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

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

[Out]

(2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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Rubi [A]  time = 0.0047368, antiderivative size = 21, normalized size of antiderivative = 1., 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} \[ \frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

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*}

Mathematica [A]  time = 0.0077376, size = 21, normalized size = 1. \[ \frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

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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Maple [A]  time = 0.044, size = 16, normalized size = 0.8 \begin{align*}{\frac{14\,c{x}^{2}+22\,b}{77}{x}^{{\frac{7}{2}}}} \end{align*}

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 = 0.965287, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} \end{align*}

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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Fricas [A]  time = 1.25043, size = 47, normalized size = 2.24 \begin{align*} \frac{2}{77} \,{\left (7 \, c x^{5} + 11 \, b x^{3}\right )} \sqrt{x} \end{align*}

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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Sympy [A]  time = 1.6907, size = 19, normalized size = 0.9 \begin{align*} \frac{2 b x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{11}{2}}}{11} \end{align*}

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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Giac [A]  time = 1.11819, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} \end{align*}

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)