3.80 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{17}} \, dx\)

Optimal. Leaf size=251 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.15832, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.8369, size = 211, normalized size = 0.84 \[ \frac{729 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{1456 x^{4} \left (a + b x^{3}\right )} + \frac{81 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{728 x^{10}} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{208 x^{16}} - \frac{243 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{364 x^{4}} - \frac{18 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{91 x^{10}} - \frac{7 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{52 x^{16}} \]

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**17,x)

[Out]

729*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(1456*x**4*(a + b*x**3)) + 81*a*b
**2*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(728*x**10) + 15*a*(a + b*x
**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(208*x**16) - 243*b**4*sqrt(a**2 + 2
*a*b*x**3 + b**2*x**6)/(364*x**4) - 18*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/
2)/(91*x**10) - 7*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(52*x**16)

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Mathematica [A]  time = 0.0378834, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (91 a^5+560 a^4 b x^3+1456 a^3 b^2 x^6+2080 a^2 b^3 x^9+1820 a b^4 x^{12}+1456 b^5 x^{15}\right )}{1456 x^{16} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(91*a^5 + 560*a^4*b*x^3 + 1456*a^3*b^2*x^6 + 2080*a^2*b^3*
x^9 + 1820*a*b^4*x^12 + 1456*b^5*x^15))/(1456*x^16*(a + b*x^3))

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Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[ -{\frac{1456\,{b}^{5}{x}^{15}+1820\,a{b}^{4}{x}^{12}+2080\,{a}^{2}{b}^{3}{x}^{9}+1456\,{a}^{3}{b}^{2}{x}^{6}+560\,{a}^{4}b{x}^{3}+91\,{a}^{5}}{1456\,{x}^{16} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^17,x)

[Out]

-1/1456*(1456*b^5*x^15+1820*a*b^4*x^12+2080*a^2*b^3*x^9+1456*a^3*b^2*x^6+560*a^4
*b*x^3+91*a^5)*((b*x^3+a)^2)^(5/2)/x^16/(b*x^3+a)^5

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Maxima [A]  time = 0.803367, size = 80, normalized size = 0.32 \[ -\frac{1456 \, b^{5} x^{15} + 1820 \, a b^{4} x^{12} + 2080 \, a^{2} b^{3} x^{9} + 1456 \, a^{3} b^{2} x^{6} + 560 \, a^{4} b x^{3} + 91 \, a^{5}}{1456 \, x^{16}} \]

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^17,x, algorithm="maxima")

[Out]

-1/1456*(1456*b^5*x^15 + 1820*a*b^4*x^12 + 2080*a^2*b^3*x^9 + 1456*a^3*b^2*x^6 +
560*a^4*b*x^3 + 91*a^5)/x^16

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Fricas [A]  time = 0.269164, size = 80, normalized size = 0.32 \[ -\frac{1456 \, b^{5} x^{15} + 1820 \, a b^{4} x^{12} + 2080 \, a^{2} b^{3} x^{9} + 1456 \, a^{3} b^{2} x^{6} + 560 \, a^{4} b x^{3} + 91 \, a^{5}}{1456 \, x^{16}} \]

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^17,x, algorithm="fricas")

[Out]

-1/1456*(1456*b^5*x^15 + 1820*a*b^4*x^12 + 2080*a^2*b^3*x^9 + 1456*a^3*b^2*x^6 +
560*a^4*b*x^3 + 91*a^5)/x^16

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{17}}\, dx \]

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**17,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**17, x)

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GIAC/XCAS [A]  time = 0.274717, size = 144, normalized size = 0.57 \[ -\frac{1456 \, b^{5} x^{15}{\rm sign}\left (b x^{3} + a\right ) + 1820 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 2080 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 1456 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 560 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 91 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{1456 \, x^{16}} \]

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^17,x, algorithm="giac")

[Out]

-1/1456*(1456*b^5*x^15*sign(b*x^3 + a) + 1820*a*b^4*x^12*sign(b*x^3 + a) + 2080*
a^2*b^3*x^9*sign(b*x^3 + a) + 1456*a^3*b^2*x^6*sign(b*x^3 + a) + 560*a^4*b*x^3*s
ign(b*x^3 + a) + 91*a^5*sign(b*x^3 + a))/x^16