3.267 \(\int \sqrt{x} (a+b x^2) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

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Rubi [A]  time = 0.0042427, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

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 (a+b x^2\right ) \, dx &=\int \left (a \sqrt{x}+b x^{5/2}\right ) \, dx\\ &=\frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0041893, size = 21, normalized size = 1. \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^2),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

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Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \begin{align*}{\frac{6\,b{x}^{2}+14\,a}{21}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*x^(3/2)*(3*b*x^2+7*a)

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Maxima [A]  time = 2.46075, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)

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Fricas [A]  time = 1.43141, size = 43, normalized size = 2.05 \begin{align*} \frac{2}{21} \,{\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*x^(1/2),x, algorithm="fricas")

[Out]

2/21*(3*b*x^3 + 7*a*x)*sqrt(x)

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Sympy [A]  time = 1.14283, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*x**(1/2),x)

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7

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Giac [A]  time = 1.40552, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)