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

Optimal. Leaf size=36 \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{11} b^2 x^{11/2} \]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*b*x^(7/2))/7 + (2*b^2*x^(11/2))/11

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Rubi [A]  time = 0.0082896, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{11} b^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2)^2,x]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*b*x^(7/2))/7 + (2*b^2*x^(11/2))/11

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sqrt{x} \left (a+b x^2\right )^2 \, dx &=\int \left (a^2 \sqrt{x}+2 a b x^{5/2}+b^2 x^{9/2}\right ) \, dx\\ &=\frac{2}{3} a^2 x^{3/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{11} b^2 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.007235, size = 30, normalized size = 0.83 \[ \frac{2}{231} x^{3/2} \left (77 a^2+66 a b x^2+21 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^2)^2,x]

[Out]

(2*x^(3/2)*(77*a^2 + 66*a*b*x^2 + 21*b^2*x^4))/231

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{42\,{b}^{2}{x}^{4}+132\,ab{x}^{2}+154\,{a}^{2}}{231}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*x^(3/2)*(21*b^2*x^4+66*a*b*x^2+77*a^2)

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Maxima [A]  time = 2.45789, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b^2*x^(11/2) + 4/7*a*b*x^(7/2) + 2/3*a^2*x^(3/2)

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Fricas [A]  time = 1.30838, size = 70, normalized size = 1.94 \begin{align*} \frac{2}{231} \,{\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*b^2*x^5 + 66*a*b*x^3 + 77*a^2*x)*sqrt(x)

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Sympy [A]  time = 1.62027, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 a b x^{\frac{7}{2}}}{7} + \frac{2 b^{2} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*b*x**(7/2)/7 + 2*b**2*x**(11/2)/11

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Giac [A]  time = 3.11783, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*b^2*x^(11/2) + 4/7*a*b*x^(7/2) + 2/3*a^2*x^(3/2)