3.404 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]

[Out]

-(a + b*x^2)^(7/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi [A]  time = 0.0452174, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^10,x]

[Out]

-(a + b*x^2)^(7/2)/(9*a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi in Sympy [A]  time = 5.20474, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{2}\right )^{\frac{7}{2}}}{9 a x^{9}} + \frac{2 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{63 a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**10,x)

[Out]

-(a + b*x**2)**(7/2)/(9*a*x**9) + 2*b*(a + b*x**2)**(7/2)/(63*a**2*x**7)

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Mathematica [A]  time = 0.041712, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^2\right )^{7/2} \left (2 b x^2-7 a\right )}{63 a^2 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^10,x]

[Out]

((a + b*x^2)^(7/2)*(-7*a + 2*b*x^2))/(63*a^2*x^9)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-2\,b{x}^{2}+7\,a}{63\,{x}^{9}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^10,x)

[Out]

-1/63*(b*x^2+a)^(7/2)*(-2*b*x^2+7*a)/x^9/a^2

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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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255372, size = 81, normalized size = 1.84 \[ \frac{{\left (2 \, b^{4} x^{8} - a b^{3} x^{6} - 15 \, a^{2} b^{2} x^{4} - 19 \, a^{3} b x^{2} - 7 \, a^{4}\right )} \sqrt{b x^{2} + a}}{63 \, a^{2} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^10,x, algorithm="fricas")

[Out]

1/63*(2*b^4*x^8 - a*b^3*x^6 - 15*a^2*b^2*x^4 - 19*a^3*b*x^2 - 7*a^4)*sqrt(b*x^2
+ a)/(a^2*x^9)

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Sympy [A]  time = 10.2129, size = 121, normalized size = 2.75 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{9 x^{8}} - \frac{19 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 x^{6}} - \frac{5 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{21 x^{4}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a x^{2}} + \frac{2 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**10,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(9*x**8) - 19*a*b**(3/2)*sqrt(a/(b*x**2) + 1)
/(63*x**6) - 5*b**(5/2)*sqrt(a/(b*x**2) + 1)/(21*x**4) - b**(7/2)*sqrt(a/(b*x**2
) + 1)/(63*a*x**2) + 2*b**(9/2)*sqrt(a/(b*x**2) + 1)/(63*a**2)

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GIAC/XCAS [A]  time = 0.215922, size = 297, normalized size = 6.75 \[ \frac{4 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} b^{\frac{9}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{2} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{3} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{4} b^{\frac{9}{2}} + 27 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{5} b^{\frac{9}{2}} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{6} b^{\frac{9}{2}} - a^{7} b^{\frac{9}{2}}\right )}}{63 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^10,x, algorithm="giac")

[Out]

4/63*(63*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(9/2) + 105*(sqrt(b)*x - sqrt(b*x^2
+ a))^12*a*b^(9/2) + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(9/2) + 189*(sqr
t(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(9/2) + 189*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^
4*b^(9/2) + 27*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(9/2) + 9*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*a^6*b^(9/2) - a^7*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^
9