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

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

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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}{7} a x^{7/2}+\frac {2}{11} b x^{11/2}+\frac {2}{15} c x^{15/2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.81 \begin {gather*} \frac {2 x^{7/2} \left (165 a+105 b x^2+77 c x^4\right )}{1155} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 29, normalized size = 0.94 \begin {gather*} \frac {2 \left (165 a x^{7/2}+105 b x^{11/2}+77 c x^{15/2}\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(165*a*x^(7/2) + 105*b*x^(11/2) + 77*c*x^(15/2)))/1155

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fricas [A]  time = 1.38, size = 24, normalized size = 0.77 \begin {gather*} \frac {2}{1155} \, {\left (77 \, c x^{7} + 105 \, b x^{5} + 165 \, a x^{3}\right )} \sqrt {x} \end {gather*}

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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giac [A]  time = 0.16, size = 19, normalized size = 0.61 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 22, normalized size = 0.71 \begin {gather*} \frac {2 \left (77 c \,x^{4}+105 b \,x^{2}+165 a \right ) x^{\frac {7}{2}}}{1155} \end {gather*}

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 = 0.98, size = 19, normalized size = 0.61 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 4.29, size = 21, normalized size = 0.68 \begin {gather*} \frac {2\,x^{7/2}\,\left (77\,c\,x^4+105\,b\,x^2+165\,a\right )}{1155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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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sympy [A]  time = 6.72, size = 29, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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