3.480 \(\int \frac{1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=268 \[ -\frac{60 a^2}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{6 \sqrt [6]{x} \left (a+b \sqrt [6]{x}\right )}{b^5 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]

[Out]

(-60*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (3*a^5)/(2*b^6*(a + b*
x^(1/6))^3*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) - (10*a^4)/(b^6*(a + b*x^(1/
6))^2*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (30*a^3)/(b^6*(a + b*x^(1/6))*S
qrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (6*(a + b*x^(1/6))*x^(1/6))/(b^5*Sqrt[
a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) - (30*a*(a + b*x^(1/6))*Log[a + b*x^(1/6)])/
(b^6*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)])

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Rubi [A]  time = 0.294771, antiderivative size = 268, 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 \[ -\frac{60 a^2}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{6 \sqrt [6]{x} \left (a+b \sqrt [6]{x}\right )}{b^5 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(-5/2),x]

[Out]

(-60*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (3*a^5)/(2*b^6*(a + b*
x^(1/6))^3*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) - (10*a^4)/(b^6*(a + b*x^(1/
6))^2*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (30*a^3)/(b^6*(a + b*x^(1/6))*S
qrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) + (6*(a + b*x^(1/6))*x^(1/6))/(b^5*Sqrt[
a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)]) - (30*a*(a + b*x^(1/6))*Log[a + b*x^(1/6)])/
(b^6*Sqrt[a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3)])

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Rubi in Sympy [A]  time = 27.63, size = 250, normalized size = 0.93 \[ - \frac{30 a \left (a + b \sqrt [6]{x}\right ) \log{\left (a + b \sqrt [6]{x} \right )}}{b^{6} \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}} - \frac{3 x^{\frac{5}{6}} \left (2 a + 2 b \sqrt [6]{x}\right )}{4 b \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{5}{2}}} - \frac{5 x^{\frac{2}{3}}}{2 b^{2} \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{3}{2}}} - \frac{5 \sqrt{x} \left (2 a + 2 b \sqrt [6]{x}\right )}{2 b^{3} \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{3}{2}}} - \frac{15 \sqrt [3]{x}}{b^{4} \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}} + \frac{30 \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a**2+2*a*b*x**(1/6)+b**2*x**(1/3))**(5/2),x)

[Out]

-30*a*(a + b*x**(1/6))*log(a + b*x**(1/6))/(b**6*sqrt(a**2 + 2*a*b*x**(1/6) + b*
*2*x**(1/3))) - 3*x**(5/6)*(2*a + 2*b*x**(1/6))/(4*b*(a**2 + 2*a*b*x**(1/6) + b*
*2*x**(1/3))**(5/2)) - 5*x**(2/3)/(2*b**2*(a**2 + 2*a*b*x**(1/6) + b**2*x**(1/3)
)**(3/2)) - 5*sqrt(x)*(2*a + 2*b*x**(1/6))/(2*b**3*(a**2 + 2*a*b*x**(1/6) + b**2
*x**(1/3))**(3/2)) - 15*x**(1/3)/(b**4*sqrt(a**2 + 2*a*b*x**(1/6) + b**2*x**(1/3
))) + 30*sqrt(a**2 + 2*a*b*x**(1/6) + b**2*x**(1/3))/b**6

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Mathematica [A]  time = 0.0911913, size = 121, normalized size = 0.45 \[ \frac{-77 a^5-248 a^4 b \sqrt [6]{x}-252 a^3 b^2 \sqrt [3]{x}-48 a^2 b^3 \sqrt{x}+48 a b^4 x^{2/3}-60 a \left (a+b \sqrt [6]{x}\right )^4 \log \left (a+b \sqrt [6]{x}\right )+12 b^5 x^{5/6}}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{\left (a+b \sqrt [6]{x}\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(-5/2),x]

[Out]

(-77*a^5 - 248*a^4*b*x^(1/6) - 252*a^3*b^2*x^(1/3) - 48*a^2*b^3*Sqrt[x] + 48*a*b
^4*x^(2/3) + 12*b^5*x^(5/6) - 60*a*(a + b*x^(1/6))^4*Log[a + b*x^(1/6)])/(2*b^6*
(a + b*x^(1/6))^3*Sqrt[(a + b*x^(1/6))^2])

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Maple [A]  time = 0.022, size = 174, normalized size = 0.7 \[{\frac{1}{2\,{b}^{6}}\sqrt{{a}^{2}+2\,ab\sqrt [6]{x}+{b}^{2}\sqrt [3]{x}} \left ( 12\,{x}^{5/6}{b}^{5}-60\,{x}^{2/3}\ln \left ( a+b\sqrt [6]{x} \right ) a{b}^{4}+48\,{x}^{2/3}a{b}^{4}-240\,\sqrt{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{2}{b}^{3}-48\,\sqrt{x}{a}^{2}{b}^{3}-360\,\sqrt [3]{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{3}{b}^{2}-252\,\sqrt [3]{x}{a}^{3}{b}^{2}-240\,\sqrt [6]{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{4}b-248\,\sqrt [6]{x}{a}^{4}b-60\,\ln \left ( a+b\sqrt [6]{x} \right ){a}^{5}-77\,{a}^{5} \right ) \left ( a+b\sqrt [6]{x} \right ) ^{-5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a^2+2*a*b*x^(1/6)+b^2*x^(1/3))^(5/2),x)

[Out]

1/2*(a^2+2*a*b*x^(1/6)+b^2*x^(1/3))^(1/2)*(12*x^(5/6)*b^5-60*x^(2/3)*ln(a+b*x^(1
/6))*a*b^4+48*x^(2/3)*a*b^4-240*x^(1/2)*ln(a+b*x^(1/6))*a^2*b^3-48*x^(1/2)*a^2*b
^3-360*x^(1/3)*ln(a+b*x^(1/6))*a^3*b^2-252*x^(1/3)*a^3*b^2-240*x^(1/6)*ln(a+b*x^
(1/6))*a^4*b-248*x^(1/6)*a^4*b-60*ln(a+b*x^(1/6))*a^5-77*a^5)/(a+b*x^(1/6))^5/b^
6

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Maxima [A]  time = 0.794399, size = 161, normalized size = 0.6 \[ \frac{12 \, b^{5} x^{\frac{5}{6}} + 48 \, a b^{4} x^{\frac{2}{3}} - 48 \, a^{2} b^{3} \sqrt{x} - 252 \, a^{3} b^{2} x^{\frac{1}{3}} - 248 \, a^{4} b x^{\frac{1}{6}} - 77 \, a^{5}}{2 \,{\left (b^{10} x^{\frac{2}{3}} + 4 \, a b^{9} \sqrt{x} + 6 \, a^{2} b^{8} x^{\frac{1}{3}} + 4 \, a^{3} b^{7} x^{\frac{1}{6}} + a^{4} b^{6}\right )}} - \frac{30 \, a \log \left (b x^{\frac{1}{6}} + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="maxima")

[Out]

1/2*(12*b^5*x^(5/6) + 48*a*b^4*x^(2/3) - 48*a^2*b^3*sqrt(x) - 252*a^3*b^2*x^(1/3
) - 248*a^4*b*x^(1/6) - 77*a^5)/(b^10*x^(2/3) + 4*a*b^9*sqrt(x) + 6*a^2*b^8*x^(1
/3) + 4*a^3*b^7*x^(1/6) + a^4*b^6) - 30*a*log(b*x^(1/6) + a)/b^6

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Fricas [A]  time = 0.276934, size = 212, normalized size = 0.79 \[ \frac{12 \, b^{5} x^{\frac{5}{6}} + 48 \, a b^{4} x^{\frac{2}{3}} - 48 \, a^{2} b^{3} \sqrt{x} - 252 \, a^{3} b^{2} x^{\frac{1}{3}} - 248 \, a^{4} b x^{\frac{1}{6}} - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{\frac{2}{3}} + 4 \, a^{2} b^{3} \sqrt{x} + 6 \, a^{3} b^{2} x^{\frac{1}{3}} + 4 \, a^{4} b x^{\frac{1}{6}} + a^{5}\right )} \log \left (b x^{\frac{1}{6}} + a\right )}{2 \,{\left (b^{10} x^{\frac{2}{3}} + 4 \, a b^{9} \sqrt{x} + 6 \, a^{2} b^{8} x^{\frac{1}{3}} + 4 \, a^{3} b^{7} x^{\frac{1}{6}} + a^{4} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="fricas")

[Out]

1/2*(12*b^5*x^(5/6) + 48*a*b^4*x^(2/3) - 48*a^2*b^3*sqrt(x) - 252*a^3*b^2*x^(1/3
) - 248*a^4*b*x^(1/6) - 77*a^5 - 60*(a*b^4*x^(2/3) + 4*a^2*b^3*sqrt(x) + 6*a^3*b
^2*x^(1/3) + 4*a^4*b*x^(1/6) + a^5)*log(b*x^(1/6) + a))/(b^10*x^(2/3) + 4*a*b^9*
sqrt(x) + 6*a^2*b^8*x^(1/3) + 4*a^3*b^7*x^(1/6) + a^4*b^6)

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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**(1/6)+b**2*x**(1/3))**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError