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

Optimal. Leaf size=163 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (3*a^2*b*x*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (3*a*b^2*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x
^6])/(4*(a + b*x^3)) + (b^3*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3))

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Rubi [A]  time = 0.112662, antiderivative size = 163, 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 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (3*a^2*b*x*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (3*a*b^2*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x
^6])/(4*(a + b*x^3)) + (b^3*x^7*Sqrt[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 = 16.8656, size = 134, normalized size = 0.82 \[ - \frac{81 a^{3} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{28 x^{2} \left (a + b x^{3}\right )} + \frac{27 a^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{14 x^{2}} + \frac{9 a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{28 x^{2}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{7 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81*a**3*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(28*x**2*(a + b*x**3)) + 27*a**2*sq
rt(a**2 + 2*a*b*x**3 + b**2*x**6)/(14*x**2) + 9*a*(a + b*x**3)*sqrt(a**2 + 2*a*b
*x**3 + b**2*x**6)/(28*x**2) + (a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(7*x**2)

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Mathematica [A]  time = 0.0321532, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-14 a^3+84 a^2 b x^3+21 a b^2 x^6+4 b^3 x^9\right )}{28 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-14*a^3 + 84*a^2*b*x^3 + 21*a*b^2*x^6 + 4*b^3*x^9))/(28*x^
2*(a + b*x^3))

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Maple [A]  time = 0.009, size = 58, normalized size = 0.4 \[ -{\frac{-4\,{b}^{3}{x}^{9}-21\,a{x}^{6}{b}^{2}-84\,{x}^{3}{a}^{2}b+14\,{a}^{3}}{28\,{x}^{2} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/28*(-4*b^3*x^9-21*a*b^2*x^6-84*a^2*b*x^3+14*a^3)*((b*x^3+a)^2)^(3/2)/x^2/(b*x
^3+a)^3

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Maxima [A]  time = 0.784053, size = 50, normalized size = 0.31 \[ \frac{4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^3,x, algorithm="maxima")

[Out]

1/28*(4*b^3*x^9 + 21*a*b^2*x^6 + 84*a^2*b*x^3 - 14*a^3)/x^2

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Fricas [A]  time = 0.254784, size = 50, normalized size = 0.31 \[ \frac{4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^3,x, algorithm="fricas")

[Out]

1/28*(4*b^3*x^9 + 21*a*b^2*x^6 + 84*a^2*b*x^3 - 14*a^3)/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.294515, size = 88, normalized size = 0.54 \[ \frac{1}{7} \, b^{3} x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{4} \, a b^{2} x^{4}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a^{2} b x{\rm sign}\left (b x^{3} + a\right ) - \frac{a^{3}{\rm sign}\left (b x^{3} + a\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^3,x, algorithm="giac")

[Out]

1/7*b^3*x^7*sign(b*x^3 + a) + 3/4*a*b^2*x^4*sign(b*x^3 + a) + 3*a^2*b*x*sign(b*x
^3 + a) - 1/2*a^3*sign(b*x^3 + a)/x^2