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

Optimal. Leaf size=410 \[ \frac{3 b^7 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac{7 b^6 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac{63 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{105 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{105 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{45 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^7 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{15 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

[Out]

(3*b^7*(a + b/x^(1/3)))/(4*a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*
x^(1/3))^4) - (7*b^6*(a + b/x^(1/3)))/(a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1
/3)]*(b + a*x^(1/3))^3) + (63*b^5*(a + b/x^(1/3)))/(2*a^8*Sqrt[a^2 + b^2/x^(2/3)
 + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))^2) - (105*b^4*(a + b/x^(1/3)))/(a^8*Sqrt[a^2
 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))) + (45*b^2*(a + b/x^(1/3))*x^(
1/3))/(a^7*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) - (15*b*(a + b/x^(1/3))*x^
(2/3))/(2*a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) + ((a + b/x^(1/3))*x)/(
a^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) - (105*b^3*(a + b/x^(1/3))*Log[b
+ a*x^(1/3)])/(a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])

_______________________________________________________________________________________

Rubi [A]  time = 0.521337, antiderivative size = 410, 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 \[ \frac{3 b^7 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac{7 b^6 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac{63 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{105 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{105 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{45 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^7 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{15 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^7*(a + b/x^(1/3)))/(4*a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*
x^(1/3))^4) - (7*b^6*(a + b/x^(1/3)))/(a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1
/3)]*(b + a*x^(1/3))^3) + (63*b^5*(a + b/x^(1/3)))/(2*a^8*Sqrt[a^2 + b^2/x^(2/3)
 + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))^2) - (105*b^4*(a + b/x^(1/3)))/(a^8*Sqrt[a^2
 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*x^(1/3))) + (45*b^2*(a + b/x^(1/3))*x^(
1/3))/(a^7*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) - (15*b*(a + b/x^(1/3))*x^
(2/3))/(2*a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) + ((a + b/x^(1/3))*x)/(
a^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]) - (105*b^3*(a + b/x^(1/3))*Log[b
+ a*x^(1/3)])/(a^8*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 68.3089, size = 396, normalized size = 0.97 \[ - \frac{3 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{8 a \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{5}{2}}} - \frac{7 x}{4 a^{2} \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}} - \frac{21 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{8 a^{3} \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}} - \frac{105 x}{4 a^{4} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{35 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{2 a^{5} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} - \frac{105 b x^{\frac{2}{3}} \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{4 a^{6} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{105 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{a^{8} \left (a + \frac{b}{\sqrt [3]{x}}\right )} - \frac{105 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (a + \frac{b}{\sqrt [3]{x}} \right )}}{a^{8} \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{105 b^{2} \sqrt [3]{x} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x*(2*a + 2*b/x**(1/3))/(8*a*(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(5/2)) -
 7*x/(4*a**2*(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(3/2)) - 21*x*(2*a + 2*b/x
**(1/3))/(8*a**3*(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(3/2)) - 105*x/(4*a**4
*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) + 35*x*(2*a + 2*b/x**(1/3))/(2*a**
5*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) - 105*b*x**(2/3)*(2*a + 2*b/x**(1
/3))/(4*a**6*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) + 105*b**3*sqrt(a**2 +
 2*a*b/x**(1/3) + b**2/x**(2/3))*log(x**(-1/3))/(a**8*(a + b/x**(1/3))) - 105*b*
*3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(a + b/x**(1/3))/(a**8*(a + b/
x**(1/3))) + 105*b**2*x**(1/3)*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))/a**8

_______________________________________________________________________________________

Mathematica [A]  time = 0.128563, size = 152, normalized size = 0.37 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (4 a^7 x^{7/3}-14 a^6 b x^2+84 a^5 b^2 x^{5/3}+556 a^4 b^3 x^{4/3}+544 a^3 b^4 x-444 a^2 b^5 x^{2/3}-856 a b^6 \sqrt [3]{x}-420 b^3 \left (a \sqrt [3]{x}+b\right )^4 \log \left (a \sqrt [3]{x}+b\right )-319 b^7\right )}{4 a^8 x^{5/3} \left (\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b + a*x^(1/3))*(-319*b^7 - 856*a*b^6*x^(1/3) - 444*a^2*b^5*x^(2/3) + 544*a^3*b
^4*x + 556*a^4*b^3*x^(4/3) + 84*a^5*b^2*x^(5/3) - 14*a^6*b*x^2 + 4*a^7*x^(7/3) -
 420*b^3*(b + a*x^(1/3))^4*Log[b + a*x^(1/3)]))/(4*a^8*((b + a*x^(1/3))^2/x^(2/3
))^(5/2)*x^(5/3))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 199, normalized size = 0.5 \[{\frac{1}{4\,{a}^{8}} \left ( 4\,{x}^{7/3}{a}^{7}+84\,{a}^{5}{b}^{2}{x}^{5/3}-420\,{x}^{4/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{4}{b}^{3}+556\,{x}^{4/3}{a}^{4}{b}^{3}-2520\,{x}^{2/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{2}{b}^{5}-444\,{x}^{2/3}{a}^{2}{b}^{5}-1680\,\sqrt [3]{x}\ln \left ( b+a\sqrt [3]{x} \right ) a{b}^{6}-1680\,x\ln \left ( b+a\sqrt [3]{x} \right ){a}^{3}{b}^{4}-14\,{a}^{6}b{x}^{2}-856\,\sqrt [3]{x}a{b}^{6}-420\,\ln \left ( b+a\sqrt [3]{x} \right ){b}^{7}+544\,x{a}^{3}{b}^{4}-319\,{b}^{7} \right ) \left ( b+a\sqrt [3]{x} \right ) \left ({1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{-{\frac{5}{2}}}{x}^{-{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(5/2)/x^(5/3)*(4*x^(7/3)*a^7+84*a^
5*b^2*x^(5/3)-420*x^(4/3)*ln(b+a*x^(1/3))*a^4*b^3+556*x^(4/3)*a^4*b^3-2520*x^(2/
3)*ln(b+a*x^(1/3))*a^2*b^5-444*x^(2/3)*a^2*b^5-1680*x^(1/3)*ln(b+a*x^(1/3))*a*b^
6-1680*x*ln(b+a*x^(1/3))*a^3*b^4-14*a^6*b*x^2-856*x^(1/3)*a*b^6-420*ln(b+a*x^(1/
3))*b^7+544*x*a^3*b^4-319*b^7)*(b+a*x^(1/3))/a^8

_______________________________________________________________________________________

Maxima [A]  time = 0.750663, size = 188, normalized size = 0.46 \[ \frac{4 \, a^{7} x^{\frac{7}{3}} - 14 \, a^{6} b x^{2} + 84 \, a^{5} b^{2} x^{\frac{5}{3}} + 556 \, a^{4} b^{3} x^{\frac{4}{3}} + 544 \, a^{3} b^{4} x - 444 \, a^{2} b^{5} x^{\frac{2}{3}} - 856 \, a b^{6} x^{\frac{1}{3}} - 319 \, b^{7}}{4 \,{\left (a^{12} x^{\frac{4}{3}} + 4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac{2}{3}} + 4 \, a^{9} b^{3} x^{\frac{1}{3}} + a^{8} b^{4}\right )}} - \frac{105 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{8}} \]

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))^(-5/2),x, algorithm="maxima")

[Out]

1/4*(4*a^7*x^(7/3) - 14*a^6*b*x^2 + 84*a^5*b^2*x^(5/3) + 556*a^4*b^3*x^(4/3) + 5
44*a^3*b^4*x - 444*a^2*b^5*x^(2/3) - 856*a*b^6*x^(1/3) - 319*b^7)/(a^12*x^(4/3)
+ 4*a^11*b*x + 6*a^10*b^2*x^(2/3) + 4*a^9*b^3*x^(1/3) + a^8*b^4) - 105*b^3*log(a
*x^(1/3) + b)/a^8

_______________________________________________________________________________________

Fricas [A]  time = 0.275565, size = 238, normalized size = 0.58 \[ -\frac{14 \, a^{6} b x^{2} - 544 \, a^{3} b^{4} x + 319 \, b^{7} + 420 \,{\left (4 \, a^{3} b^{4} x + 6 \, a^{2} b^{5} x^{\frac{2}{3}} + b^{7} +{\left (a^{4} b^{3} x + 4 \, a b^{6}\right )} x^{\frac{1}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 12 \,{\left (7 \, a^{5} b^{2} x - 37 \, a^{2} b^{5}\right )} x^{\frac{2}{3}} - 4 \,{\left (a^{7} x^{2} + 139 \, a^{4} b^{3} x - 214 \, a b^{6}\right )} x^{\frac{1}{3}}}{4 \,{\left (4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac{2}{3}} + a^{8} b^{4} +{\left (a^{12} x + 4 \, a^{9} b^{3}\right )} x^{\frac{1}{3}}\right )}} \]

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))^(-5/2),x, algorithm="fricas")

[Out]

-1/4*(14*a^6*b*x^2 - 544*a^3*b^4*x + 319*b^7 + 420*(4*a^3*b^4*x + 6*a^2*b^5*x^(2
/3) + b^7 + (a^4*b^3*x + 4*a*b^6)*x^(1/3))*log(a*x^(1/3) + b) - 12*(7*a^5*b^2*x
- 37*a^2*b^5)*x^(2/3) - 4*(a^7*x^2 + 139*a^4*b^3*x - 214*a*b^6)*x^(1/3))/(4*a^11
*b*x + 6*a^10*b^2*x^(2/3) + a^8*b^4 + (a^12*x + 4*a^9*b^3)*x^(1/3))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.319103, size = 190, normalized size = 0.46 \[ -\frac{105 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{8}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} - \frac{420 \, a^{3} b^{4} x + 1134 \, a^{2} b^{5} x^{\frac{2}{3}} + 1036 \, a b^{6} x^{\frac{1}{3}} + 319 \, b^{7}}{4 \,{\left (a x^{\frac{1}{3}} + b\right )}^{4} a^{8}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} + \frac{2 \, a^{10} x - 15 \, a^{9} b x^{\frac{2}{3}} + 90 \, a^{8} b^{2} x^{\frac{1}{3}}}{2 \, a^{15}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} \]

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))^(-5/2),x, algorithm="giac")

[Out]

-105*b^3*ln(abs(a*x^(1/3) + b))/(a^8*sign(a*x^(2/3) + b*x^(1/3))) - 1/4*(420*a^3
*b^4*x + 1134*a^2*b^5*x^(2/3) + 1036*a*b^6*x^(1/3) + 319*b^7)/((a*x^(1/3) + b)^4
*a^8*sign(a*x^(2/3) + b*x^(1/3))) + 1/2*(2*a^10*x - 15*a^9*b*x^(2/3) + 90*a^8*b^
2*x^(1/3))/(a^15*sign(a*x^(2/3) + b*x^(1/3)))