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

Optimal. Leaf size=21 \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

________________________________________________________________________________________

Rubi [A]  time = 0.0041378, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0054825, size = 21, normalized size = 1. \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \begin{align*} -{\frac{-6\,c{x}^{4}+10\,a}{15}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*c*x^4+5*a)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.01066, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*c*x^(5/2) - 2/3*a/x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.43483, size = 41, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (3 \, c x^{4} - 5 \, a\right )}}{15 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c*x^4 - 5*a)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 1.71812, size = 19, normalized size = 0.9 \begin{align*} - \frac{2 a}{3 x^{\frac{3}{2}}} + \frac{2 c x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(3*x**(3/2)) + 2*c*x**(5/2)/5

________________________________________________________________________________________

Giac [A]  time = 1.15063, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*c*x^(5/2) - 2/3*a/x^(3/2)