3.72 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=247 \[ \frac{b^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (a^4*b*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(x^5*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x
^6])/(x^2*(a + b*x^3)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x
^3) + (5*a*b^4*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b^5*x^7*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3))

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Rubi [A]  time = 0.157919, 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 \[ \frac{b^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (a^4*b*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(x^5*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x
^6])/(x^2*(a + b*x^3)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x
^3) + (5*a*b^4*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b^5*x^7*S
qrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3))

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Rubi in Sympy [A]  time = 27.1662, size = 214, normalized size = 0.87 \[ - \frac{729 a^{3} b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{56 x^{2} \left (a + b x^{3}\right )} + \frac{243 a^{2} b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{28 x^{2}} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{56 x^{2}} + \frac{3 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{8 x^{8}} + \frac{9 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{14 x^{2}} - \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{2 x^{8}} \]

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**9,x)

[Out]

-729*a**3*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(56*x**2*(a + b*x**3)) + 243*
a**2*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(28*x**2) + 81*a*b**2*(a + b*x**3)
*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(56*x**2) + 3*a*(a + b*x**3)*(a**2 + 2*a*b*
x**3 + b**2*x**6)**(3/2)/(8*x**8) + 9*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2
)/(14*x**2) - (a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(2*x**8)

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Mathematica [A]  time = 0.0366966, size = 83, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-7 a^5-56 a^4 b x^3-280 a^3 b^2 x^6+560 a^2 b^3 x^9+70 a b^4 x^{12}+8 b^5 x^{15}\right )}{56 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-7*a^5 - 56*a^4*b*x^3 - 280*a^3*b^2*x^6 + 560*a^2*b^3*x^9
+ 70*a*b^4*x^12 + 8*b^5*x^15))/(56*x^8*(a + b*x^3))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[ -{\frac{-8\,{b}^{5}{x}^{15}-70\,a{b}^{4}{x}^{12}-560\,{a}^{2}{b}^{3}{x}^{9}+280\,{a}^{3}{b}^{2}{x}^{6}+56\,{a}^{4}b{x}^{3}+7\,{a}^{5}}{56\,{x}^{8} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^9,x)

[Out]

-1/56*(-8*b^5*x^15-70*a*b^4*x^12-560*a^2*b^3*x^9+280*a^3*b^2*x^6+56*a^4*b*x^3+7*
a^5)*((b*x^3+a)^2)^(5/2)/x^8/(b*x^3+a)^5

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Maxima [A]  time = 0.793914, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \]

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^9,x, algorithm="maxima")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*
x^3 - 7*a^5)/x^8

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Fricas [A]  time = 0.261778, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \]

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^9,x, algorithm="fricas")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*
x^3 - 7*a^5)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{9}}\, dx \]

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**9,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.257658, size = 142, normalized size = 0.57 \[ \frac{1}{7} \, b^{5} x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{4} \, a b^{4} x^{4}{\rm sign}\left (b x^{3} + a\right ) + 10 \, a^{2} b^{3} x{\rm sign}\left (b x^{3} + a\right ) - \frac{40 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 8 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm sign}\left (b x^{3} + a\right )}{8 \, x^{8}} \]

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^9,x, algorithm="giac")

[Out]

1/7*b^5*x^7*sign(b*x^3 + a) + 5/4*a*b^4*x^4*sign(b*x^3 + a) + 10*a^2*b^3*x*sign(
b*x^3 + a) - 1/8*(40*a^3*b^2*x^6*sign(b*x^3 + a) + 8*a^4*b*x^3*sign(b*x^3 + a) +
a^5*sign(b*x^3 + a))/x^8