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

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{15} a c x^{15/2}+\frac{2}{23} c^2 x^{23/2} \]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*c*x^(15/2))/15 + (2*c^2*x^(23/2))/23

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Rubi [A]  time = 0.0264287, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{15} a c x^{15/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*c*x^(15/2))/15 + (2*c^2*x^(23/2))/23

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Rubi in Sympy [A]  time = 4.51766, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a c x^{\frac{15}{2}}}{15} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*c*x**(15/2)/15 + 2*c**2*x**(23/2)/23

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Mathematica [A]  time = 0.012895, size = 30, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (345 a^2+322 a c x^4+105 c^2 x^8\right )}{2415} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*(345*a^2 + 322*a*c*x^4 + 105*c^2*x^8))/2415

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Maple [A]  time = 0.006, size = 27, normalized size = 0.8 \[{\frac{210\,{c}^{2}{x}^{8}+644\,ac{x}^{4}+690\,{a}^{2}}{2415}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(c*x^4+a)^2,x)

[Out]

2/2415*x^(7/2)*(105*c^2*x^8+322*a*c*x^4+345*a^2)

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Maxima [A]  time = 1.43973, size = 32, normalized size = 0.89 \[ \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/23*c^2*x^(23/2) + 4/15*a*c*x^(15/2) + 2/7*a^2*x^(7/2)

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Fricas [A]  time = 0.226314, size = 39, normalized size = 1.08 \[ \frac{2}{2415} \,{\left (105 \, c^{2} x^{11} + 322 \, a c x^{7} + 345 \, a^{2} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2415*(105*c^2*x^11 + 322*a*c*x^7 + 345*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 70.4942, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a c x^{\frac{15}{2}}}{15} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*c*x**(15/2)/15 + 2*c**2*x**(23/2)/23

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GIAC/XCAS [A]  time = 0.217067, size = 32, normalized size = 0.89 \[ \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/23*c^2*x^(23/2) + 4/15*a*c*x^(15/2) + 2/7*a^2*x^(7/2)