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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{3} a x^{3/2}+\frac {2}{7} b x^{7/2}+\frac {2}{11} c x^{11/2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.81 \begin {gather*} \frac {2}{231} x^{3/2} \left (77 a+33 b x^2+21 c x^4\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 29, normalized size = 0.94 \begin {gather*} \frac {2}{231} \left (77 a x^{3/2}+33 b x^{7/2}+21 c x^{11/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(77*a*x^(3/2) + 33*b*x^(7/2) + 21*c*x^(11/2)))/231

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fricas [A]  time = 2.21, size = 22, normalized size = 0.71 \begin {gather*} \frac {2}{231} \, {\left (21 \, c x^{5} + 33 \, b x^{3} + 77 \, a x\right )} \sqrt {x} \end {gather*}

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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giac [A]  time = 0.17, size = 19, normalized size = 0.61 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 22, normalized size = 0.71 \begin {gather*} \frac {2 \left (21 c \,x^{4}+33 b \,x^{2}+77 a \right ) x^{\frac {3}{2}}}{231} \end {gather*}

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 = 1.01, size = 19, normalized size = 0.61 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 21, normalized size = 0.68 \begin {gather*} \frac {2\,x^{3/2}\,\left (21\,c\,x^4+33\,b\,x^2+77\,a\right )}{231} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(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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sympy [A]  time = 2.10, size = 29, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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