3.1045 \(\int \frac{a+b x^2+c x^4}{x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 a}{\sqrt{x}}+\frac{2}{3} b x^{3/2}+\frac{2}{7} c x^{7/2} \]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

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Rubi [A]  time = 0.0062198, 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} \[ -\frac{2 a}{\sqrt{x}}+\frac{2}{3} b x^{3/2}+\frac{2}{7} c x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

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

Mathematica [A]  time = 0.0082624, size = 25, normalized size = 0.86 \[ \frac{2 \left (-21 a+7 b x^2+3 c x^4\right )}{21 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*(-21*a + 7*b*x^2 + 3*c*x^4))/(21*Sqrt[x])

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Maple [A]  time = 0.045, size = 22, normalized size = 0.8 \begin{align*} -{\frac{-6\,c{x}^{4}-14\,b{x}^{2}+42\,a}{21}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^(3/2),x)

[Out]

-2/21*(-3*c*x^4-7*b*x^2+21*a)/x^(1/2)

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Maxima [A]  time = 0.948077, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{7} \, c x^{\frac{7}{2}} + \frac{2}{3} \, b x^{\frac{3}{2}} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="maxima")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)

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Fricas [A]  time = 1.21412, size = 55, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (3 \, c x^{4} + 7 \, b x^{2} - 21 \, a\right )}}{21 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*c*x^4 + 7*b*x^2 - 21*a)/sqrt(x)

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Sympy [A]  time = 0.969157, size = 27, normalized size = 0.93 \begin{align*} - \frac{2 a}{\sqrt{x}} + \frac{2 b x^{\frac{3}{2}}}{3} + \frac{2 c x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**(3/2),x)

[Out]

-2*a/sqrt(x) + 2*b*x**(3/2)/3 + 2*c*x**(7/2)/7

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Giac [A]  time = 1.14864, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{7} \, c x^{\frac{7}{2}} + \frac{2}{3} \, b x^{\frac{3}{2}} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="giac")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)