3.462 \(\int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^9}{10 b^3}-\frac{2 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^8}{3 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3} \]

[Out]

(3*a^2*(a + b*x^(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3) - (2*a
*(a + b*x^(1/3))^8*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(3*b^3) + (3*(a + b*
x^(1/3))^9*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(10*b^3)

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Rubi [A]  time = 0.171075, 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 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^9}{10 b^3}-\frac{2 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^8}{3 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(7/2),x]

[Out]

(3*a^2*(a + b*x^(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3) - (2*a
*(a + b*x^(1/3))^8*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(3*b^3) + (3*(a + b*
x^(1/3))^9*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(10*b^3)

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Rubi in Sympy [A]  time = 15.006, size = 126, normalized size = 0.92 \[ \frac{3 a^{2} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}{80 b^{3}} - \frac{a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{9}{2}}}{15 b^{3}} + \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}{20 b} \]

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))**(7/2),x)

[Out]

3*a**2*(2*a + 2*b*x**(1/3))*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(7/2)/(80*b
**3) - a*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(9/2)/(15*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))**(7/2)/(20*b)

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Mathematica [A]  time = 0.0630229, size = 115, normalized size = 0.84 \[ \frac{x \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (120 a^7+630 a^6 b \sqrt [3]{x}+1512 a^5 b^2 x^{2/3}+2100 a^4 b^3 x+1800 a^3 b^4 x^{4/3}+945 a^2 b^5 x^{5/3}+280 a b^6 x^2+36 b^7 x^{7/3}\right )}{120 \left (a+b \sqrt [3]{x}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(7/2),x]

[Out]

(Sqrt[(a + b*x^(1/3))^2]*x*(120*a^7 + 630*a^6*b*x^(1/3) + 1512*a^5*b^2*x^(2/3) +
2100*a^4*b^3*x + 1800*a^3*b^4*x^(4/3) + 945*a^2*b^5*x^(5/3) + 280*a*b^6*x^2 + 3
6*b^7*x^(7/3)))/(120*(a + b*x^(1/3)))

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Maple [A]  time = 0.014, size = 109, normalized size = 0.8 \[{\frac{1}{120}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 36\,{b}^{7}{x}^{10/3}+945\,{a}^{2}{b}^{5}{x}^{8/3}+1800\,{a}^{3}{b}^{4}{x}^{7/3}+1512\,{a}^{5}{b}^{2}{x}^{5/3}+630\,{a}^{6}b{x}^{4/3}+280\,a{b}^{6}{x}^{3}+2100\,{a}^{4}{b}^{3}{x}^{2}+120\,{a}^{7}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]

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))^(7/2),x)

[Out]

1/120*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(36*b^7*x^(10/3)+945*a^2*b^5*x^(8/3)
+1800*a^3*b^4*x^(7/3)+1512*a^5*b^2*x^(5/3)+630*a^6*b*x^(4/3)+280*a*b^6*x^3+2100*
a^4*b^3*x^2+120*a^7*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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274036, size = 113, normalized size = 0.82 \[ \frac{7}{3} \, a b^{6} x^{3} + \frac{35}{2} \, a^{4} b^{3} x^{2} + a^{7} x + \frac{63}{40} \,{\left (5 \, a^{2} b^{5} x^{2} + 8 \, a^{5} b^{2} x\right )} x^{\frac{2}{3}} + \frac{3}{20} \,{\left (2 \, b^{7} x^{3} + 100 \, a^{3} b^{4} x^{2} + 35 \, a^{6} b x\right )} x^{\frac{1}{3}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

7/3*a*b^6*x^3 + 35/2*a^4*b^3*x^2 + a^7*x + 63/40*(5*a^2*b^5*x^2 + 8*a^5*b^2*x)*x
^(2/3) + 3/20*(2*b^7*x^3 + 100*a^3*b^4*x^2 + 35*a^6*b*x)*x^(1/3)

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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((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304417, size = 189, normalized size = 1.38 \[ \frac{3}{10} \, b^{7} x^{\frac{10}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{7}{3} \, a b^{6} x^{3}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{63}{8} \, a^{2} b^{5} x^{\frac{8}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + 15 \, a^{3} b^{4} x^{\frac{7}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{35}{2} \, a^{4} b^{3} x^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{63}{5} \, a^{5} b^{2} x^{\frac{5}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{21}{4} \, a^{6} b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a^{7} x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]

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)^(7/2),x, algorithm="giac")

[Out]

3/10*b^7*x^(10/3)*sign(b*x^(1/3) + a) + 7/3*a*b^6*x^3*sign(b*x^(1/3) + a) + 63/8
*a^2*b^5*x^(8/3)*sign(b*x^(1/3) + a) + 15*a^3*b^4*x^(7/3)*sign(b*x^(1/3) + a) +
35/2*a^4*b^3*x^2*sign(b*x^(1/3) + a) + 63/5*a^5*b^2*x^(5/3)*sign(b*x^(1/3) + a)
+ 21/4*a^6*b*x^(4/3)*sign(b*x^(1/3) + a) + a^7*x*sign(b*x^(1/3) + a)