3.1042 \(\int x^{3/2} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2} \]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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Rubi [A]  time = 0.0065908, 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} \[ \frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2 + c*x^4),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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 x^{3/2} \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a x^{3/2}+b x^{7/2}+c x^{11/2}\right ) \, dx\\ &=\frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0064249, size = 25, normalized size = 0.81 \[ \frac{2}{585} x^{5/2} \left (117 a+65 b x^2+45 c x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(117*a + 65*b*x^2 + 45*c*x^4))/585

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Maple [A]  time = 0.043, size = 22, normalized size = 0.7 \begin{align*}{\frac{90\,c{x}^{4}+130\,b{x}^{2}+234\,a}{585}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*x^(5/2)*(45*c*x^4+65*b*x^2+117*a)

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Maxima [A]  time = 0.964942, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2) + 2/5*a*x^(5/2)

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Fricas [A]  time = 1.30514, size = 66, normalized size = 2.13 \begin{align*} \frac{2}{585} \,{\left (45 \, c x^{6} + 65 \, b x^{4} + 117 \, a x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(45*c*x^6 + 65*b*x^4 + 117*a*x^2)*sqrt(x)

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Sympy [A]  time = 2.73244, size = 29, normalized size = 0.94 \begin{align*} \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{9}{2}}}{9} + \frac{2 c x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(9/2)/9 + 2*c*x**(13/2)/13

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Giac [A]  time = 1.15261, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2) + 2/5*a*x^(5/2)