3.471 \(\int \frac{1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.156081, 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 \[ -\frac{3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-11/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.3736, size = 126, normalized size = 0.92 \[ - \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right )}{20 b \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{11}{2}}} - \frac{\sqrt [3]{x}}{15 b^{2} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{9}{2}}} - \frac{2 a + 2 b \sqrt [3]{x}}{240 b^{3} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{9}{2}}} \]

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))**(11/2),x)

[Out]

-3*x**(2/3)*(2*a + 2*b*x**(1/3))/(20*b*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**
(11/2)) - x**(1/3)/(15*b**2*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(9/2)) - (2
*a + 2*b*x**(1/3))/(240*b**3*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(9/2))

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Mathematica [A]  time = 0.0385327, size = 58, normalized size = 0.42 \[ \frac{-a^2-10 a b \sqrt [3]{x}-45 b^2 x^{2/3}}{120 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-11/2),x]

[Out]

(-a^2 - 10*a*b*x^(1/3) - 45*b^2*x^(2/3))/(120*b^3*(a + b*x^(1/3))^9*Sqrt[(a + b*
x^(1/3))^2])

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Maple [A]  time = 0.011, size = 54, normalized size = 0.4 \[ -{\frac{1}{120\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 45\,{b}^{2}{x}^{2/3}+10\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-11}} \]

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))^(11/2),x)

[Out]

-1/120*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(45*b^2*x^(2/3)+10*a*b*x^(1/3)+a^2)
/(a+b*x^(1/3))^11/b^3

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Maxima [A]  time = 0.767573, size = 85, normalized size = 0.62 \[ -\frac{3 \, a^{2} b^{2}}{10 \,{\left (b^{2}\right )}^{\frac{15}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{10}} + \frac{2 \, a b}{3 \,{\left (b^{2}\right )}^{\frac{13}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{9}} - \frac{3}{8 \,{\left (b^{2}\right )}^{\frac{11}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{8}} \]

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)^(-11/2),x, algorithm="maxima")

[Out]

-3/10*a^2*b^2/((b^2)^(15/2)*(x^(1/3) + a/b)^10) + 2/3*a*b/((b^2)^(13/2)*(x^(1/3)
+ a/b)^9) - 3/8/((b^2)^(11/2)*(x^(1/3) + a/b)^8)

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Fricas [A]  time = 0.276703, size = 181, normalized size = 1.32 \[ -\frac{45 \, b^{2} x^{\frac{2}{3}} + 10 \, a b x^{\frac{1}{3}} + a^{2}}{120 \,{\left (10 \, a b^{12} x^{3} + 210 \, a^{4} b^{9} x^{2} + 120 \, a^{7} b^{6} x + a^{10} b^{3} + 9 \,{\left (5 \, a^{2} b^{11} x^{2} + 28 \, a^{5} b^{8} x + 5 \, a^{8} b^{5}\right )} x^{\frac{2}{3}} +{\left (b^{13} x^{3} + 120 \, a^{3} b^{10} x^{2} + 210 \, a^{6} b^{7} x + 10 \, a^{9} b^{4}\right )} x^{\frac{1}{3}}\right )}} \]

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)^(-11/2),x, algorithm="fricas")

[Out]

-1/120*(45*b^2*x^(2/3) + 10*a*b*x^(1/3) + a^2)/(10*a*b^12*x^3 + 210*a^4*b^9*x^2
+ 120*a^7*b^6*x + a^10*b^3 + 9*(5*a^2*b^11*x^2 + 28*a^5*b^8*x + 5*a^8*b^5)*x^(2/
3) + (b^13*x^3 + 120*a^3*b^10*x^2 + 210*a^6*b^7*x + 10*a^9*b^4)*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(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(11/2),x)

[Out]

Timed out

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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)^(-11/2),x, algorithm="giac")

[Out]

undef