3.1046 \(\int \frac{a+b x^2+c x^4}{x^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0059993, 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} \[ -\frac{2 a}{3 x^{3/2}}+2 b \sqrt{x}+\frac{2}{5} c x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0082028, size = 25, normalized size = 0.86 \[ \frac{2 \left (-5 a+15 b x^2+3 c x^4\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.044, size = 22, normalized size = 0.8 \begin{align*} -{\frac{-6\,c{x}^{4}-30\,b{x}^{2}+10\,a}{15}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948782, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} + 2 \, b \sqrt{x} - \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+b*x^2+a)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24963, size = 55, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (3 \, c x^{4} + 15 \, b x^{2} - 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+b*x^2+a)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.21285, size = 27, normalized size = 0.93 \begin{align*} - \frac{2 a}{3 x^{\frac{3}{2}}} + 2 b \sqrt{x} + \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+b*x**2+a)/x**(5/2),x)

[Out]

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

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Giac [A]  time = 1.17868, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} + 2 \, b \sqrt{x} - \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+b*x^2+a)/x^(5/2),x, algorithm="giac")

[Out]

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