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

Optimal. Leaf size=21 \[ \frac {2}{7} b x^{7/2}+\frac {2}{11} c x^{11/2} \]

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \begin {gather*} \frac {2}{7} b x^{7/2}+\frac {2}{11} c x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(b*x^2 + c*x^4),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{7} b x^{7/2}+\frac {2}{11} c x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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IntegrateAlgebraic [A]  time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{77} \left (11 b x^{7/2}+7 c x^{11/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(11*b*x^(7/2) + 7*c*x^(11/2)))/77

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fricas [A]  time = 0.58, size = 18, normalized size = 0.86 \begin {gather*} \frac {2}{77} \, {\left (7 \, c x^{5} + 11 \, b x^{3}\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),x, algorithm="fricas")

[Out]

2/77*(7*c*x^5 + 11*b*x^3)*sqrt(x)

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giac [A]  time = 0.21, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{11} \, c x^{\frac {11}{2}} + \frac {2}{7} \, b x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \begin {gather*} \frac {2 \left (7 c \,x^{2}+11 b \right ) x^{\frac {7}{2}}}{77} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(c*x^4+b*x^2),x)

[Out]

2/77*x^(7/2)*(7*c*x^2+11*b)

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maxima [A]  time = 1.33, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{11} \, c x^{\frac {11}{2}} + \frac {2}{7} \, b x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.71 \begin {gather*} \frac {2\,x^{7/2}\,\left (7\,c\,x^2+11\,b\right )}{77} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(b*x^2 + c*x^4),x)

[Out]

(2*x^(7/2)*(11*b + 7*c*x^2))/77

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sympy [A]  time = 1.73, size = 19, normalized size = 0.90 \begin {gather*} \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),x)

[Out]

2*b*x**(7/2)/7 + 2*c*x**(11/2)/11

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