3.728 \(\int \frac{\left (a+c x^4\right )^2}{x^{7/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{4}{3} a c x^{3/2}+\frac{2}{11} c^2 x^{11/2} \]

[Out]

(-2*a^2)/(5*x^(5/2)) + (4*a*c*x^(3/2))/3 + (2*c^2*x^(11/2))/11

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Rubi [A]  time = 0.0263919, 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 \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{4}{3} a c x^{3/2}+\frac{2}{11} c^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^2/x^(7/2),x]

[Out]

(-2*a^2)/(5*x^(5/2)) + (4*a*c*x^(3/2))/3 + (2*c^2*x^(11/2))/11

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Rubi in Sympy [A]  time = 4.50867, size = 34, normalized size = 0.94 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**2/x**(7/2),x)

[Out]

-2*a**2/(5*x**(5/2)) + 4*a*c*x**(3/2)/3 + 2*c**2*x**(11/2)/11

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Mathematica [A]  time = 0.0141404, size = 30, normalized size = 0.83 \[ \frac{2 \left (-33 a^2+110 a c x^4+15 c^2 x^8\right )}{165 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^2/x^(7/2),x]

[Out]

(2*(-33*a^2 + 110*a*c*x^4 + 15*c^2*x^8))/(165*x^(5/2))

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Maple [A]  time = 0.007, size = 27, normalized size = 0.8 \[ -{\frac{-30\,{c}^{2}{x}^{8}-220\,ac{x}^{4}+66\,{a}^{2}}{165}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^2/x^(7/2),x)

[Out]

-2/165*(-15*c^2*x^8-110*a*c*x^4+33*a^2)/x^(5/2)

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Maxima [A]  time = 1.44146, size = 32, normalized size = 0.89 \[ \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{3} \, a c x^{\frac{3}{2}} - \frac{2 \, a^{2}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/x^(7/2),x, algorithm="maxima")

[Out]

2/11*c^2*x^(11/2) + 4/3*a*c*x^(3/2) - 2/5*a^2/x^(5/2)

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Fricas [A]  time = 0.232838, size = 35, normalized size = 0.97 \[ \frac{2 \,{\left (15 \, c^{2} x^{8} + 110 \, a c x^{4} - 33 \, a^{2}\right )}}{165 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/x^(7/2),x, algorithm="fricas")

[Out]

2/165*(15*c^2*x^8 + 110*a*c*x^4 - 33*a^2)/x^(5/2)

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Sympy [A]  time = 31.7333, size = 34, normalized size = 0.94 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**2/x**(7/2),x)

[Out]

-2*a**2/(5*x**(5/2)) + 4*a*c*x**(3/2)/3 + 2*c**2*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.216221, size = 32, normalized size = 0.89 \[ \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{3} \, a c x^{\frac{3}{2}} - \frac{2 \, a^{2}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/x^(7/2),x, algorithm="giac")

[Out]

2/11*c^2*x^(11/2) + 4/3*a*c*x^(3/2) - 2/5*a^2/x^(5/2)