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]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.82, size = 27, normalized size = 0.75 \[ \frac {2}{231} \, {\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt {x} \]

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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giac [A]  time = 0.58, size = 24, normalized size = 0.67 \[ \frac {2}{11} \, b^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 27, normalized size = 0.75 \[ \frac {2 \left (21 b^{2} x^{4}+66 a b \,x^{2}+77 a^{2}\right ) x^{\frac {3}{2}}}{231} \]

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 = 1.34, size = 24, normalized size = 0.67 \[ \frac {2}{11} \, b^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]

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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mupad [B]  time = 0.04, size = 26, normalized size = 0.72 \[ \frac {2\,x^{3/2}\,\left (77\,a^2+66\,a\,b\,x^2+21\,b^2\,x^4\right )}{231} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x^2)^2,x)

[Out]

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

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sympy [A]  time = 1.70, size = 34, normalized size = 0.94 \[ \frac {2 a^{2} x^{\frac {3}{2}}}{3} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {2 b^{2} x^{\frac {11}{2}}}{11} \]

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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