3.1043 \(\int \sqrt{x} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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Rubi [A]  time = 0.0066384, antiderivative size = 31, 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} \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2 + c*x^4),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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 \sqrt{x} \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a \sqrt{x}+b x^{5/2}+c x^{9/2}\right ) \, dx\\ &=\frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0061114, size = 25, normalized size = 0.81 \[ \frac{2}{231} x^{3/2} \left (77 a+33 b x^2+21 c x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(77*a + 33*b*x^2 + 21*c*x^4))/231

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Maple [A]  time = 0.042, size = 22, normalized size = 0.7 \begin{align*}{\frac{42\,c{x}^{4}+66\,b{x}^{2}+154\,a}{231}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*x^(3/2)*(21*c*x^4+33*b*x^2+77*a)

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Maxima [A]  time = 0.945647, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2) + 2/3*a*x^(3/2)

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Fricas [A]  time = 1.22361, size = 62, normalized size = 2. \begin{align*} \frac{2}{231} \,{\left (21 \, c x^{5} + 33 \, b x^{3} + 77 \, a x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*c*x^5 + 33*b*x^3 + 77*a*x)*sqrt(x)

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Sympy [A]  time = 1.69798, size = 29, normalized size = 0.94 \begin{align*} \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7 + 2*c*x**(11/2)/11

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Giac [A]  time = 1.17217, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2) + 2/3*a*x^(3/2)