3.430 \(\int x^{3/2} (a+b x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{7} b x^{7/2} \]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(7/2))/7

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Rubi [A]  time = 0.0131158, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{5} a x^{5/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a + b*x),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(7/2))/7

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Rubi in Sympy [A]  time = 2.32616, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(b*x+a),x)

[Out]

2*a*x**(5/2)/5 + 2*b*x**(7/2)/7

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Mathematica [A]  time = 0.00503557, size = 17, normalized size = 0.81 \[ \frac{2}{35} x^{5/2} (7 a+5 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(a + b*x),x]

[Out]

(2*x^(5/2)*(7*a + 5*b*x))/35

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \[{\frac{10\,bx+14\,a}{35}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34405, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(3/2),x, algorithm="maxima")

[Out]

2/7*b*x^(7/2) + 2/5*a*x^(5/2)

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Fricas [A]  time = 0.207841, size = 24, normalized size = 1.14 \[ \frac{2}{35} \,{\left (5 \, b x^{3} + 7 \, a x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b*x^3 + 7*a*x^2)*sqrt(x)

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Sympy [A]  time = 1.02849, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(b*x+a),x)

[Out]

2*a*x**(5/2)/5 + 2*b*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.200362, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^(3/2),x, algorithm="giac")

[Out]

2/7*b*x^(7/2) + 2/5*a*x^(5/2)