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

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

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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Rubi [A]  time = 0.0041887, 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} \[ \frac{2}{3} c x^{3/2}-\frac{2 a}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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

Mathematica [A]  time = 0.005415, size = 21, normalized size = 1. \[ \frac{2}{3} c x^{3/2}-\frac{2 a}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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Maple [A]  time = 0.001, size = 16, normalized size = 0.8 \begin{align*} -{\frac{-10\,c{x}^{4}+6\,a}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.01152, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, a}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*c*x^(3/2) - 2/5*a/x^(5/2)

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Fricas [A]  time = 1.44679, size = 41, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (5 \, c x^{4} - 3 \, a\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*c*x^4 - 3*a)/x^(5/2)

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Sympy [A]  time = 2.38774, size = 19, normalized size = 0.9 \begin{align*} - \frac{2 a}{5 x^{\frac{5}{2}}} + \frac{2 c x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(5*x**(5/2)) + 2*c*x**(3/2)/3

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Giac [A]  time = 1.13963, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, a}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*c*x^(3/2) - 2/5*a/x^(5/2)