3.715 \(\int x^{5/2} \left (a+c x^4\right ) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{7} a x^{7/2}+\frac{2}{15} c x^{15/2} \]

[Out]

(2*a*x^(7/2))/7 + (2*c*x^(15/2))/15

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Rubi [A]  time = 0.0119542, 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 \[ \frac{2}{7} a x^{7/2}+\frac{2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + c*x^4),x]

[Out]

(2*a*x^(7/2))/7 + (2*c*x^(15/2))/15

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Rubi in Sympy [A]  time = 2.61917, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(c*x**4+a),x)

[Out]

2*a*x**(7/2)/7 + 2*c*x**(15/2)/15

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Mathematica [A]  time = 0.00762455, size = 21, normalized size = 1. \[ \frac{2}{7} a x^{7/2}+\frac{2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + c*x^4),x]

[Out]

(2*a*x^(7/2))/7 + (2*c*x^(15/2))/15

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Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \[{\frac{14\,c{x}^{4}+30\,a}{105}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(c*x^4+a),x)

[Out]

2/105*x^(7/2)*(7*c*x^4+15*a)

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Maxima [A]  time = 1.44899, size = 18, normalized size = 0.86 \[ \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)*x^(5/2),x, algorithm="maxima")

[Out]

2/15*c*x^(15/2) + 2/7*a*x^(7/2)

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Fricas [A]  time = 0.227247, size = 24, normalized size = 1.14 \[ \frac{2}{105} \,{\left (7 \, c x^{7} + 15 \, a x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)*x^(5/2),x, algorithm="fricas")

[Out]

2/105*(7*c*x^7 + 15*a*x^3)*sqrt(x)

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Sympy [A]  time = 18.9275, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(c*x**4+a),x)

[Out]

2*a*x**(7/2)/7 + 2*c*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.215617, size = 18, normalized size = 0.86 \[ \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)*x^(5/2),x, algorithm="giac")

[Out]

2/15*c*x^(15/2) + 2/7*a*x^(7/2)