3.1041 \(\int x^{5/2} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11 + (2*c*x^(15/2))/15

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Rubi [A]  time = 0.0061682, 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} \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2 + c*x^4),x]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11 + (2*c*x^(15/2))/15

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^{5/2} \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a x^{5/2}+b x^{9/2}+c x^{13/2}\right ) \, dx\\ &=\frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.006172, size = 25, normalized size = 0.81 \[ \frac{2 x^{7/2} \left (165 a+105 b x^2+77 c x^4\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2 + c*x^4),x]

[Out]

(2*x^(7/2)*(165*a + 105*b*x^2 + 77*c*x^4))/1155

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Maple [A]  time = 0.045, size = 22, normalized size = 0.7 \begin{align*}{\frac{154\,c{x}^{4}+210\,b{x}^{2}+330\,a}{1155}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+b*x^2+a),x)

[Out]

2/1155*x^(7/2)*(77*c*x^4+105*b*x^2+165*a)

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Maxima [A]  time = 0.990398, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

2/15*c*x^(15/2) + 2/11*b*x^(11/2) + 2/7*a*x^(7/2)

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Fricas [A]  time = 1.22206, size = 69, normalized size = 2.23 \begin{align*} \frac{2}{1155} \,{\left (77 \, c x^{7} + 105 \, b x^{5} + 165 \, a x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

2/1155*(77*c*x^7 + 105*b*x^5 + 165*a*x^3)*sqrt(x)

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Sympy [A]  time = 5.76865, size = 29, normalized size = 0.94 \begin{align*} \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{11}{2}}}{11} + \frac{2 c x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**4+b*x**2+a),x)

[Out]

2*a*x**(7/2)/7 + 2*b*x**(11/2)/11 + 2*c*x**(15/2)/15

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Giac [A]  time = 1.1642, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

2/15*c*x^(15/2) + 2/11*b*x^(11/2) + 2/7*a*x^(7/2)