3.1047 \(\int \frac{a+b x^2+c x^4}{x^{7/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2} \]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x] + (2*c*x^(3/2))/3

________________________________________________________________________________________

Rubi [A]  time = 0.0062878, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.007854, size = 25, normalized size = 0.86 \[ \frac{2 \left (-3 a-15 b x^2+5 c x^4\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 22, normalized size = 0.8 \begin{align*} -{\frac{-10\,c{x}^{4}+30\,b{x}^{2}+6\,a}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.952902, size = 27, normalized size = 0.93 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.25229, size = 55, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (5 \, c x^{4} - 15 \, b x^{2} - 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+b*x^2+a)/x^(7/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.88949, size = 27, normalized size = 0.93 \begin{align*} - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} + \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+b*x**2+a)/x**(7/2),x)

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x) + 2*c*x**(3/2)/3

________________________________________________________________________________________

Giac [A]  time = 1.13365, size = 27, normalized size = 0.93 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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