3.53 \(\int x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac{5 a b^4 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )} \]

[Out]

(a^5*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a^4*b*x^14*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a + b*x^3)) + (10*a^3*b^2*x^17*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(17*(a + b*x^3)) + (a^2*b^3*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])/(2*(a + b*x^3)) + (5*a*b^4*x^23*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*(a + b*
x^3)) + (b^5*x^26*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(26*(a + b*x^3))

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Rubi [A]  time = 0.170755, 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 \[ \frac{b^5 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left (a+b x^3\right )}+\frac{5 a b^4 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a^4*b*x^14*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a + b*x^3)) + (10*a^3*b^2*x^17*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(17*(a + b*x^3)) + (a^2*b^3*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])/(2*(a + b*x^3)) + (5*a*b^4*x^23*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(23*(a + b*
x^3)) + (b^5*x^26*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(26*(a + b*x^3))

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Rubi in Sympy [A]  time = 27.1124, size = 207, normalized size = 0.81 \[ \frac{729 a^{5} x^{11} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{782782 \left (a + b x^{3}\right )} + \frac{243 a^{4} x^{11} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{71162} + \frac{81 a^{3} x^{11} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{10166} + \frac{9 a^{2} x^{11} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{598} + \frac{15 a x^{11} \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{598} + \frac{x^{11} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{26} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

729*a**5*x**11*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(782782*(a + b*x**3)) + 243*a
**4*x**11*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/71162 + 81*a**3*x**11*(a + b*x**3)
*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/10166 + 9*a**2*x**11*(a**2 + 2*a*b*x**3 + b
**2*x**6)**(3/2)/598 + 15*a*x**11*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**
(3/2)/598 + x**11*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/26

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Mathematica [A]  time = 0.0370438, size = 83, normalized size = 0.33 \[ \frac{x^{11} \sqrt{\left (a+b x^3\right )^2} \left (71162 a^5+279565 a^4 b x^3+460460 a^3 b^2 x^6+391391 a^2 b^3 x^9+170170 a b^4 x^{12}+30107 b^5 x^{15}\right )}{782782 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^11*Sqrt[(a + b*x^3)^2]*(71162*a^5 + 279565*a^4*b*x^3 + 460460*a^3*b^2*x^6 + 3
91391*a^2*b^3*x^9 + 170170*a*b^4*x^12 + 30107*b^5*x^15))/(782782*(a + b*x^3))

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Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[{\frac{{x}^{11} \left ( 30107\,{b}^{5}{x}^{15}+170170\,a{b}^{4}{x}^{12}+391391\,{a}^{2}{b}^{3}{x}^{9}+460460\,{a}^{3}{b}^{2}{x}^{6}+279565\,{a}^{4}b{x}^{3}+71162\,{a}^{5} \right ) }{782782\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/782782*x^11*(30107*b^5*x^15+170170*a*b^4*x^12+391391*a^2*b^3*x^9+460460*a^3*b^
2*x^6+279565*a^4*b*x^3+71162*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 0.775507, size = 77, normalized size = 0.3 \[ \frac{1}{26} \, b^{5} x^{26} + \frac{5}{23} \, a b^{4} x^{23} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{10}{17} \, a^{3} b^{2} x^{17} + \frac{5}{14} \, a^{4} b x^{14} + \frac{1}{11} \, a^{5} x^{11} \]

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^10,x, algorithm="maxima")

[Out]

1/26*b^5*x^26 + 5/23*a*b^4*x^23 + 1/2*a^2*b^3*x^20 + 10/17*a^3*b^2*x^17 + 5/14*a
^4*b*x^14 + 1/11*a^5*x^11

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Fricas [A]  time = 0.266822, size = 77, normalized size = 0.3 \[ \frac{1}{26} \, b^{5} x^{26} + \frac{5}{23} \, a b^{4} x^{23} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{10}{17} \, a^{3} b^{2} x^{17} + \frac{5}{14} \, a^{4} b x^{14} + \frac{1}{11} \, a^{5} x^{11} \]

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^10,x, algorithm="fricas")

[Out]

1/26*b^5*x^26 + 5/23*a*b^4*x^23 + 1/2*a^2*b^3*x^20 + 10/17*a^3*b^2*x^17 + 5/14*a
^4*b*x^14 + 1/11*a^5*x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{10} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**10*((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.266949, size = 142, normalized size = 0.56 \[ \frac{1}{26} \, b^{5} x^{26}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{23} \, a b^{4} x^{23}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{2} b^{3} x^{20}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{17} \, a^{3} b^{2} x^{17}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{14} \, a^{4} b x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{11} \, a^{5} x^{11}{\rm sign}\left (b x^{3} + a\right ) \]

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^10,x, algorithm="giac")

[Out]

1/26*b^5*x^26*sign(b*x^3 + a) + 5/23*a*b^4*x^23*sign(b*x^3 + a) + 1/2*a^2*b^3*x^
20*sign(b*x^3 + a) + 10/17*a^3*b^2*x^17*sign(b*x^3 + a) + 5/14*a^4*b*x^14*sign(b
*x^3 + a) + 1/11*a^5*x^11*sign(b*x^3 + a)