3.492 \(\int \left (a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx\)

Optimal. Leaf size=391 \[ -\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]

[Out]

(-6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)])/((a + b/x^(1/6))*x^(1/6)) + (
126*a^2*b^5*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1/6))/(a + b/x^(1/6)) +
(105*a^3*b^4*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1/3))/(a + b/x^(1/6))
+ (70*a^4*b^3*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Sqrt[x])/(a + b/x^(1/6)
) + (63*a^5*b^2*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(2/3))/(2*(a + b/x^(
1/6))) + (42*a^6*b*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(5/6))/(5*(a + b/
x^(1/6))) + (a^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x)/(a + b/x^(1/6)) +
(42*a*b^6*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Log[x^(1/6)])/(a + b/x^(1/6)
)

_______________________________________________________________________________________

Rubi [A]  time = 0.391675, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(-6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)])/((a + b/x^(1/6))*x^(1/6)) + (
126*a^2*b^5*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1/6))/(a + b/x^(1/6)) +
(105*a^3*b^4*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1/3))/(a + b/x^(1/6))
+ (70*a^4*b^3*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Sqrt[x])/(a + b/x^(1/6)
) + (63*a^5*b^2*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(2/3))/(2*(a + b/x^(
1/6))) + (42*a^6*b*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(5/6))/(5*(a + b/
x^(1/6))) + (a^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x)/(a + b/x^(1/6)) +
(42*a*b^6*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Log[x^(1/6)])/(a + b/x^(1/6)
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.4084, size = 313, normalized size = 0.8 \[ - \frac{42 a b^{6} \sqrt{a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}} \log{\left (\frac{1}{\sqrt [6]{x}} \right )}}{a + \frac{b}{\sqrt [6]{x}}} - 42 b^{6} \sqrt{a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}} + 21 b^{5} \sqrt [6]{x} \left (a + \frac{b}{\sqrt [6]{x}}\right ) \sqrt{a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}} + 7 b^{4} \sqrt [3]{x} \left (a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}\right )^{\frac{3}{2}} + \frac{7 b^{3} \sqrt{x} \left (a + \frac{b}{\sqrt [6]{x}}\right ) \left (a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}\right )^{\frac{3}{2}}}{2} + \frac{21 b^{2} x^{\frac{2}{3}} \left (a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}\right )^{\frac{5}{2}}}{10} + \frac{7 b x^{\frac{5}{6}} \left (a + \frac{b}{\sqrt [6]{x}}\right ) \left (a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}\right )^{\frac{5}{2}}}{5} + x \left (a^{2} + \frac{2 a b}{\sqrt [6]{x}} + \frac{b^{2}}{\sqrt [3]{x}}\right )^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**2+b**2/x**(1/3)+2*a*b/x**(1/6))**(7/2),x)

[Out]

-42*a*b**6*sqrt(a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3))*log(x**(-1/6))/(a + b/x**
(1/6)) - 42*b**6*sqrt(a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3)) + 21*b**5*x**(1/6)*
(a + b/x**(1/6))*sqrt(a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3)) + 7*b**4*x**(1/3)*(
a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3))**(3/2) + 7*b**3*sqrt(x)*(a + b/x**(1/6))*
(a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3))**(3/2)/2 + 21*b**2*x**(2/3)*(a**2 + 2*a*
b/x**(1/6) + b**2/x**(1/3))**(5/2)/10 + 7*b*x**(5/6)*(a + b/x**(1/6))*(a**2 + 2*
a*b/x**(1/6) + b**2/x**(1/3))**(5/2)/5 + x*(a**2 + 2*a*b/x**(1/6) + b**2/x**(1/3
))**(7/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0857193, size = 124, normalized size = 0.32 \[ \frac{\sqrt{\frac{\left (a \sqrt [6]{x}+b\right )^2}{\sqrt [3]{x}}} \left (10 a^7 x^{7/6}+84 a^6 b x+315 a^5 b^2 x^{5/6}+700 a^4 b^3 x^{2/3}+1050 a^3 b^4 \sqrt{x}+1260 a^2 b^5 \sqrt [3]{x}+70 a b^6 \sqrt [6]{x} \log (x)-60 b^7\right )}{10 \left (a \sqrt [6]{x}+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(Sqrt[(b + a*x^(1/6))^2/x^(1/3)]*(-60*b^7 + 1260*a^2*b^5*x^(1/3) + 1050*a^3*b^4*
Sqrt[x] + 700*a^4*b^3*x^(2/3) + 315*a^5*b^2*x^(5/6) + 84*a^6*b*x + 10*a^7*x^(7/6
) + 70*a*b^6*x^(1/6)*Log[x]))/(10*(b + a*x^(1/6)))

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 116, normalized size = 0.3 \[{\frac{1}{10}\sqrt{{1 \left ({a}^{2}\sqrt{x}+2\,ab\sqrt [3]{x}+{b}^{2}\sqrt [6]{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 84\,{a}^{6}bx+315\,{a}^{5}{b}^{2}{x}^{5/6}+1260\,{a}^{2}{b}^{5}\sqrt [3]{x}+700\,{a}^{4}{b}^{3}{x}^{2/3}+1050\,\sqrt{x}{a}^{3}{b}^{4}+70\,a{b}^{6}\ln \left ( x \right ) \sqrt [6]{x}+10\,{a}^{7}{x}^{7/6}-60\,{b}^{7} \right ) \left ( \sqrt [6]{x}a+b \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x)

[Out]

1/10*((a^2*x^(1/2)+2*a*b*x^(1/3)+b^2*x^(1/6))/x^(1/2))^(1/2)*(84*a^6*b*x+315*a^5
*b^2*x^(5/6)+1260*a^2*b^5*x^(1/3)+700*a^4*b^3*x^(2/3)+1050*x^(1/2)*a^3*b^4+70*a*
b^6*ln(x)*x^(1/6)+10*a^7*x^(7/6)-60*b^7)/(x^(1/6)*a+b)

_______________________________________________________________________________________

Maxima [A]  time = 0.757155, size = 107, normalized size = 0.27 \[ 7 \, a b^{6} \log \left (x\right ) + \frac{10 \, a^{7} x^{\frac{7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac{5}{6}} + 700 \, a^{4} b^{3} x^{\frac{2}{3}} + 1050 \, a^{3} b^{4} \sqrt{x} + 1260 \, a^{2} b^{5} x^{\frac{1}{3}} - 60 \, b^{7}}{10 \, x^{\frac{1}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/6) + b^2/x^(1/3))^(7/2),x, algorithm="maxima")

[Out]

7*a*b^6*log(x) + 1/10*(10*a^7*x^(7/6) + 84*a^6*b*x + 315*a^5*b^2*x^(5/6) + 700*a
^4*b^3*x^(2/3) + 1050*a^3*b^4*sqrt(x) + 1260*a^2*b^5*x^(1/3) - 60*b^7)/x^(1/6)

_______________________________________________________________________________________

Fricas [A]  time = 0.274199, size = 112, normalized size = 0.29 \[ \frac{10 \, a^{7} x^{\frac{7}{6}} + 420 \, a b^{6} x^{\frac{1}{6}} \log \left (x^{\frac{1}{6}}\right ) + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac{5}{6}} + 700 \, a^{4} b^{3} x^{\frac{2}{3}} + 1050 \, a^{3} b^{4} \sqrt{x} + 1260 \, a^{2} b^{5} x^{\frac{1}{3}} - 60 \, b^{7}}{10 \, x^{\frac{1}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/6) + b^2/x^(1/3))^(7/2),x, algorithm="fricas")

[Out]

1/10*(10*a^7*x^(7/6) + 420*a*b^6*x^(1/6)*log(x^(1/6)) + 84*a^6*b*x + 315*a^5*b^2
*x^(5/6) + 700*a^4*b^3*x^(2/3) + 1050*a^3*b^4*sqrt(x) + 1260*a^2*b^5*x^(1/3) - 6
0*b^7)/x^(1/6)

_______________________________________________________________________________________

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+b**2/x**(1/3)+2*a*b/x**(1/6))**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.389965, size = 232, normalized size = 0.59 \[ a^{7} x{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + 7 \, a b^{6}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + \frac{42}{5} \, a^{6} b x^{\frac{5}{6}}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + \frac{63}{2} \, a^{5} b^{2} x^{\frac{2}{3}}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt{x}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac{1}{3}}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac{1}{6}}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right ) - \frac{6 \, b^{7}{\rm sign}\left (a x + b x^{\frac{5}{6}}\right ){\rm sign}\left (x\right )}{x^{\frac{1}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/6) + b^2/x^(1/3))^(7/2),x, algorithm="giac")

[Out]

a^7*x*sign(a*x + b*x^(5/6))*sign(x) + 7*a*b^6*ln(abs(x))*sign(a*x + b*x^(5/6))*s
ign(x) + 42/5*a^6*b*x^(5/6)*sign(a*x + b*x^(5/6))*sign(x) + 63/2*a^5*b^2*x^(2/3)
*sign(a*x + b*x^(5/6))*sign(x) + 70*a^4*b^3*sqrt(x)*sign(a*x + b*x^(5/6))*sign(x
) + 105*a^3*b^4*x^(1/3)*sign(a*x + b*x^(5/6))*sign(x) + 126*a^2*b^5*x^(1/6)*sign
(a*x + b*x^(5/6))*sign(x) - 6*b^7*sign(a*x + b*x^(5/6))*sign(x)/x^(1/6)