3.264 \(\int x^{7/2} \left (a+b x^2\right ) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

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Rubi [A]  time = 0.0137314, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

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Rubi in Sympy [A]  time = 2.9219, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{9}{2}}}{9} + \frac{2 b x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x**2+a),x)

[Out]

2*a*x**(9/2)/9 + 2*b*x**(13/2)/13

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Mathematica [A]  time = 0.00804533, size = 21, normalized size = 1. \[ \frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[{\frac{18\,b{x}^{2}+26\,a}{117}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a),x)

[Out]

2/117*x^(9/2)*(9*b*x^2+13*a)

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Maxima [A]  time = 1.34918, size = 18, normalized size = 0.86 \[ \frac{2}{13} \, b x^{\frac{13}{2}} + \frac{2}{9} \, a x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^(7/2),x, algorithm="maxima")

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)

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Fricas [A]  time = 0.217804, size = 24, normalized size = 1.14 \[ \frac{2}{117} \,{\left (9 \, b x^{6} + 13 \, a x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^(7/2),x, algorithm="fricas")

[Out]

2/117*(9*b*x^6 + 13*a*x^4)*sqrt(x)

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Sympy [A]  time = 19.8947, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{9}{2}}}{9} + \frac{2 b x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(b*x**2+a),x)

[Out]

2*a*x**(9/2)/9 + 2*b*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.205132, size = 18, normalized size = 0.86 \[ \frac{2}{13} \, b x^{\frac{13}{2}} + \frac{2}{9} \, a x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^(7/2),x, algorithm="giac")

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)