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

Optimal. Leaf size=251 \[ \frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

[Out]

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

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Rubi [A]  time = 0.157973, antiderivative size = 251, 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 \[ \frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.7104, size = 196, normalized size = 0.78 \[ - \frac{729 a^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{308 x \left (a + b x^{3}\right )} + \frac{243 a^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{308 x} + \frac{81 a^{3} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{308 x} + \frac{45 a^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{308 x} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{154 x} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{14 x} \]

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**2,x)

[Out]

-729*a**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(308*x*(a + b*x**3)) + 243*a**4*sq
rt(a**2 + 2*a*b*x**3 + b**2*x**6)/(308*x) + 81*a**3*(a + b*x**3)*sqrt(a**2 + 2*a
*b*x**3 + b**2*x**6)/(308*x) + 45*a**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(3
08*x) + 15*a*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(154*x) + (a**2
+ 2*a*b*x**3 + b**2*x**6)**(5/2)/(14*x)

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Mathematica [A]  time = 0.0381417, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-308 a^5+770 a^4 b x^3+616 a^3 b^2 x^6+385 a^2 b^3 x^9+140 a b^4 x^{12}+22 b^5 x^{15}\right )}{308 x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-308*a^5 + 770*a^4*b*x^3 + 616*a^3*b^2*x^6 + 385*a^2*b^3*x
^9 + 140*a*b^4*x^12 + 22*b^5*x^15))/(308*x*(a + b*x^3))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[ -{\frac{-22\,{b}^{5}{x}^{15}-140\,a{b}^{4}{x}^{12}-385\,{a}^{2}{b}^{3}{x}^{9}-616\,{a}^{3}{b}^{2}{x}^{6}-770\,{a}^{4}b{x}^{3}+308\,{a}^{5}}{308\,x \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^2,x)

[Out]

-1/308*(-22*b^5*x^15-140*a*b^4*x^12-385*a^2*b^3*x^9-616*a^3*b^2*x^6-770*a^4*b*x^
3+308*a^5)*((b*x^3+a)^2)^(5/2)/x/(b*x^3+a)^5

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Maxima [A]  time = 0.788433, size = 80, normalized size = 0.32 \[ \frac{22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \]

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^2,x, algorithm="maxima")

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^
4*b*x^3 - 308*a^5)/x

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Fricas [A]  time = 0.263446, size = 80, normalized size = 0.32 \[ \frac{22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^
4*b*x^3 - 308*a^5)/x

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27993, size = 142, normalized size = 0.57 \[ \frac{1}{14} \, b^{5} x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{11} \, a b^{4} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{2} b^{3} x^{8}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{3} b^{2} x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{2} \, a^{4} b x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{a^{5}{\rm sign}\left (b x^{3} + a\right )}{x} \]

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^2,x, algorithm="giac")

[Out]

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