3.729 \(\int \frac{(a+c x^4)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{2 a^2}{3 x^{3/2}}+\frac{4}{5} a c x^{5/2}+\frac{2}{13} c^2 x^{13/2} \]

[Out]

(-2*a^2)/(3*x^(3/2)) + (4*a*c*x^(5/2))/5 + (2*c^2*x^(13/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0094453, 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} \[ -\frac{2 a^2}{3 x^{3/2}}+\frac{4}{5} a c x^{5/2}+\frac{2}{13} c^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) + (4*a*c*x^(5/2))/5 + (2*c^2*x^(13/2))/13

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^{5/2}} \, dx &=\int \left (\frac{a^2}{x^{5/2}}+2 a c x^{3/2}+c^2 x^{11/2}\right ) \, dx\\ &=-\frac{2 a^2}{3 x^{3/2}}+\frac{4}{5} a c x^{5/2}+\frac{2}{13} c^2 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0083106, size = 30, normalized size = 0.83 \[ \frac{2 \left (-65 a^2+78 a c x^4+15 c^2 x^8\right )}{195 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/x^(5/2),x]

[Out]

(2*(-65*a^2 + 78*a*c*x^4 + 15*c^2*x^8))/(195*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{8}-156\,ac{x}^{4}+130\,{a}^{2}}{195}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^2/x^(5/2),x)

[Out]

-2/195*(-15*c^2*x^8-78*a*c*x^4+65*a^2)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.978341, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{5} \, a c x^{\frac{5}{2}} - \frac{2 \, a^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c^2*x^(13/2) + 4/5*a*c*x^(5/2) - 2/3*a^2/x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.49152, size = 68, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (15 \, c^{2} x^{8} + 78 \, a c x^{4} - 65 \, a^{2}\right )}}{195 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/195*(15*c^2*x^8 + 78*a*c*x^4 - 65*a^2)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 8.27607, size = 34, normalized size = 0.94 \begin{align*} - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} + \frac{4 a c x^{\frac{5}{2}}}{5} + \frac{2 c^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(3*x**(3/2)) + 4*a*c*x**(5/2)/5 + 2*c**2*x**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 1.14113, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{5} \, a c x^{\frac{5}{2}} - \frac{2 \, a^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c^2*x^(13/2) + 4/5*a*c*x^(5/2) - 2/3*a^2/x^(3/2)