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

Optimal. Leaf size=178 \[ -\frac{21 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{128 b^{9/2}}+\frac{21 a^4 \sqrt{a x+b x^{2/3}}}{128 b^4 x^{2/3}}-\frac{7 a^3 \sqrt{a x+b x^{2/3}}}{64 b^3 x}+\frac{7 a^2 \sqrt{a x+b x^{2/3}}}{80 b^2 x^{4/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{40 b x^{5/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{5 x^2} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(5*x^2) - (3*a*Sqrt[b*x^(2/3) + a*x])/(40*b*x^(5/3))
+ (7*a^2*Sqrt[b*x^(2/3) + a*x])/(80*b^2*x^(4/3)) - (7*a^3*Sqrt[b*x^(2/3) + a*x])
/(64*b^3*x) + (21*a^4*Sqrt[b*x^(2/3) + a*x])/(128*b^4*x^(2/3)) - (21*a^5*ArcTanh
[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(128*b^(9/2))

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Rubi [A]  time = 0.506702, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{21 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{128 b^{9/2}}+\frac{21 a^4 \sqrt{a x+b x^{2/3}}}{128 b^4 x^{2/3}}-\frac{7 a^3 \sqrt{a x+b x^{2/3}}}{64 b^3 x}+\frac{7 a^2 \sqrt{a x+b x^{2/3}}}{80 b^2 x^{4/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{40 b x^{5/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{5 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(5*x^2) - (3*a*Sqrt[b*x^(2/3) + a*x])/(40*b*x^(5/3))
+ (7*a^2*Sqrt[b*x^(2/3) + a*x])/(80*b^2*x^(4/3)) - (7*a^3*Sqrt[b*x^(2/3) + a*x])
/(64*b^3*x) + (21*a^4*Sqrt[b*x^(2/3) + a*x])/(128*b^4*x^(2/3)) - (21*a^5*ArcTanh
[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(128*b^(9/2))

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Rubi in Sympy [A]  time = 43.3736, size = 165, normalized size = 0.93 \[ - \frac{21 a^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{128 b^{\frac{9}{2}}} + \frac{21 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{128 b^{4} x^{\frac{2}{3}}} - \frac{7 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{64 b^{3} x} + \frac{7 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{80 b^{2} x^{\frac{4}{3}}} - \frac{3 a \sqrt{a x + b x^{\frac{2}{3}}}}{40 b x^{\frac{5}{3}}} - \frac{3 \sqrt{a x + b x^{\frac{2}{3}}}}{5 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-21*a**5*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(128*b**(9/2)) + 21*a**4
*sqrt(a*x + b*x**(2/3))/(128*b**4*x**(2/3)) - 7*a**3*sqrt(a*x + b*x**(2/3))/(64*
b**3*x) + 7*a**2*sqrt(a*x + b*x**(2/3))/(80*b**2*x**(4/3)) - 3*a*sqrt(a*x + b*x*
*(2/3))/(40*b*x**(5/3)) - 3*sqrt(a*x + b*x**(2/3))/(5*x**2)

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Mathematica [A]  time = 0.177362, size = 112, normalized size = 0.63 \[ \frac{\sqrt{a x+b x^{2/3}} \left (105 a^4 x^{4/3}-70 a^3 b x+56 a^2 b^2 x^{2/3}-48 a b^3 \sqrt [3]{x}-384 b^4\right )}{640 b^4 x^2}-\frac{21 a^5 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{128 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-384*b^4 - 48*a*b^3*x^(1/3) + 56*a^2*b^2*x^(2/3) - 70*a^
3*b*x + 105*a^4*x^(4/3)))/(640*b^4*x^2) - (21*a^5*ArcTanh[Sqrt[b*x^(2/3) + a*x]/
(Sqrt[b]*x^(1/3))])/(128*b^(9/2))

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Maple [A]  time = 0.018, size = 125, normalized size = 0.7 \[{\frac{1}{640\,{x}^{2}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( 105\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{9/2}-490\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{11/2}+896\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{13/2}-790\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{15/2}-105\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{4}{a}^{5}{x}^{5/3}-105\,\sqrt{b+a\sqrt [3]{x}}{b}^{17/2} \right ){\frac{1}{\sqrt{b+a\sqrt [3]{x}}}}{b}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*(b*x^(2/3)+a*x)^(1/2)*(105*(b+a*x^(1/3))^(9/2)*b^(9/2)-490*(b+a*x^(1/3))^(
7/2)*b^(11/2)+896*(b+a*x^(1/3))^(5/2)*b^(13/2)-790*(b+a*x^(1/3))^(3/2)*b^(15/2)-
105*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b^4*a^5*x^(5/3)-105*(b+a*x^(1/3))^(1/2)
*b^(17/2))/x^2/(b+a*x^(1/3))^(1/2)/b^(17/2)

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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(sqrt(a*x + b*x^(2/3))/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(sqrt(a*x + b*x^(2/3))/x^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284706, size = 176, normalized size = 0.99 \[ \frac{{\left (\frac{105 \, a^{6} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{105 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{6} - 490 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{6} b + 896 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{6} b^{2} - 790 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{6} b^{3} - 105 \, \sqrt{a x^{\frac{1}{3}} + b} a^{6} b^{4}}{a^{5} b^{4} x^{\frac{5}{3}}}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{640 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*(105*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^4) + (105*(a*x^(
1/3) + b)^(9/2)*a^6 - 490*(a*x^(1/3) + b)^(7/2)*a^6*b + 896*(a*x^(1/3) + b)^(5/2
)*a^6*b^2 - 790*(a*x^(1/3) + b)^(3/2)*a^6*b^3 - 105*sqrt(a*x^(1/3) + b)*a^6*b^4)
/(a^5*b^4*x^(5/3)))*sign(x^(1/3))/a