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

Optimal. Leaf size=255 \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )} \]

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (a^4*b*x^10*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (10*a^3*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(13*(a + b*x^3)) + (5*a^2*b^3*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(8*(a + b*x^3)) + (5*a*b^4*x^19*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(a + b*x^3
)) + (b^5*x^22*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*(a + b*x^3))

_______________________________________________________________________________________

Rubi [A]  time = 0.165741, antiderivative size = 255, 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 \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (a^4*b*x^10*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (10*a^3*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(13*(a + b*x^3)) + (5*a^2*b^3*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(8*(a + b*x^3)) + (5*a*b^4*x^19*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(a + b*x^3
)) + (b^5*x^22*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*(a + b*x^3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.1131, size = 207, normalized size = 0.81 \[ \frac{729 a^{5} x^{7} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{152152 \left (a + b x^{3}\right )} + \frac{243 a^{4} x^{7} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{21736} + \frac{405 a^{3} x^{7} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{21736} + \frac{45 a^{2} x^{7} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{1672} + \frac{15 a x^{7} \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{418} + \frac{x^{7} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

729*a**5*x**7*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(152152*(a + b*x**3)) + 243*a*
*4*x**7*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/21736 + 405*a**3*x**7*(a + b*x**3)*s
qrt(a**2 + 2*a*b*x**3 + b**2*x**6)/21736 + 45*a**2*x**7*(a**2 + 2*a*b*x**3 + b**
2*x**6)**(3/2)/1672 + 15*a*x**7*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3
/2)/418 + x**7*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/22

_______________________________________________________________________________________

Mathematica [A]  time = 0.0360826, size = 83, normalized size = 0.33 \[ \frac{x^7 \sqrt{\left (a+b x^3\right )^2} \left (21736 a^5+76076 a^4 b x^3+117040 a^3 b^2 x^6+95095 a^2 b^3 x^9+40040 a b^4 x^{12}+6916 b^5 x^{15}\right )}{152152 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^7*Sqrt[(a + b*x^3)^2]*(21736*a^5 + 76076*a^4*b*x^3 + 117040*a^3*b^2*x^6 + 950
95*a^2*b^3*x^9 + 40040*a*b^4*x^12 + 6916*b^5*x^15))/(152152*(a + b*x^3))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[{\frac{{x}^{7} \left ( 6916\,{b}^{5}{x}^{15}+40040\,a{b}^{4}{x}^{12}+95095\,{a}^{2}{b}^{3}{x}^{9}+117040\,{a}^{3}{b}^{2}{x}^{6}+76076\,{a}^{4}b{x}^{3}+21736\,{a}^{5} \right ) }{152152\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/152152*x^7*(6916*b^5*x^15+40040*a*b^4*x^12+95095*a^2*b^3*x^9+117040*a^3*b^2*x^
6+76076*a^4*b*x^3+21736*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

_______________________________________________________________________________________

Maxima [A]  time = 0.802327, size = 77, normalized size = 0.3 \[ \frac{1}{22} \, b^{5} x^{22} + \frac{5}{19} \, a b^{4} x^{19} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{7} \, a^{5} x^{7} \]

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]

1/22*b^5*x^22 + 5/19*a*b^4*x^19 + 5/8*a^2*b^3*x^16 + 10/13*a^3*b^2*x^13 + 1/2*a^
4*b*x^10 + 1/7*a^5*x^7

_______________________________________________________________________________________

Fricas [A]  time = 0.250694, size = 77, normalized size = 0.3 \[ \frac{1}{22} \, b^{5} x^{22} + \frac{5}{19} \, a b^{4} x^{19} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{7} \, a^{5} x^{7} \]

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]

1/22*b^5*x^22 + 5/19*a*b^4*x^19 + 5/8*a^2*b^3*x^16 + 10/13*a^3*b^2*x^13 + 1/2*a^
4*b*x^10 + 1/7*a^5*x^7

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300202, size = 142, normalized size = 0.56 \[ \frac{1}{22} \, b^{5} x^{22}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{19} \, a b^{4} x^{19}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{8} \, a^{2} b^{3} x^{16}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{13} \, a^{3} b^{2} x^{13}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{4} b x^{10}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{7} \, a^{5} x^{7}{\rm sign}\left (b x^{3} + a\right ) \]

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]

1/22*b^5*x^22*sign(b*x^3 + a) + 5/19*a*b^4*x^19*sign(b*x^3 + a) + 5/8*a^2*b^3*x^
16*sign(b*x^3 + a) + 10/13*a^3*b^2*x^13*sign(b*x^3 + a) + 1/2*a^4*b*x^10*sign(b*
x^3 + a) + 1/7*a^5*x^7*sign(b*x^3 + a)