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

Optimal. Leaf size=18 \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

[Out]

(a + b*x^2)^(11/2)/(11*b)

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Rubi [A]  time = 0.0112103, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^(9/2),x]

[Out]

(a + b*x^2)^(11/2)/(11*b)

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Rubi in Sympy [A]  time = 2.1749, size = 12, normalized size = 0.67 \[ \frac{\left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**(11/2)/(11*b)

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Mathematica [A]  time = 0.0135474, size = 18, normalized size = 1. \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^2)^(11/2)/(11*b)

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Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \[{\frac{1}{11\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^(9/2),x)

[Out]

1/11*(b*x^2+a)^(11/2)/b

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Maxima [A]  time = 1.32404, size = 19, normalized size = 1.06 \[ \frac{{\left (b x^{2} + a\right )}^{\frac{11}{2}}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*(b*x^2 + a)^(11/2)/b

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Fricas [A]  time = 0.25506, size = 88, normalized size = 4.89 \[ \frac{{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{b x^{2} + a}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a
^5)*sqrt(b*x^2 + a)/b

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Sympy [A]  time = 43.375, size = 133, normalized size = 7.39 \[ \begin{cases} \frac{a^{5} \sqrt{a + b x^{2}}}{11 b} + \frac{5 a^{4} x^{2} \sqrt{a + b x^{2}}}{11} + \frac{10 a^{3} b x^{4} \sqrt{a + b x^{2}}}{11} + \frac{10 a^{2} b^{2} x^{6} \sqrt{a + b x^{2}}}{11} + \frac{5 a b^{3} x^{8} \sqrt{a + b x^{2}}}{11} + \frac{b^{4} x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**5*sqrt(a + b*x**2)/(11*b) + 5*a**4*x**2*sqrt(a + b*x**2)/11 + 10*a
**3*b*x**4*sqrt(a + b*x**2)/11 + 10*a**2*b**2*x**6*sqrt(a + b*x**2)/11 + 5*a*b**
3*x**8*sqrt(a + b*x**2)/11 + b**4*x**10*sqrt(a + b*x**2)/11, Ne(b, 0)), (a**(9/2
)*x**2/2, True))

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GIAC/XCAS [A]  time = 0.208728, size = 267, normalized size = 14.83 \[ \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 2310 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} + 924 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a^{3} + 198 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{2} + 44 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3465*(315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/
2)*a^2 - 2772*(b*x^2 + a)^(5/2)*a^3 + 2310*(b*x^2 + a)^(3/2)*a^4 + 924*(3*(b*x^2
+ a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*a^3 + 198*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2
+ a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*a^2 + 44*(35*(b*x^2 + a)^(9/2) - 135*(
b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*a)/b