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

Optimal. Leaf size=160 \[ \frac{512 b^4 \sqrt{a x+b x^{2/3}}}{21 a^6 \sqrt [3]{x}}-\frac{256 b^3 \sqrt{a x+b x^{2/3}}}{21 a^5}+\frac{64 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^3}+\frac{20 x \sqrt{a x+b x^{2/3}}}{3 a^2}-\frac{6 x^2}{a \sqrt{a x+b x^{2/3}}} \]

[Out]

(-6*x^2)/(a*Sqrt[b*x^(2/3) + a*x]) - (256*b^3*Sqrt[b*x^(2/3) + a*x])/(21*a^5) +
(512*b^4*Sqrt[b*x^(2/3) + a*x])/(21*a^6*x^(1/3)) + (64*b^2*x^(1/3)*Sqrt[b*x^(2/3
) + a*x])/(7*a^4) - (160*b*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(21*a^3) + (20*x*Sqrt[
b*x^(2/3) + a*x])/(3*a^2)

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Rubi [A]  time = 0.397586, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{512 b^4 \sqrt{a x+b x^{2/3}}}{21 a^6 \sqrt [3]{x}}-\frac{256 b^3 \sqrt{a x+b x^{2/3}}}{21 a^5}+\frac{64 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{7 a^4}-\frac{160 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^3}+\frac{20 x \sqrt{a x+b x^{2/3}}}{3 a^2}-\frac{6 x^2}{a \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*x^2)/(a*Sqrt[b*x^(2/3) + a*x]) - (256*b^3*Sqrt[b*x^(2/3) + a*x])/(21*a^5) +
(512*b^4*Sqrt[b*x^(2/3) + a*x])/(21*a^6*x^(1/3)) + (64*b^2*x^(1/3)*Sqrt[b*x^(2/3
) + a*x])/(7*a^4) - (160*b*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(21*a^3) + (20*x*Sqrt[
b*x^(2/3) + a*x])/(3*a^2)

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Rubi in Sympy [A]  time = 34.945, size = 150, normalized size = 0.94 \[ - \frac{6 x^{2}}{a \sqrt{a x + b x^{\frac{2}{3}}}} + \frac{20 x \sqrt{a x + b x^{\frac{2}{3}}}}{3 a^{2}} - \frac{160 b x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{21 a^{3}} + \frac{64 b^{2} \sqrt [3]{x} \sqrt{a x + b x^{\frac{2}{3}}}}{7 a^{4}} - \frac{256 b^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{21 a^{5}} + \frac{512 b^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{21 a^{6} \sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*x**2/(a*sqrt(a*x + b*x**(2/3))) + 20*x*sqrt(a*x + b*x**(2/3))/(3*a**2) - 160*
b*x**(2/3)*sqrt(a*x + b*x**(2/3))/(21*a**3) + 64*b**2*x**(1/3)*sqrt(a*x + b*x**(
2/3))/(7*a**4) - 256*b**3*sqrt(a*x + b*x**(2/3))/(21*a**5) + 512*b**4*sqrt(a*x +
b*x**(2/3))/(21*a**6*x**(1/3))

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Mathematica [A]  time = 0.0563666, size = 98, normalized size = 0.61 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (7 a^5 x^{5/3}-10 a^4 b x^{4/3}+16 a^3 b^2 x-32 a^2 b^3 x^{2/3}+128 a b^4 \sqrt [3]{x}+256 b^5\right )}{21 a^6 \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(256*b^5 + 128*a*b^4*x^(1/3) - 32*a^2*b^3*x^(2/3) + 16*
a^3*b^2*x - 10*a^4*b*x^(4/3) + 7*a^5*x^(5/3)))/(21*a^6*(b + a*x^(1/3))*x^(1/3))

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Maple [A]  time = 0.01, size = 77, normalized size = 0.5 \[{\frac{2\,x}{21\,{a}^{6}} \left ( b+a\sqrt [3]{x} \right ) \left ( 7\,{x}^{5/3}{a}^{5}-10\,{a}^{4}b{x}^{4/3}+16\,{a}^{3}{b}^{2}x-32\,{x}^{2/3}{a}^{2}{b}^{3}+128\,a{b}^{4}\sqrt [3]{x}+256\,{b}^{5} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*x*(b+a*x^(1/3))*(7*x^(5/3)*a^5-10*a^4*b*x^(4/3)+16*a^3*b^2*x-32*x^(2/3)*a^2
*b^3+128*a*b^4*x^(1/3)+256*b^5)/(b*x^(2/3)+a*x)^(3/2)/a^6

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Maxima [A]  time = 1.43693, size = 132, normalized size = 0.82 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}}}{3 \, a^{6}} - \frac{30 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b}{7 \, a^{6}} + \frac{12 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{2}}{a^{6}} - \frac{20 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{3}}{a^{6}} + \frac{30 \, \sqrt{a x^{\frac{1}{3}} + b} b^{4}}{a^{6}} + \frac{6 \, b^{5}}{\sqrt{a x^{\frac{1}{3}} + b} a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a*x^(1/3) + b)^(9/2)/a^6 - 30/7*(a*x^(1/3) + b)^(7/2)*b/a^6 + 12*(a*x^(1/3)
+ b)^(5/2)*b^2/a^6 - 20*(a*x^(1/3) + b)^(3/2)*b^3/a^6 + 30*sqrt(a*x^(1/3) + b)*
b^4/a^6 + 6*b^5/(sqrt(a*x^(1/3) + b)*a^6)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229823, size = 173, normalized size = 1.08 \[ -\frac{512 \, b^{\frac{9}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{21 \, a^{6}} + \frac{6 \, b^{5}}{\sqrt{a x^{\frac{1}{3}} + b} a^{6}{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{2 \,{\left (7 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{48} - 45 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{48} b + 126 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{48} b^{2} - 210 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{48} b^{3} + 315 \, \sqrt{a x^{\frac{1}{3}} + b} a^{48} b^{4}\right )}}{21 \, a^{54}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-512/21*b^(9/2)*sign(x^(1/3))/a^6 + 6*b^5/(sqrt(a*x^(1/3) + b)*a^6*sign(x^(1/3))
) + 2/21*(7*(a*x^(1/3) + b)^(9/2)*a^48 - 45*(a*x^(1/3) + b)^(7/2)*a^48*b + 126*(
a*x^(1/3) + b)^(5/2)*a^48*b^2 - 210*(a*x^(1/3) + b)^(3/2)*a^48*b^3 + 315*sqrt(a*
x^(1/3) + b)*a^48*b^4)/(a^54*sign(x^(1/3)))