + 503730990 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{9} b^{6} - 169799070 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{9} b^{7} + 26738145 \, \sqrt{a x^{\frac{1}{3}} + b} a^{9} b^{8}}{688128 \, a^{9} b^{10} x^{3}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-692835/32768*a^9*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10*sign(x^(1/
3))) - 6*a^9/(sqrt(a*x^(1/3) + b)*b^10*sign(x^(1/3))) - 1/688128*(10420767*(a*x^
(1/3) + b)^(17/2)*a^9 - 88937058*(a*x^(1/3) + b)^(15/2)*a^9*b + 334408914*(a*x^(
1/3) + b)^(13/2)*a^9*b^2 - 724860666*(a*x^(1/3) + b)^(11/2)*a^9*b^3 + 993296384*
(a*x^(1/3) + b)^(9/2)*a^9*b^4 - 884769030*(a*x^(1/3) + b)^(7/2)*a^9*b^5 + 503730
990*(a*x^(1/3) + b)^(5/2)*a^9*b^6 - 169799070*(a*x^(1/3) + b)^(3/2)*a^9*b^7 + 26
738145*sqrt(a*x^(1/3) + b)*a^9*b^8)/(a^9*b^10*x^3*sign(x^(1/3)))