3.86 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{23}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*x^22*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(19*x^19*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(8*x^16*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(13*x^13*(a + b*x^3)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^10*(a + b*
x^3)) - (b^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3))

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Rubi [A]  time = 0.161161, antiderivative size = 255, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^23,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*x^22*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(19*x^19*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(8*x^16*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
/(13*x^13*(a + b*x^3)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^10*(a + b*
x^3)) - (b^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3))

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Rubi in Sympy [A]  time = 26.8141, size = 211, normalized size = 0.83 \[ \frac{729 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{152152 x^{10} \left (a + b x^{3}\right )} + \frac{405 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{21736 x^{16}} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{418 x^{22}} - \frac{1215 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{76076 x^{10}} - \frac{45 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{988 x^{16}} - \frac{17 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{209 x^{22}} \]

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**23,x)

[Out]

729*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(152152*x**10*(a + b*x**3)) + 405
*a*b**2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(21736*x**16) + 15*a*(a
+ b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(418*x**22) - 1215*b**4*sqrt(a
**2 + 2*a*b*x**3 + b**2*x**6)/(76076*x**10) - 45*b**2*(a**2 + 2*a*b*x**3 + b**2*
x**6)**(3/2)/(988*x**16) - 17*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(209*x**22)

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Mathematica [A]  time = 0.0312652, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (6916 a^5+40040 a^4 b x^3+95095 a^3 b^2 x^6+117040 a^2 b^3 x^9+76076 a b^4 x^{12}+21736 b^5 x^{15}\right )}{152152 x^{22} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^23,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(6916*a^5 + 40040*a^4*b*x^3 + 95095*a^3*b^2*x^6 + 117040*a
^2*b^3*x^9 + 76076*a*b^4*x^12 + 21736*b^5*x^15))/(152152*x^22*(a + b*x^3))

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Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[ -{\frac{21736\,{b}^{5}{x}^{15}+76076\,a{b}^{4}{x}^{12}+117040\,{a}^{2}{b}^{3}{x}^{9}+95095\,{a}^{3}{b}^{2}{x}^{6}+40040\,{a}^{4}b{x}^{3}+6916\,{a}^{5}}{152152\,{x}^{22} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^23,x)

[Out]

-1/152152*(21736*b^5*x^15+76076*a*b^4*x^12+117040*a^2*b^3*x^9+95095*a^3*b^2*x^6+
40040*a^4*b*x^3+6916*a^5)*((b*x^3+a)^2)^(5/2)/x^22/(b*x^3+a)^5

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Maxima [A]  time = 0.785126, size = 80, normalized size = 0.31 \[ -\frac{21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \]

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^23,x, algorithm="maxima")

[Out]

-1/152152*(21736*b^5*x^15 + 76076*a*b^4*x^12 + 117040*a^2*b^3*x^9 + 95095*a^3*b^
2*x^6 + 40040*a^4*b*x^3 + 6916*a^5)/x^22

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Fricas [A]  time = 0.252037, size = 80, normalized size = 0.31 \[ -\frac{21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \]

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^23,x, algorithm="fricas")

[Out]

-1/152152*(21736*b^5*x^15 + 76076*a*b^4*x^12 + 117040*a^2*b^3*x^9 + 95095*a^3*b^
2*x^6 + 40040*a^4*b*x^3 + 6916*a^5)/x^22

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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**23,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280151, size = 144, normalized size = 0.56 \[ -\frac{21736 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 76076 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 117040 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 95095 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 40040 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 6916 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{152152 \, x^{22}} \]

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^23,x, algorithm="giac")

[Out]

-1/152152*(21736*b^5*x^15*sign(b*x^3 + a) + 76076*a*b^4*x^12*sign(b*x^3 + a) + 1
17040*a^2*b^3*x^9*sign(b*x^3 + a) + 95095*a^3*b^2*x^6*sign(b*x^3 + a) + 40040*a^
4*b*x^3*sign(b*x^3 + a) + 6916*a^5*sign(b*x^3 + a))/x^22