3.733 \(\int \frac{\left (a+c x^4\right )^3}{x^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 a^3}{\sqrt{x}}+\frac{6}{7} a^2 c x^{7/2}+\frac{2}{5} a c^2 x^{15/2}+\frac{2}{23} c^3 x^{23/2} \]

[Out]

(-2*a^3)/Sqrt[x] + (6*a^2*c*x^(7/2))/7 + (2*a*c^2*x^(15/2))/5 + (2*c^3*x^(23/2))
/23

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Rubi [A]  time = 0.0354327, antiderivative size = 49, 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 \[ -\frac{2 a^3}{\sqrt{x}}+\frac{6}{7} a^2 c x^{7/2}+\frac{2}{5} a c^2 x^{15/2}+\frac{2}{23} c^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + (6*a^2*c*x^(7/2))/7 + (2*a*c^2*x^(15/2))/5 + (2*c^3*x^(23/2))
/23

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Rubi in Sympy [A]  time = 5.78201, size = 48, normalized size = 0.98 \[ - \frac{2 a^{3}}{\sqrt{x}} + \frac{6 a^{2} c x^{\frac{7}{2}}}{7} + \frac{2 a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 c^{3} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**3/x**(3/2),x)

[Out]

-2*a**3/sqrt(x) + 6*a**2*c*x**(7/2)/7 + 2*a*c**2*x**(15/2)/5 + 2*c**3*x**(23/2)/
23

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Mathematica [A]  time = 0.0162779, size = 41, normalized size = 0.84 \[ \frac{2 \left (-805 a^3+345 a^2 c x^4+161 a c^2 x^8+35 c^3 x^{12}\right )}{805 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^3/x^(3/2),x]

[Out]

(2*(-805*a^3 + 345*a^2*c*x^4 + 161*a*c^2*x^8 + 35*c^3*x^12))/(805*Sqrt[x])

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Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[ -{\frac{-70\,{c}^{3}{x}^{12}-322\,a{c}^{2}{x}^{8}-690\,{a}^{2}c{x}^{4}+1610\,{a}^{3}}{805}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^3/x^(3/2),x)

[Out]

-2/805*(-35*c^3*x^12-161*a*c^2*x^8-345*a^2*c*x^4+805*a^3)/x^(1/2)

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Maxima [A]  time = 1.43543, size = 47, normalized size = 0.96 \[ \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^3/x^(3/2),x, algorithm="maxima")

[Out]

2/23*c^3*x^(23/2) + 2/5*a*c^2*x^(15/2) + 6/7*a^2*c*x^(7/2) - 2*a^3/sqrt(x)

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Fricas [A]  time = 0.225161, size = 50, normalized size = 1.02 \[ \frac{2 \,{\left (35 \, c^{3} x^{12} + 161 \, a c^{2} x^{8} + 345 \, a^{2} c x^{4} - 805 \, a^{3}\right )}}{805 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^3/x^(3/2),x, algorithm="fricas")

[Out]

2/805*(35*c^3*x^12 + 161*a*c^2*x^8 + 345*a^2*c*x^4 - 805*a^3)/sqrt(x)

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Sympy [A]  time = 69.2825, size = 48, normalized size = 0.98 \[ - \frac{2 a^{3}}{\sqrt{x}} + \frac{6 a^{2} c x^{\frac{7}{2}}}{7} + \frac{2 a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 c^{3} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**3/x**(3/2),x)

[Out]

-2*a**3/sqrt(x) + 6*a**2*c*x**(7/2)/7 + 2*a*c**2*x**(15/2)/5 + 2*c**3*x**(23/2)/
23

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GIAC/XCAS [A]  time = 0.211472, size = 47, normalized size = 0.96 \[ \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^3/x^(3/2),x, algorithm="giac")

[Out]

2/23*c^3*x^(23/2) + 2/5*a*c^2*x^(15/2) + 6/7*a^2*c*x^(7/2) - 2*a^3/sqrt(x)