3.193 \(\int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx\)

Optimal. Leaf size=329 \[ -\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) + (19*a*Sqrt[b*x^(2/3) + a*x])/(60*b^
2*x^(10/3)) - (323*a^2*Sqrt[b*x^(2/3) + a*x])/(960*b^3*x^3) + (323*a^3*Sqrt[b*x^
(2/3) + a*x])/(896*b^4*x^(8/3)) - (4199*a^4*Sqrt[b*x^(2/3) + a*x])/(10752*b^5*x^
(7/3)) + (46189*a^5*Sqrt[b*x^(2/3) + a*x])/(107520*b^6*x^2) - (138567*a^6*Sqrt[b
*x^(2/3) + a*x])/(286720*b^7*x^(5/3)) + (46189*a^7*Sqrt[b*x^(2/3) + a*x])/(81920
*b^8*x^(4/3)) - (46189*a^8*Sqrt[b*x^(2/3) + a*x])/(65536*b^9*x) + (138567*a^9*Sq
rt[b*x^(2/3) + a*x])/(131072*b^10*x^(2/3)) - (138567*a^10*ArcTanh[(Sqrt[b]*x^(1/
3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))

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Rubi [A]  time = 1.01321, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) + (19*a*Sqrt[b*x^(2/3) + a*x])/(60*b^
2*x^(10/3)) - (323*a^2*Sqrt[b*x^(2/3) + a*x])/(960*b^3*x^3) + (323*a^3*Sqrt[b*x^
(2/3) + a*x])/(896*b^4*x^(8/3)) - (4199*a^4*Sqrt[b*x^(2/3) + a*x])/(10752*b^5*x^
(7/3)) + (46189*a^5*Sqrt[b*x^(2/3) + a*x])/(107520*b^6*x^2) - (138567*a^6*Sqrt[b
*x^(2/3) + a*x])/(286720*b^7*x^(5/3)) + (46189*a^7*Sqrt[b*x^(2/3) + a*x])/(81920
*b^8*x^(4/3)) - (46189*a^8*Sqrt[b*x^(2/3) + a*x])/(65536*b^9*x) + (138567*a^9*Sq
rt[b*x^(2/3) + a*x])/(131072*b^10*x^(2/3)) - (138567*a^10*ArcTanh[(Sqrt[b]*x^(1/
3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))

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Rubi in Sympy [A]  time = 96.8438, size = 311, normalized size = 0.95 \[ - \frac{138567 a^{10} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{131072 b^{\frac{21}{2}}} + \frac{138567 a^{9} \sqrt{a x + b x^{\frac{2}{3}}}}{131072 b^{10} x^{\frac{2}{3}}} - \frac{46189 a^{8} \sqrt{a x + b x^{\frac{2}{3}}}}{65536 b^{9} x} + \frac{46189 a^{7} \sqrt{a x + b x^{\frac{2}{3}}}}{81920 b^{8} x^{\frac{4}{3}}} - \frac{138567 a^{6} \sqrt{a x + b x^{\frac{2}{3}}}}{286720 b^{7} x^{\frac{5}{3}}} + \frac{46189 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{107520 b^{6} x^{2}} - \frac{4199 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{10752 b^{5} x^{\frac{7}{3}}} + \frac{323 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{896 b^{4} x^{\frac{8}{3}}} - \frac{323 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{960 b^{3} x^{3}} + \frac{19 a \sqrt{a x + b x^{\frac{2}{3}}}}{60 b^{2} x^{\frac{10}{3}}} - \frac{3 \sqrt{a x + b x^{\frac{2}{3}}}}{10 b x^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**(2/3)+a*x)**(1/2),x)

[Out]

-138567*a**10*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(131072*b**(21/2))
+ 138567*a**9*sqrt(a*x + b*x**(2/3))/(131072*b**10*x**(2/3)) - 46189*a**8*sqrt(a
*x + b*x**(2/3))/(65536*b**9*x) + 46189*a**7*sqrt(a*x + b*x**(2/3))/(81920*b**8*
x**(4/3)) - 138567*a**6*sqrt(a*x + b*x**(2/3))/(286720*b**7*x**(5/3)) + 46189*a*
*5*sqrt(a*x + b*x**(2/3))/(107520*b**6*x**2) - 4199*a**4*sqrt(a*x + b*x**(2/3))/
(10752*b**5*x**(7/3)) + 323*a**3*sqrt(a*x + b*x**(2/3))/(896*b**4*x**(8/3)) - 32
3*a**2*sqrt(a*x + b*x**(2/3))/(960*b**3*x**3) + 19*a*sqrt(a*x + b*x**(2/3))/(60*
b**2*x**(10/3)) - 3*sqrt(a*x + b*x**(2/3))/(10*b*x**(11/3))

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Mathematica [A]  time = 0.334833, size = 175, normalized size = 0.53 \[ \frac{\sqrt{a x+b x^{2/3}} \left (14549535 a^9 x^3-9699690 a^8 b x^{8/3}+7759752 a^7 b^2 x^{7/3}-6651216 a^6 b^3 x^2+5912192 a^5 b^4 x^{5/3}-5374720 a^4 b^5 x^{4/3}+4961280 a^3 b^6 x-4630528 a^2 b^7 x^{2/3}+4358144 a b^8 \sqrt [3]{x}-4128768 b^9\right )}{13762560 b^{10} x^{11/3}}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{131072 b^{21/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-4128768*b^9 + 4358144*a*b^8*x^(1/3) - 4630528*a^2*b^7*x
^(2/3) + 4961280*a^3*b^6*x - 5374720*a^4*b^5*x^(4/3) + 5912192*a^5*b^4*x^(5/3) -
6651216*a^6*b^3*x^2 + 7759752*a^7*b^2*x^(7/3) - 9699690*a^8*b*x^(8/3) + 1454953
5*a^9*x^3))/(13762560*b^10*x^(11/3)) - (138567*a^10*ArcTanh[Sqrt[b*x^(2/3) + a*x
]/(Sqrt[b]*x^(1/3))])/(131072*b^(21/2))

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Maple [A]  time = 0.011, size = 248, normalized size = 0.8 \[ -{\frac{1}{13762560\,{x}^{6}}\sqrt{b+a\sqrt [3]{x}} \left ( 9699690\,\sqrt{b+a\sqrt [3]{x}}{x}^{{\frac{17}{3}}}{b}^{5/2}{a}^{8}-7759752\,\sqrt{b+a\sqrt [3]{x}}{x}^{16/3}{b}^{7/2}{a}^{7}-5912192\,\sqrt{b+a\sqrt [3]{x}}{x}^{14/3}{b}^{11/2}{a}^{5}+5374720\,\sqrt{b+a\sqrt [3]{x}}{x}^{13/3}{b}^{13/2}{a}^{4}+4630528\,\sqrt{b+a\sqrt [3]{x}}{x}^{11/3}{b}^{17/2}{a}^{2}-4358144\,\sqrt{b+a\sqrt [3]{x}}{x}^{10/3}{b}^{19/2}a+14549535\,{x}^{{\frac{19}{3}}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){a}^{10}b+4128768\,\sqrt{b+a\sqrt [3]{x}}{b}^{21/2}{x}^{3}-4961280\,\sqrt{b+a\sqrt [3]{x}}{x}^{4}{b}^{15/2}{a}^{3}+6651216\,\sqrt{b+a\sqrt [3]{x}}{x}^{5}{b}^{9/2}{a}^{6}-14549535\,\sqrt{b+a\sqrt [3]{x}}{x}^{6}{b}^{3/2}{a}^{9} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}{b}^{-{\frac{23}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^(2/3)+a*x)^(1/2),x)

[Out]

-1/13762560*(b+a*x^(1/3))^(1/2)*(9699690*(b+a*x^(1/3))^(1/2)*x^(17/3)*b^(5/2)*a^
8-7759752*(b+a*x^(1/3))^(1/2)*x^(16/3)*b^(7/2)*a^7-5912192*(b+a*x^(1/3))^(1/2)*x
^(14/3)*b^(11/2)*a^5+5374720*(b+a*x^(1/3))^(1/2)*x^(13/3)*b^(13/2)*a^4+4630528*(
b+a*x^(1/3))^(1/2)*x^(11/3)*b^(17/2)*a^2-4358144*(b+a*x^(1/3))^(1/2)*x^(10/3)*b^
(19/2)*a+14549535*x^(19/3)*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*a^10*b+4128768*(
b+a*x^(1/3))^(1/2)*b^(21/2)*x^3-4961280*(b+a*x^(1/3))^(1/2)*x^4*b^(15/2)*a^3+665
1216*(b+a*x^(1/3))^(1/2)*x^5*b^(9/2)*a^6-14549535*(b+a*x^(1/3))^(1/2)*x^6*b^(3/2
)*a^9)/x^6/(b*x^(2/3)+a*x)^(1/2)/b^(23/2)

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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/(sqrt(a*x + b*x^(2/3))*x^4),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/(sqrt(a*x + b*x^(2/3))*x^4),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**4/(b*x**(2/3)+a*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.473336, size = 293, normalized size = 0.89 \[ \frac{\frac{14549535 \, a^{11} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{11} - 140645505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{11} b + 609140532 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{11} b^{2} - 1554721740 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{11} b^{3} + 2585198330 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{11} b^{4} - 2918514950 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{11} b^{5} + 2255541300 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{11} b^{6} - 1168982220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{11} b^{7} + 382331775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{11} b^{8} - 68025825 \, \sqrt{a x^{\frac{1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac{10}{3}}}}{13762560 \, a{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a*x + b*x^(2/3))*x^4),x, algorithm="giac")

[Out]

1/13762560*(14549535*a^11*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) +
(14549535*(a*x^(1/3) + b)^(19/2)*a^11 - 140645505*(a*x^(1/3) + b)^(17/2)*a^11*b
+ 609140532*(a*x^(1/3) + b)^(15/2)*a^11*b^2 - 1554721740*(a*x^(1/3) + b)^(13/2)
*a^11*b^3 + 2585198330*(a*x^(1/3) + b)^(11/2)*a^11*b^4 - 2918514950*(a*x^(1/3) +
b)^(9/2)*a^11*b^5 + 2255541300*(a*x^(1/3) + b)^(7/2)*a^11*b^6 - 1168982220*(a*x
^(1/3) + b)^(5/2)*a^11*b^7 + 382331775*(a*x^(1/3) + b)^(3/2)*a^11*b^8 - 68025825
*sqrt(a*x^(1/3) + b)*a^11*b^9)/(a^10*b^10*x^(10/3)))/(a*sign(x^(1/3)))