3.69 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=251 \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (10*a^3*b^2*x*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (b^5
*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3))

_______________________________________________________________________________________

Rubi [A]  time = 0.157908, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (10*a^3*b^2*x*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (b^5
*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.2078, size = 211, normalized size = 0.84 \[ \frac{729 a^{3} b^{2} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{70 \left (a + b x^{3}\right )} + \frac{243 a^{2} b^{2} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{70} + \frac{81 a b^{2} x \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{35} + \frac{3 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{2 x^{5}} + \frac{9 b^{2} x \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{5} - \frac{17 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

729*a**3*b**2*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(70*(a + b*x**3)) + 243*a**2
*b**2*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/70 + 81*a*b**2*x*(a + b*x**3)*sqrt(a
**2 + 2*a*b*x**3 + b**2*x**6)/35 + 3*a*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x*
*6)**(3/2)/(2*x**5) + 9*b**2*x*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/5 - 17*(a*
*2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(10*x**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0381785, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-14 a^5-175 a^4 b x^3+700 a^3 b^2 x^6+175 a^2 b^3 x^9+50 a b^4 x^{12}+7 b^5 x^{15}\right )}{70 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-14*a^5 - 175*a^4*b*x^3 + 700*a^3*b^2*x^6 + 175*a^2*b^3*x^
9 + 50*a*b^4*x^12 + 7*b^5*x^15))/(70*x^5*(a + b*x^3))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[ -{\frac{-7\,{b}^{5}{x}^{15}-50\,a{b}^{4}{x}^{12}-175\,{a}^{2}{b}^{3}{x}^{9}-700\,{a}^{3}{b}^{2}{x}^{6}+175\,{a}^{4}b{x}^{3}+14\,{a}^{5}}{70\,{x}^{5} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/70*(-7*b^5*x^15-50*a*b^4*x^12-175*a^2*b^3*x^9-700*a^3*b^2*x^6+175*a^4*b*x^3+1
4*a^5)*((b*x^3+a)^2)^(5/2)/x^5/(b*x^3+a)^5

_______________________________________________________________________________________

Maxima [A]  time = 0.761959, size = 80, normalized size = 0.32 \[ \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b
*x^3 - 14*a^5)/x^5

_______________________________________________________________________________________

Fricas [A]  time = 0.267945, size = 80, normalized size = 0.32 \[ \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b
*x^3 - 14*a^5)/x^5

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27047, size = 143, normalized size = 0.57 \[ \frac{1}{10} \, b^{5} x^{10}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{7} \, a b^{4} x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{2} \, a^{2} b^{3} x^{4}{\rm sign}\left (b x^{3} + a\right ) + 10 \, a^{3} b^{2} x{\rm sign}\left (b x^{3} + a\right ) - \frac{25 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^5*x^10*sign(b*x^3 + a) + 5/7*a*b^4*x^7*sign(b*x^3 + a) + 5/2*a^2*b^3*x^4*
sign(b*x^3 + a) + 10*a^3*b^2*x*sign(b*x^3 + a) - 1/10*(25*a^4*b*x^3*sign(b*x^3 +
a) + 2*a^5*sign(b*x^3 + a))/x^5