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]

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

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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} \[ \frac {2}{5} a x^{5/2}+\frac {2}{9} b x^{9/2}+\frac {2}{13} c x^{13/2} \]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.80, size = 24, normalized size = 0.77 \[ \frac {2}{585} \, {\left (45 \, c x^{6} + 65 \, b x^{4} + 117 \, a x^{2}\right )} \sqrt {x} \]

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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giac [A]  time = 0.21, size = 19, normalized size = 0.61 \[ \frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]

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)

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maple [A]  time = 0.00, size = 22, normalized size = 0.71 \[ \frac {2 \left (45 c \,x^{4}+65 b \,x^{2}+117 a \right ) x^{\frac {5}{2}}}{585} \]

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 = 1.04, size = 19, normalized size = 0.61 \[ \frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]

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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mupad [B]  time = 0.04, size = 21, normalized size = 0.68 \[ \frac {2\,x^{5/2}\,\left (45\,c\,x^4+65\,b\,x^2+117\,a\right )}{585} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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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sympy [A]  time = 2.67, size = 29, normalized size = 0.94 \[ \frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 b x^{\frac {9}{2}}}{9} + \frac {2 c x^{\frac {13}{2}}}{13} \]

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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