3.403 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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Rubi [A]  time = 0.0222983, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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Rubi in Sympy [A]  time = 3.19426, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**8,x)

[Out]

-(a + b*x**2)**(7/2)/(7*a*x**7)

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Mathematica [A]  time = 0.0328821, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \[ -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^8,x)

[Out]

-1/7*(b*x^2+a)^(7/2)/a/x^7

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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*x^2 + a)^(5/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236962, size = 62, normalized size = 2.95 \[ -\frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{7 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^8,x, algorithm="fricas")

[Out]

-1/7*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)/(a*x^7)

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Sympy [A]  time = 6.4575, size = 95, normalized size = 4.52 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{6}} - \frac{3 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{4}} - \frac{3 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{2}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**8,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(7*x**6) - 3*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/
(7*x**4) - 3*b**(5/2)*sqrt(a/(b*x**2) + 1)/(7*x**2) - b**(7/2)*sqrt(a/(b*x**2) +
1)/(7*a)

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GIAC/XCAS [A]  time = 0.216156, size = 153, normalized size = 7.29 \[ \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} b^{\frac{7}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} b^{\frac{7}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} b^{\frac{7}{2}} + a^{6} b^{\frac{7}{2}}\right )}}{7 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^8,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(7/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a
))^8*a^2*b^(7/2) + 21*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(7/2) + a^6*b^(7/2))
/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7