3.463 \(\int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{5/2} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]

[Out]

(a^2*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(2*b^3) - (6*a*(
a + b*x^(1/3))^6*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(7*b^3) + (3*(a + b*x^
(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3)

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Rubi [A]  time = 0.156943, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]

[Out]

(a^2*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(2*b^3) - (6*a*(
a + b*x^(1/3))^6*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(7*b^3) + (3*(a + b*x^
(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3)

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Rubi in Sympy [A]  time = 14.9486, size = 126, normalized size = 0.92 \[ \frac{a^{2} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{16 b^{3}} - \frac{3 a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}{28 b^{3}} + \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{16 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)

[Out]

a**2*(2*a + 2*b*x**(1/3))*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2)/(16*b**
3) - 3*a*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(7/2)/(28*b**3) + 3*x**(2/3)*(
2*a + 2*b*x**(1/3))*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2)/(16*b)

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Mathematica [A]  time = 0.0423923, size = 91, normalized size = 0.66 \[ \frac{x \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (56 a^5+210 a^4 b \sqrt [3]{x}+336 a^3 b^2 x^{2/3}+280 a^2 b^3 x+120 a b^4 x^{4/3}+21 b^5 x^{5/3}\right )}{56 \left (a+b \sqrt [3]{x}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]

[Out]

(Sqrt[(a + b*x^(1/3))^2]*x*(56*a^5 + 210*a^4*b*x^(1/3) + 336*a^3*b^2*x^(2/3) + 2
80*a^2*b^3*x + 120*a*b^4*x^(4/3) + 21*b^5*x^(5/3)))/(56*(a + b*x^(1/3)))

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Maple [A]  time = 0.004, size = 87, normalized size = 0.6 \[{\frac{1}{56}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 21\,{b}^{5}{x}^{8/3}+120\,a{b}^{4}{x}^{7/3}+336\,{a}^{3}{b}^{2}{x}^{5/3}+210\,{a}^{4}b{x}^{4/3}+280\,{a}^{2}{b}^{3}{x}^{2}+56\,{a}^{5}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x)

[Out]

1/56*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(21*b^5*x^(8/3)+120*a*b^4*x^(7/3)+336
*a^3*b^2*x^(5/3)+210*a^4*b*x^(4/3)+280*a^2*b^3*x^2+56*a^5*x)/(a+b*x^(1/3))

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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((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274681, size = 82, normalized size = 0.6 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac{3}{8} \,{\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac{2}{3}} + \frac{15}{28} \,{\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(5/2),x, algorithm="fricas")

[Out]

5*a^2*b^3*x^2 + a^5*x + 3/8*(b^5*x^2 + 16*a^3*b^2*x)*x^(2/3) + 15/28*(4*a*b^4*x^
2 + 7*a^4*b*x)*x^(1/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2), x)

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GIAC/XCAS [A]  time = 0.2843, size = 138, normalized size = 1.01 \[ \frac{3}{8} \, b^{5} x^{\frac{8}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + 5 \, a^{2} b^{3} x^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + 6 \, a^{3} b^{2} x^{\frac{5}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a^{5} x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(5/2),x, algorithm="giac")

[Out]

3/8*b^5*x^(8/3)*sign(b*x^(1/3) + a) + 15/7*a*b^4*x^(7/3)*sign(b*x^(1/3) + a) + 5
*a^2*b^3*x^2*sign(b*x^(1/3) + a) + 6*a^3*b^2*x^(5/3)*sign(b*x^(1/3) + a) + 15/4*
a^4*b*x^(4/3)*sign(b*x^(1/3) + a) + a^5*x*sign(b*x^(1/3) + a)