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

Optimal. Leaf size=252 \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.148092, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4538, size = 207, normalized size = 0.82 \[ \frac{729 a^{5} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5236 \left (a + b x^{3}\right )} + \frac{243 a^{4} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{2618} + \frac{405 a^{3} x^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5236} + \frac{90 a^{2} x^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{1309} + \frac{15 a x^{2} \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{238} + \frac{x^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

729*a**5*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(5236*(a + b*x**3)) + 243*a**4
*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/2618 + 405*a**3*x**2*(a + b*x**3)*sqrt
(a**2 + 2*a*b*x**3 + b**2*x**6)/5236 + 90*a**2*x**2*(a**2 + 2*a*b*x**3 + b**2*x*
*6)**(3/2)/1309 + 15*a*x**2*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/
238 + x**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/17

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Mathematica [A]  time = 0.0352532, size = 83, normalized size = 0.33 \[ \frac{x^2 \sqrt{\left (a+b x^3\right )^2} \left (2618 a^5+5236 a^4 b x^3+6545 a^3 b^2 x^6+4760 a^2 b^3 x^9+1870 a b^4 x^{12}+308 b^5 x^{15}\right )}{5236 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x^3)^2]*(2618*a^5 + 5236*a^4*b*x^3 + 6545*a^3*b^2*x^6 + 4760*a^
2*b^3*x^9 + 1870*a*b^4*x^12 + 308*b^5*x^15))/(5236*(a + b*x^3))

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Maple [A]  time = 0.008, size = 80, normalized size = 0.3 \[{\frac{{x}^{2} \left ( 308\,{b}^{5}{x}^{15}+1870\,a{b}^{4}{x}^{12}+4760\,{a}^{2}{b}^{3}{x}^{9}+6545\,{a}^{3}{b}^{2}{x}^{6}+5236\,{a}^{4}b{x}^{3}+2618\,{a}^{5} \right ) }{5236\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5236*x^2*(308*b^5*x^15+1870*a*b^4*x^12+4760*a^2*b^3*x^9+6545*a^3*b^2*x^6+5236*
a^4*b*x^3+2618*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 0.807452, size = 76, normalized size = 0.3 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.263643, size = 76, normalized size = 0.3 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{5}{14} \, a b^{4} x^{14} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac{1}{2} \, a^{5} x^{2} \]

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,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277485, size = 140, normalized size = 0.56 \[ \frac{1}{17} \, b^{5} x^{17}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{14} \, a b^{4} x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{11} \, a^{2} b^{3} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{3} b^{2} x^{8}{\rm sign}\left (b x^{3} + a\right ) + a^{4} b x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{5} x^{2}{\rm sign}\left (b x^{3} + a\right ) \]

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,x, algorithm="giac")

[Out]

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