3.469 \(\int \frac{1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

[Out]

-a^2/(2*b^3*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) + (6*a)/(
5*b^3*(a + b*x^(1/3))^4*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) - 3/(4*b^3*(a +
b*x^(1/3))^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])

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Rubi [A]  time = 0.159789, 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 \[ -\frac{a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(2*b^3*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) + (6*a)/(
5*b^3*(a + b*x^(1/3))^4*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) - 3/(4*b^3*(a +
b*x^(1/3))^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])

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Rubi in Sympy [A]  time = 10.9411, size = 124, normalized size = 0.91 \[ - \frac{x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right )}{4 b \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}} - \frac{\sqrt [3]{x}}{5 b^{2} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}} - \frac{2 a + 2 b \sqrt [3]{x}}{40 b^{3} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(2/3)*(2*a + 2*b*x**(1/3))/(4*b*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(7/
2)) - x**(1/3)/(5*b**2*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2)) - (2*a +
2*b*x**(1/3))/(40*b**3*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2))

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Mathematica [A]  time = 0.0410084, size = 58, normalized size = 0.42 \[ \frac{-a^2-6 a b \sqrt [3]{x}-15 b^2 x^{2/3}}{20 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-a^2 - 6*a*b*x^(1/3) - 15*b^2*x^(2/3))/(20*b^3*(a + b*x^(1/3))^5*Sqrt[(a + b*x^
(1/3))^2])

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Maple [A]  time = 0.01, size = 54, normalized size = 0.4 \[ -{\frac{1}{20\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 15\,{b}^{2}{x}^{2/3}+6\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(15*b^2*x^(2/3)+6*a*b*x^(1/3)+a^2)/(
a+b*x^(1/3))^7/b^3

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Maxima [A]  time = 0.744911, size = 85, normalized size = 0.62 \[ -\frac{a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{11}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{6}} + \frac{6 \, a b}{5 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{5}} - \frac{3}{4 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{4}} \]

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]

-1/2*a^2*b^2/((b^2)^(11/2)*(x^(1/3) + a/b)^6) + 6/5*a*b/((b^2)^(9/2)*(x^(1/3) +
a/b)^5) - 3/4/((b^2)^(7/2)*(x^(1/3) + a/b)^4)

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Fricas [A]  time = 0.275195, size = 123, normalized size = 0.9 \[ -\frac{15 \, b^{2} x^{\frac{2}{3}} + 6 \, a b x^{\frac{1}{3}} + a^{2}}{20 \,{\left (b^{9} x^{2} + 20 \, a^{3} b^{6} x + a^{6} b^{3} + 3 \,{\left (2 \, a b^{8} x + 5 \, a^{4} b^{5}\right )} x^{\frac{2}{3}} + 3 \,{\left (5 \, a^{2} b^{7} x + 2 \, a^{5} b^{4}\right )} x^{\frac{1}{3}}\right )}} \]

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="fricas")

[Out]

-1/20*(15*b^2*x^(2/3) + 6*a*b*x^(1/3) + a^2)/(b^9*x^2 + 20*a^3*b^6*x + a^6*b^3 +
3*(2*a*b^8*x + 5*a^4*b^5)*x^(2/3) + 3*(5*a^2*b^7*x + 2*a^5*b^4)*x^(1/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

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]

undef