3.600 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{17}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{16 a x^{16}}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{56 a^2 x^{14}}-\frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{336 a^3 x^{12}} \]

[Out]

-((a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*a*x^16) + (b*(a + b*x^2)^5*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(56*a^2*x^14) - (b^2*(a + b*x^2)^5*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(336*a^3*x^12)

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Rubi [A]  time = 0.228735, 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 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{16 a x^{16}}+\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{56 a^2 x^{14}}-\frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{336 a^3 x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^17,x]

[Out]

-((a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*a*x^16) + (b*(a + b*x^2)^5*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(56*a^2*x^14) - (b^2*(a + b*x^2)^5*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(336*a^3*x^12)

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Rubi in Sympy [A]  time = 15.2611, size = 112, normalized size = 0.88 \[ - \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{32 a x^{16}} + \frac{b \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{96 a^{2} x^{14}} - \frac{b \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{336 a^{3} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**17,x)

[Out]

-(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(32*a*x**16) + b*(2*a +
2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(96*a**2*x**14) - b*(a**2 + 2*
a*b*x**2 + b**2*x**4)**(7/2)/(336*a**3*x**14)

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Mathematica [A]  time = 0.0365187, size = 83, normalized size = 0.65 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (21 a^5+120 a^4 b x^2+280 a^3 b^2 x^4+336 a^2 b^3 x^6+210 a b^4 x^8+56 b^5 x^{10}\right )}{336 x^{16} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^17,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(21*a^5 + 120*a^4*b*x^2 + 280*a^3*b^2*x^4 + 336*a^2*b^3*x^
6 + 210*a*b^4*x^8 + 56*b^5*x^10))/(336*x^16*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.6 \[ -{\frac{56\,{b}^{5}{x}^{10}+210\,a{b}^{4}{x}^{8}+336\,{a}^{2}{b}^{3}{x}^{6}+280\,{a}^{3}{b}^{2}{x}^{4}+120\,{a}^{4}b{x}^{2}+21\,{a}^{5}}{336\,{x}^{16} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^17,x)

[Out]

-1/336*(56*b^5*x^10+210*a*b^4*x^8+336*a^2*b^3*x^6+280*a^3*b^2*x^4+120*a^4*b*x^2+
21*a^5)*((b*x^2+a)^2)^(5/2)/x^16/(b*x^2+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^4 + 2*a*b*x^2 + a^2)^(5/2)/x^17,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267972, size = 80, normalized size = 0.62 \[ -\frac{56 \, b^{5} x^{10} + 210 \, a b^{4} x^{8} + 336 \, a^{2} b^{3} x^{6} + 280 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} + 21 \, a^{5}}{336 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^17,x, algorithm="fricas")

[Out]

-1/336*(56*b^5*x^10 + 210*a*b^4*x^8 + 336*a^2*b^3*x^6 + 280*a^3*b^2*x^4 + 120*a^
4*b*x^2 + 21*a^5)/x^16

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**17,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272898, size = 144, normalized size = 1.12 \[ -\frac{56 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 210 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 336 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 280 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 120 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 21 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{336 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^17,x, algorithm="giac")

[Out]

-1/336*(56*b^5*x^10*sign(b*x^2 + a) + 210*a*b^4*x^8*sign(b*x^2 + a) + 336*a^2*b^
3*x^6*sign(b*x^2 + a) + 280*a^3*b^2*x^4*sign(b*x^2 + a) + 120*a^4*b*x^2*sign(b*x
^2 + a) + 21*a^5*sign(b*x^2 + a))/x^16