3.604 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{25}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(24*x^24*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(22*x^22*(a + b*x^2)) - (a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(2*x^20*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9
*x^18*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x
^2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2))

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Rubi [A]  time = 0.366966, antiderivative size = 255, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^25,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(24*x^24*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(22*x^22*(a + b*x^2)) - (a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(2*x^20*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9
*x^18*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x
^2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.4063, size = 204, normalized size = 0.8 \[ \frac{a b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{11088 x^{16} \left (a + b x^{2}\right )} + \frac{a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{396 x^{20}} + \frac{5 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{264 x^{24}} - \frac{b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1386 x^{16}} - \frac{b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{99 x^{20}} - \frac{2 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{33 x^{24}} \]

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**25,x)

[Out]

a*b**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(11088*x**16*(a + b*x**2)) + a*b**2*(
a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(396*x**20) + 5*a*(a + b*x**2)*(
a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(264*x**24) - b**4*sqrt(a**2 + 2*a*b*x**2
+ b**2*x**4)/(1386*x**16) - b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(99*x**2
0) - 2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(33*x**24)

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Mathematica [A]  time = 0.0399038, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (462 a^5+2520 a^4 b x^2+5544 a^3 b^2 x^4+6160 a^2 b^3 x^6+3465 a b^4 x^8+792 b^5 x^{10}\right )}{11088 x^{24} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^25,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(462*a^5 + 2520*a^4*b*x^2 + 5544*a^3*b^2*x^4 + 6160*a^2*b^
3*x^6 + 3465*a*b^4*x^8 + 792*b^5*x^10))/(11088*x^24*(a + b*x^2))

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Maple [A]  time = 0.013, size = 80, normalized size = 0.3 \[ -{\frac{792\,{b}^{5}{x}^{10}+3465\,a{b}^{4}{x}^{8}+6160\,{a}^{2}{b}^{3}{x}^{6}+5544\,{a}^{3}{b}^{2}{x}^{4}+2520\,{a}^{4}b{x}^{2}+462\,{a}^{5}}{11088\,{x}^{24} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^25,x)

[Out]

-1/11088*(792*b^5*x^10+3465*a*b^4*x^8+6160*a^2*b^3*x^6+5544*a^3*b^2*x^4+2520*a^4
*b*x^2+462*a^5)*((b*x^2+a)^2)^(5/2)/x^24/(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^25,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263522, size = 80, normalized size = 0.31 \[ -\frac{792 \, b^{5} x^{10} + 3465 \, a b^{4} x^{8} + 6160 \, a^{2} b^{3} x^{6} + 5544 \, a^{3} b^{2} x^{4} + 2520 \, a^{4} b x^{2} + 462 \, a^{5}}{11088 \, x^{24}} \]

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^25,x, algorithm="fricas")

[Out]

-1/11088*(792*b^5*x^10 + 3465*a*b^4*x^8 + 6160*a^2*b^3*x^6 + 5544*a^3*b^2*x^4 +
2520*a^4*b*x^2 + 462*a^5)/x^24

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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**4+2*a*b*x**2+a**2)**(5/2)/x**25,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272949, size = 144, normalized size = 0.56 \[ -\frac{792 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 3465 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 6160 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 5544 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 2520 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 462 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{11088 \, x^{24}} \]

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^25,x, algorithm="giac")

[Out]

-1/11088*(792*b^5*x^10*sign(b*x^2 + a) + 3465*a*b^4*x^8*sign(b*x^2 + a) + 6160*a
^2*b^3*x^6*sign(b*x^2 + a) + 5544*a^3*b^2*x^4*sign(b*x^2 + a) + 2520*a^4*b*x^2*s
ign(b*x^2 + a) + 462*a^5*sign(b*x^2 + a))/x^24