3.88 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{25}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}}+\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac{b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}} \]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(24*a*x^24) + (b*(a + b*x^3)^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(84*a^2*x^21) - (b^2*(a + b*x^3)^5*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(504*a^3*x^18)

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Rubi [A]  time = 0.145238, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{24 a x^{24}}+\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{84 a^2 x^{21}}-\frac{b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{504 a^3 x^{18}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25,x]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(24*a*x^24) + (b*(a + b*x^3)^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(84*a^2*x^21) - (b^2*(a + b*x^3)^5*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(504*a^3*x^18)

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Rubi in Sympy [A]  time = 15.6485, size = 112, normalized size = 0.88 \[ - \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{48 a x^{24}} + \frac{b \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{144 a^{2} x^{21}} - \frac{b \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{7}{2}}}{504 a^{3} x^{21}} \]

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**25,x)

[Out]

-(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(48*a*x**24) + b*(2*a +
2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(144*a**2*x**21) - b*(a**2 + 2
*a*b*x**3 + b**2*x**6)**(7/2)/(504*a**3*x**21)

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Mathematica [A]  time = 0.0350055, size = 83, normalized size = 0.65 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (21 a^5+120 a^4 b x^3+280 a^3 b^2 x^6+336 a^2 b^3 x^9+210 a b^4 x^{12}+56 b^5 x^{15}\right )}{504 x^{24} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(21*a^5 + 120*a^4*b*x^3 + 280*a^3*b^2*x^6 + 336*a^2*b^3*x^
9 + 210*a*b^4*x^12 + 56*b^5*x^15))/(504*x^24*(a + b*x^3))

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Maple [A]  time = 0.015, size = 80, normalized size = 0.6 \[ -{\frac{56\,{b}^{5}{x}^{15}+210\,a{b}^{4}{x}^{12}+336\,{a}^{2}{b}^{3}{x}^{9}+280\,{a}^{3}{b}^{2}{x}^{6}+120\,{a}^{4}b{x}^{3}+21\,{a}^{5}}{504\,{x}^{24} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^25,x)

[Out]

-1/504*(56*b^5*x^15+210*a*b^4*x^12+336*a^2*b^3*x^9+280*a^3*b^2*x^6+120*a^4*b*x^3
+21*a^5)*((b*x^3+a)^2)^(5/2)/x^24/(b*x^3+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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^25,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251845, size = 80, normalized size = 0.62 \[ -\frac{56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

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^25,x, algorithm="fricas")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a
^4*b*x^3 + 21*a^5)/x^24

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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**25,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.260684, size = 144, normalized size = 1.12 \[ -\frac{56 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 210 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 336 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 280 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 120 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 21 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{504 \, x^{24}} \]

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^25,x, algorithm="giac")

[Out]

-1/504*(56*b^5*x^15*sign(b*x^3 + a) + 210*a*b^4*x^12*sign(b*x^3 + a) + 336*a^2*b
^3*x^9*sign(b*x^3 + a) + 280*a^3*b^2*x^6*sign(b*x^3 + a) + 120*a^4*b*x^3*sign(b*
x^3 + a) + 21*a^5*sign(b*x^3 + a))/x^24