3.59 \(\int x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a^4 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]

[Out]

(a^5*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (5*a^4*b*x^8*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (10*a^3*b^2*x^11*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a^2*b^3*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
)/(7*(a + b*x^3)) + (5*a*b^4*x^17*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^
3)) + (b^5*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(20*(a + b*x^3))

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Rubi [A]  time = 0.163892, 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 \[ \frac{b^5 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a^4 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (5*a^4*b*x^8*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (10*a^3*b^2*x^11*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(11*(a + b*x^3)) + (5*a^2*b^3*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
)/(7*(a + b*x^3)) + (5*a*b^4*x^17*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(17*(a + b*x^
3)) + (b^5*x^20*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(20*(a + b*x^3))

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Rubi in Sympy [A]  time = 27.0897, size = 207, normalized size = 0.81 \[ \frac{729 a^{5} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{52360 \left (a + b x^{3}\right )} + \frac{243 a^{4} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{10472} + \frac{81 a^{3} x^{5} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{2618} + \frac{9 a^{2} x^{5} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{238} + \frac{3 a x^{5} \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{68} + \frac{x^{5} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

729*a**5*x**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(52360*(a + b*x**3)) + 243*a**
4*x**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/10472 + 81*a**3*x**5*(a + b*x**3)*sqr
t(a**2 + 2*a*b*x**3 + b**2*x**6)/2618 + 9*a**2*x**5*(a**2 + 2*a*b*x**3 + b**2*x*
*6)**(3/2)/238 + 3*a*x**5*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/68
+ x**5*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/20

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Mathematica [A]  time = 0.0369222, size = 83, normalized size = 0.33 \[ \frac{x^5 \sqrt{\left (a+b x^3\right )^2} \left (10472 a^5+32725 a^4 b x^3+47600 a^3 b^2 x^6+37400 a^2 b^3 x^9+15400 a b^4 x^{12}+2618 b^5 x^{15}\right )}{52360 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^5*Sqrt[(a + b*x^3)^2]*(10472*a^5 + 32725*a^4*b*x^3 + 47600*a^3*b^2*x^6 + 3740
0*a^2*b^3*x^9 + 15400*a*b^4*x^12 + 2618*b^5*x^15))/(52360*(a + b*x^3))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[{\frac{{x}^{5} \left ( 2618\,{b}^{5}{x}^{15}+15400\,a{b}^{4}{x}^{12}+37400\,{a}^{2}{b}^{3}{x}^{9}+47600\,{a}^{3}{b}^{2}{x}^{6}+32725\,{a}^{4}b{x}^{3}+10472\,{a}^{5} \right ) }{52360\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/52360*x^5*(2618*b^5*x^15+15400*a*b^4*x^12+37400*a^2*b^3*x^9+47600*a^3*b^2*x^6+
32725*a^4*b*x^3+10472*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 0.822102, size = 77, normalized size = 0.3 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \]

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^4,x, algorithm="maxima")

[Out]

1/20*b^5*x^20 + 5/17*a*b^4*x^17 + 5/7*a^2*b^3*x^14 + 10/11*a^3*b^2*x^11 + 5/8*a^
4*b*x^8 + 1/5*a^5*x^5

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Fricas [A]  time = 0.25404, size = 77, normalized size = 0.3 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \]

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^4,x, algorithm="fricas")

[Out]

1/20*b^5*x^20 + 5/17*a*b^4*x^17 + 5/7*a^2*b^3*x^14 + 10/11*a^3*b^2*x^11 + 5/8*a^
4*b*x^8 + 1/5*a^5*x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**4*((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.265001, size = 142, normalized size = 0.56 \[ \frac{1}{20} \, b^{5} x^{20}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{17} \, a b^{4} x^{17}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{7} \, a^{2} b^{3} x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{11} \, a^{3} b^{2} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{8} \, a^{4} b x^{8}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{5} \, a^{5} x^{5}{\rm sign}\left (b x^{3} + a\right ) \]

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^4,x, algorithm="giac")

[Out]

1/20*b^5*x^20*sign(b*x^3 + a) + 5/17*a*b^4*x^17*sign(b*x^3 + a) + 5/7*a^2*b^3*x^
14*sign(b*x^3 + a) + 10/11*a^3*b^2*x^11*sign(b*x^3 + a) + 5/8*a^4*b*x^8*sign(b*x
^3 + a) + 1/5*a^5*x^5*sign(b*x^3 + a)