3.1044 \(\int \frac{a+b x^2+c x^4}{\sqrt{x}} \, dx\)

Optimal. Leaf size=29 \[ 2 a \sqrt{x}+\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2} \]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5 + (2*c*x^(9/2))/9

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Rubi [A]  time = 0.0062927, 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} \[ 2 a \sqrt{x}+\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5 + (2*c*x^(9/2))/9

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}{\sqrt{x}} \, dx &=\int \left (\frac{a}{\sqrt{x}}+b x^{3/2}+c x^{7/2}\right ) \, dx\\ &=2 a \sqrt{x}+\frac{2}{5} b x^{5/2}+\frac{2}{9} c x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0063804, size = 25, normalized size = 0.86 \[ \frac{2}{45} \sqrt{x} \left (45 a+9 b x^2+5 c x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(45*a + 9*b*x^2 + 5*c*x^4))/45

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Maple [A]  time = 0.043, size = 22, normalized size = 0.8 \begin{align*}{\frac{10\,c{x}^{4}+18\,b{x}^{2}+90\,a}{45}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^(1/2),x)

[Out]

2/45*x^(1/2)*(5*c*x^4+9*b*x^2+45*a)

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Maxima [A]  time = 0.946927, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{9} \, c x^{\frac{9}{2}} + \frac{2}{5} \, b x^{\frac{5}{2}} + 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^(1/2),x, algorithm="maxima")

[Out]

2/9*c*x^(9/2) + 2/5*b*x^(5/2) + 2*a*sqrt(x)

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Fricas [A]  time = 1.20527, size = 55, normalized size = 1.9 \begin{align*} \frac{2}{45} \,{\left (5 \, c x^{4} + 9 \, b x^{2} + 45 \, a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(1/2),x, algorithm="fricas")

[Out]

2/45*(5*c*x^4 + 9*b*x^2 + 45*a)*sqrt(x)

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Sympy [A]  time = 0.716301, size = 27, normalized size = 0.93 \begin{align*} 2 a \sqrt{x} + \frac{2 b x^{\frac{5}{2}}}{5} + \frac{2 c x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**(1/2),x)

[Out]

2*a*sqrt(x) + 2*b*x**(5/2)/5 + 2*c*x**(9/2)/9

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Giac [A]  time = 1.16637, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{9} \, c x^{\frac{9}{2}} + \frac{2}{5} \, b x^{\frac{5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/9*c*x^(9/2) + 2/5*b*x^(5/2) + 2*a*sqrt(x)