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

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 = 0.0081328, 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, Rules used = {270} \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{4}{3} a c x^{3/2}+\frac{2}{11} c^2 x^{11/2} \]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^4\right )^2}{x^{7/2}} \, dx &=\int \left (\frac{a^2}{x^{7/2}}+2 a c \sqrt{x}+c^2 x^{9/2}\right ) \, dx\\ &=-\frac{2 a^2}{5 x^{5/2}}+\frac{4}{3} a c x^{3/2}+\frac{2}{11} c^2 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0084556, 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 = 0.005, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{8}-220\,ac{x}^{4}+66\,{a}^{2}}{165}{x}^{-{\frac{5}{2}}}} \end{align*}

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 = 0.99478, size = 32, normalized size = 0.89 \begin{align*} \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}}} \end{align*}

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 = 1.46521, size = 69, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (15 \, c^{2} x^{8} + 110 \, a c x^{4} - 33 \, a^{2}\right )}}{165 \, x^{\frac{5}{2}}} \end{align*}

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 = 10.5735, size = 34, normalized size = 0.94 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.11386, size = 32, normalized size = 0.89 \begin{align*} \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}}} \end{align*}

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)