3.571 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{17}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2)) - (a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(4*x^12*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*x^10*(
a + b*x^2))

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Rubi [A]  time = 0.254031, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^17,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2)) - (a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(4*x^12*(a + b*x^2)) - (b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*x^10*(
a + b*x^2))

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Rubi in Sympy [A]  time = 16.9319, size = 136, normalized size = 0.81 \[ \frac{a b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{560 x^{12} \left (a + b x^{2}\right )} + \frac{3 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{112 x^{16}} - \frac{3 b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{280 x^{12}} - \frac{5 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{56 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**17,x)

[Out]

a*b**2*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(560*x**12*(a + b*x**2)) + 3*a*(a + b
*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(112*x**16) - 3*b**2*sqrt(a**2 + 2*a*
b*x**2 + b**2*x**4)/(280*x**12) - 5*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(56*x
**16)

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Mathematica [A]  time = 0.0237277, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (35 a^3+120 a^2 b x^2+140 a b^2 x^4+56 b^3 x^6\right )}{560 x^{16} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^17,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(35*a^3 + 120*a^2*b*x^2 + 140*a*b^2*x^4 + 56*b^3*x^6))/(56
0*x^16*(a + b*x^2))

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Maple [A]  time = 0.011, size = 58, normalized size = 0.4 \[ -{\frac{56\,{b}^{3}{x}^{6}+140\,a{x}^{4}{b}^{2}+120\,{a}^{2}b{x}^{2}+35\,{a}^{3}}{560\,{x}^{16} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^17,x)

[Out]

-1/560*(56*b^3*x^6+140*a*b^2*x^4+120*a^2*b*x^2+35*a^3)*((b*x^2+a)^2)^(3/2)/x^16/
(b*x^2+a)^3

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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^4 + 2*a*b*x^2 + a^2)^(3/2)/x^17,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258006, size = 50, normalized size = 0.3 \[ -\frac{56 \, b^{3} x^{6} + 140 \, a b^{2} x^{4} + 120 \, a^{2} b x^{2} + 35 \, a^{3}}{560 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^17,x, algorithm="fricas")

[Out]

-1/560*(56*b^3*x^6 + 140*a*b^2*x^4 + 120*a^2*b*x^2 + 35*a^3)/x^16

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**17,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270698, size = 93, normalized size = 0.56 \[ -\frac{56 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 140 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 120 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 35 \, a^{3}{\rm sign}\left (b x^{2} + a\right )}{560 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^17,x, algorithm="giac")

[Out]

-1/560*(56*b^3*x^6*sign(b*x^2 + a) + 140*a*b^2*x^4*sign(b*x^2 + a) + 120*a^2*b*x
^2*sign(b*x^2 + a) + 35*a^3*sign(b*x^2 + a))/x^16