3.75 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=247 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.157147, antiderivative size = 247, 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 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^12,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.3905, size = 209, normalized size = 0.85 \[ \frac{729 a b^{4} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88 \left (a + b x^{3}\right )} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{44 x^{5}} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{88 x^{11}} + \frac{243 b^{4} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88} - \frac{9 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{4 x^{5}} - \frac{23 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{88 x^{11}} \]

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**12,x)

[Out]

729*a*b**4*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(88*(a + b*x**3)) + 81*a*b**2*(
a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(44*x**5) + 15*a*(a + b*x**3)*(a
**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(88*x**11) + 243*b**4*x*sqrt(a**2 + 2*a*b*x
**3 + b**2*x**6)/88 - 9*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(4*x**5) - 2
3*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(88*x**11)

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Mathematica [A]  time = 0.0304854, size = 83, normalized size = 0.34 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (8 a^5+55 a^4 b x^3+176 a^3 b^2 x^6+440 a^2 b^3 x^9-440 a b^4 x^{12}-22 b^5 x^{15}\right )}{88 x^{11} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(8*a^5 + 55*a^4*b*x^3 + 176*a^3*b^2*x^6 + 440*a^2*b^3*x^9
- 440*a*b^4*x^12 - 22*b^5*x^15))/(88*x^11*(a + b*x^3))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-22\,{b}^{5}{x}^{15}-440\,a{b}^{4}{x}^{12}+440\,{a}^{2}{b}^{3}{x}^{9}+176\,{a}^{3}{b}^{2}{x}^{6}+55\,{a}^{4}b{x}^{3}+8\,{a}^{5}}{88\,{x}^{11} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^12,x)

[Out]

-1/88*(-22*b^5*x^15-440*a*b^4*x^12+440*a^2*b^3*x^9+176*a^3*b^2*x^6+55*a^4*b*x^3+
8*a^5)*((b*x^3+a)^2)^(5/2)/x^11/(b*x^3+a)^5

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Maxima [A]  time = 0.796162, size = 80, normalized size = 0.32 \[ \frac{22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \]

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^12,x, algorithm="maxima")

[Out]

1/88*(22*b^5*x^15 + 440*a*b^4*x^12 - 440*a^2*b^3*x^9 - 176*a^3*b^2*x^6 - 55*a^4*
b*x^3 - 8*a^5)/x^11

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Fricas [A]  time = 0.286694, size = 80, normalized size = 0.32 \[ \frac{22 \, b^{5} x^{15} + 440 \, a b^{4} x^{12} - 440 \, a^{2} b^{3} x^{9} - 176 \, a^{3} b^{2} x^{6} - 55 \, a^{4} b x^{3} - 8 \, a^{5}}{88 \, x^{11}} \]

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^12,x, algorithm="fricas")

[Out]

1/88*(22*b^5*x^15 + 440*a*b^4*x^12 - 440*a^2*b^3*x^9 - 176*a^3*b^2*x^6 - 55*a^4*
b*x^3 - 8*a^5)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.296663, size = 143, normalized size = 0.58 \[ \frac{1}{4} \, b^{5} x^{4}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a b^{4} x{\rm sign}\left (b x^{3} + a\right ) - \frac{440 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 176 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 55 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 8 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{88 \, x^{11}} \]

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^12,x, algorithm="giac")

[Out]

1/4*b^5*x^4*sign(b*x^3 + a) + 5*a*b^4*x*sign(b*x^3 + a) - 1/88*(440*a^2*b^3*x^9*
sign(b*x^3 + a) + 176*a^3*b^2*x^6*sign(b*x^3 + a) + 55*a^4*b*x^3*sign(b*x^3 + a)
+ 8*a^5*sign(b*x^3 + a))/x^11