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

Optimal. Leaf size=137 \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]

[Out]

(-128*b^3*Sqrt[b*x^(2/3) + a*x])/(105*a^4) + (256*b^4*Sqrt[b*x^(2/3) + a*x])/(10
5*a^5*x^(1/3)) + (32*b^2*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(35*a^3) - (16*b*x^(2/3)
*Sqrt[b*x^(2/3) + a*x])/(21*a^2) + (2*x*Sqrt[b*x^(2/3) + a*x])/(3*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.307186, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(-128*b^3*Sqrt[b*x^(2/3) + a*x])/(105*a^4) + (256*b^4*Sqrt[b*x^(2/3) + a*x])/(10
5*a^5*x^(1/3)) + (32*b^2*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(35*a^3) - (16*b*x^(2/3)
*Sqrt[b*x^(2/3) + a*x])/(21*a^2) + (2*x*Sqrt[b*x^(2/3) + a*x])/(3*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.4664, size = 128, normalized size = 0.93 \[ \frac{2 x \sqrt{a x + b x^{\frac{2}{3}}}}{3 a} - \frac{16 b x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{21 a^{2}} + \frac{32 b^{2} \sqrt [3]{x} \sqrt{a x + b x^{\frac{2}{3}}}}{35 a^{3}} - \frac{128 b^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{105 a^{4}} + \frac{256 b^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{105 a^{5} \sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*sqrt(a*x + b*x**(2/3))/(3*a) - 16*b*x**(2/3)*sqrt(a*x + b*x**(2/3))/(21*a**2
) + 32*b**2*x**(1/3)*sqrt(a*x + b*x**(2/3))/(35*a**3) - 128*b**3*sqrt(a*x + b*x*
*(2/3))/(105*a**4) + 256*b**4*sqrt(a*x + b*x**(2/3))/(105*a**5*x**(1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0424941, size = 74, normalized size = 0.54 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (35 a^4 x^{4/3}-40 a^3 b x+48 a^2 b^2 x^{2/3}-64 a b^3 \sqrt [3]{x}+128 b^4\right )}{105 a^5 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(128*b^4 - 64*a*b^3*x^(1/3) + 48*a^2*b^2*x^(2/3) - 40*a
^3*b*x + 35*a^4*x^(4/3)))/(105*a^5*x^(1/3))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 68, normalized size = 0.5 \[{\frac{2}{105\,{a}^{5}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 35\,{x}^{4/3}{a}^{4}-40\,x{a}^{3}b+48\,{x}^{2/3}{a}^{2}{b}^{2}-64\,\sqrt [3]{x}a{b}^{3}+128\,{b}^{4} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^(2/3)+a*x)^(1/2),x)

[Out]

2/105*x^(1/3)*(b+a*x^(1/3))*(35*x^(4/3)*a^4-40*x*a^3*b+48*x^(2/3)*a^2*b^2-64*x^(
1/3)*a*b^3+128*b^4)/(b*x^(2/3)+a*x)^(1/2)/a^5

_______________________________________________________________________________________

Maxima [A]  time = 1.46307, size = 109, normalized size = 0.8 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}}}{3 \, a^{5}} - \frac{24 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b}{7 \, a^{5}} + \frac{36 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{2}}{5 \, a^{5}} - \frac{8 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{3}}{a^{5}} + \frac{6 \, \sqrt{a x^{\frac{1}{3}} + b} b^{4}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a*x^(1/3) + b)^(9/2)/a^5 - 24/7*(a*x^(1/3) + b)^(7/2)*b/a^5 + 36/5*(a*x^(1/
3) + b)^(5/2)*b^2/a^5 - 8*(a*x^(1/3) + b)^(3/2)*b^3/a^5 + 6*sqrt(a*x^(1/3) + b)*
b^4/a^5

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(a*x + b*x**(2/3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224448, size = 142, normalized size = 1.04 \[ -\frac{256 \, b^{\frac{9}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{105 \, a^{5}} + \frac{2 \,{\left (35 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{32} - 180 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{32} b + 378 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{32} b^{2} - 420 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{32} b^{3} + 315 \, \sqrt{a x^{\frac{1}{3}} + b} a^{32} b^{4}\right )}}{105 \, a^{37}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256/105*b^(9/2)*sign(x^(1/3))/a^5 + 2/105*(35*(a*x^(1/3) + b)^(9/2)*a^32 - 180*
(a*x^(1/3) + b)^(7/2)*a^32*b + 378*(a*x^(1/3) + b)^(5/2)*a^32*b^2 - 420*(a*x^(1/
3) + b)^(3/2)*a^32*b^3 + 315*sqrt(a*x^(1/3) + b)*a^32*b^4)/(a^37*sign(x^(1/3)))