3.81 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{18}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.159328, antiderivative size = 253, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^18,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.8099, size = 211, normalized size = 0.83 \[ \frac{729 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5236 x^{5} \left (a + b x^{3}\right )} + \frac{405 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5236 x^{11}} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{238 x^{17}} - \frac{1215 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{5236 x^{5}} - \frac{45 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{308 x^{11}} - \frac{29 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{238 x^{17}} \]

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**18,x)

[Out]

729*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(5236*x**5*(a + b*x**3)) + 405*a*
b**2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(5236*x**11) + 15*a*(a + b
*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(238*x**17) - 1215*b**4*sqrt(a**2
+ 2*a*b*x**3 + b**2*x**6)/(5236*x**5) - 45*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)*
*(3/2)/(308*x**11) - 29*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(238*x**17)

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Mathematica [A]  time = 0.034577, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (308 a^5+1870 a^4 b x^3+4760 a^3 b^2 x^6+6545 a^2 b^3 x^9+5236 a b^4 x^{12}+2618 b^5 x^{15}\right )}{5236 x^{17} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^18,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(308*a^5 + 1870*a^4*b*x^3 + 4760*a^3*b^2*x^6 + 6545*a^2*b^
3*x^9 + 5236*a*b^4*x^12 + 2618*b^5*x^15))/(5236*x^17*(a + b*x^3))

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Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[ -{\frac{2618\,{b}^{5}{x}^{15}+5236\,a{b}^{4}{x}^{12}+6545\,{a}^{2}{b}^{3}{x}^{9}+4760\,{a}^{3}{b}^{2}{x}^{6}+1870\,{a}^{4}b{x}^{3}+308\,{a}^{5}}{5236\,{x}^{17} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^18,x)

[Out]

-1/5236*(2618*b^5*x^15+5236*a*b^4*x^12+6545*a^2*b^3*x^9+4760*a^3*b^2*x^6+1870*a^
4*b*x^3+308*a^5)*((b*x^3+a)^2)^(5/2)/x^17/(b*x^3+a)^5

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Maxima [A]  time = 0.79513, size = 80, normalized size = 0.32 \[ -\frac{2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \]

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^18,x, algorithm="maxima")

[Out]

-1/5236*(2618*b^5*x^15 + 5236*a*b^4*x^12 + 6545*a^2*b^3*x^9 + 4760*a^3*b^2*x^6 +
1870*a^4*b*x^3 + 308*a^5)/x^17

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Fricas [A]  time = 0.254866, size = 80, normalized size = 0.32 \[ -\frac{2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \]

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^18,x, algorithm="fricas")

[Out]

-1/5236*(2618*b^5*x^15 + 5236*a*b^4*x^12 + 6545*a^2*b^3*x^9 + 4760*a^3*b^2*x^6 +
1870*a^4*b*x^3 + 308*a^5)/x^17

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{18}}\, dx \]

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**18,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**18, x)

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GIAC/XCAS [A]  time = 0.274794, size = 144, normalized size = 0.57 \[ -\frac{2618 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 5236 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 6545 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 4760 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 1870 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 308 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{5236 \, x^{17}} \]

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^18,x, algorithm="giac")

[Out]

-1/5236*(2618*b^5*x^15*sign(b*x^3 + a) + 5236*a*b^4*x^12*sign(b*x^3 + a) + 6545*
a^2*b^3*x^9*sign(b*x^3 + a) + 4760*a^3*b^2*x^6*sign(b*x^3 + a) + 1870*a^4*b*x^3*
sign(b*x^3 + a) + 308*a^5*sign(b*x^3 + a))/x^17