3.523 \(\int \frac{x^4}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}} \]

[Out]

x^5/(5*a*(a + b*x^2)^(7/2)) + (2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))

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Rubi [A]  time = 0.0493929, 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 \[ \frac{2 b x^7}{35 a^2 \left (a+b x^2\right )^{7/2}}+\frac{x^5}{5 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(9/2),x]

[Out]

x^5/(5*a*(a + b*x^2)^(7/2)) + (2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))

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Rubi in Sympy [A]  time = 5.53708, size = 37, normalized size = 0.84 \[ \frac{x^{5}}{5 a \left (a + b x^{2}\right )^{\frac{7}{2}}} + \frac{2 b x^{7}}{35 a^{2} \left (a + b x^{2}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**(9/2),x)

[Out]

x**5/(5*a*(a + b*x**2)**(7/2)) + 2*b*x**7/(35*a**2*(a + b*x**2)**(7/2))

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Mathematica [A]  time = 0.0332654, size = 31, normalized size = 0.7 \[ \frac{x^5 \left (7 a+2 b x^2\right )}{35 a^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^2)^(9/2),x]

[Out]

(x^5*(7*a + 2*b*x^2))/(35*a^2*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.008, size = 28, normalized size = 0.6 \[{\frac{{x}^{5} \left ( 2\,b{x}^{2}+7\,a \right ) }{35\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(9/2),x)

[Out]

1/35*x^5*(2*b*x^2+7*a)/(b*x^2+a)^(7/2)/a^2

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Maxima [A]  time = 1.34811, size = 115, normalized size = 2.61 \[ -\frac{x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{3 \, x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

-1/4*x^3/((b*x^2 + a)^(7/2)*b) + 3/140*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*x/(sqrt(
b*x^2 + a)*a^2*b^2) + 1/35*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*a*x/((b*x^2 + a)^(
7/2)*b^2)

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Fricas [A]  time = 0.252244, size = 96, normalized size = 2.18 \[ \frac{{\left (2 \, b x^{7} + 7 \, a x^{5}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{2} b^{4} x^{8} + 4 \, a^{3} b^{3} x^{6} + 6 \, a^{4} b^{2} x^{4} + 4 \, a^{5} b x^{2} + a^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

1/35*(2*b*x^7 + 7*a*x^5)*sqrt(b*x^2 + a)/(a^2*b^4*x^8 + 4*a^3*b^3*x^6 + 6*a^4*b^
2*x^4 + 4*a^5*b*x^2 + a^6)

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Sympy [A]  time = 10.0758, size = 199, normalized size = 4.52 \[ \frac{7 a x^{5}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{7}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)**(9/2),x)

[Out]

7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/
a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 +
b*x**2/a)) + 2*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqr
t(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x
**6*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.221414, size = 39, normalized size = 0.89 \[ \frac{x^{5}{\left (\frac{2 \, b x^{2}}{a^{2}} + \frac{7}{a}\right )}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

1/35*x^5*(2*b*x^2/a^2 + 7/a)/(b*x^2 + a)^(7/2)