3.85 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{22}} \, dx\)

Optimal. Leaf size=84 \[ \frac{b \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*a*x^21) + (b*(a + b*x^3)^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(126*a^2*x^18)

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Rubi [A]  time = 0.110063, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^22,x]

[Out]

-((a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*a*x^21) + (b*(a + b*x^3)^5*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(126*a^2*x^18)

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Rubi in Sympy [A]  time = 8.64034, size = 68, normalized size = 0.81 \[ - \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{36 a x^{21}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{7}{2}}}{126 a^{2} x^{21}} \]

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**22,x)

[Out]

-(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(36*a*x**21) + (a**2 +
2*a*b*x**3 + b**2*x**6)**(7/2)/(126*a**2*x**21)

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Mathematica [A]  time = 0.0322872, size = 83, normalized size = 0.99 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (6 a^5+35 a^4 b x^3+84 a^3 b^2 x^6+105 a^2 b^3 x^9+70 a b^4 x^{12}+21 b^5 x^{15}\right )}{126 x^{21} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^22,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(6*a^5 + 35*a^4*b*x^3 + 84*a^3*b^2*x^6 + 105*a^2*b^3*x^9 +
70*a*b^4*x^12 + 21*b^5*x^15))/(126*x^21*(a + b*x^3))

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Maple [A]  time = 0.012, size = 80, normalized size = 1. \[ -{\frac{21\,{b}^{5}{x}^{15}+70\,a{b}^{4}{x}^{12}+105\,{a}^{2}{b}^{3}{x}^{9}+84\,{a}^{3}{b}^{2}{x}^{6}+35\,{a}^{4}b{x}^{3}+6\,{a}^{5}}{126\,{x}^{21} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^22,x)

[Out]

-1/126*(21*b^5*x^15+70*a*b^4*x^12+105*a^2*b^3*x^9+84*a^3*b^2*x^6+35*a^4*b*x^3+6*
a^5)*((b*x^3+a)^2)^(5/2)/x^21/(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^22,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254495, size = 80, normalized size = 0.95 \[ -\frac{21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \]

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^22,x, algorithm="fricas")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*
b*x^3 + 6*a^5)/x^21

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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**22,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280166, size = 144, normalized size = 1.71 \[ -\frac{21 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 70 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 105 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 84 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 35 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 6 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{126 \, x^{21}} \]

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^22,x, algorithm="giac")

[Out]

-1/126*(21*b^5*x^15*sign(b*x^3 + a) + 70*a*b^4*x^12*sign(b*x^3 + a) + 105*a^2*b^
3*x^9*sign(b*x^3 + a) + 84*a^3*b^2*x^6*sign(b*x^3 + a) + 35*a^4*b*x^3*sign(b*x^3
+ a) + 6*a^5*sign(b*x^3 + a))/x^21