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

Optimal. Leaf size=34 \[ 2 a^2 \sqrt {x}+\frac {4}{5} a b x^{5/2}+\frac {2}{9} b^2 x^{9/2} \]

[Out]

4/5*a*b*x^(5/2)+2/9*b^2*x^(9/2)+2*a^2*x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 34, 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} \[ 2 a^2 \sqrt {x}+\frac {4}{5} a b x^{5/2}+\frac {2}{9} b^2 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] + (4*a*b*x^(5/2))/5 + (2*b^2*x^(9/2))/9

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.88 \[ \frac {2}{45} \sqrt {x} \left (45 a^2+18 a b x^2+5 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(45*a^2 + 18*a*b*x^2 + 5*b^2*x^4))/45

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fricas [A]  time = 1.03, size = 26, normalized size = 0.76 \[ \frac {2}{45} \, {\left (5 \, b^{2} x^{4} + 18 \, a b x^{2} + 45 \, a^{2}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*b^2*x^4 + 18*a*b*x^2 + 45*a^2)*sqrt(x)

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giac [A]  time = 0.63, size = 24, normalized size = 0.71 \[ \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + 2 \, a^{2} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^2*x^(9/2) + 4/5*a*b*x^(5/2) + 2*a^2*sqrt(x)

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maple [A]  time = 0.00, size = 27, normalized size = 0.79 \[ \frac {2 \left (5 b^{2} x^{4}+18 a b \,x^{2}+45 a^{2}\right ) \sqrt {x}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*x^(1/2)*(5*b^2*x^4+18*a*b*x^2+45*a^2)

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maxima [A]  time = 1.30, size = 24, normalized size = 0.71 \[ \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + 2 \, a^{2} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^2*x^(9/2) + 4/5*a*b*x^(5/2) + 2*a^2*sqrt(x)

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mupad [B]  time = 0.03, size = 26, normalized size = 0.76 \[ \frac {2\,\sqrt {x}\,\left (45\,a^2+18\,a\,b\,x^2+5\,b^2\,x^4\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^(1/2),x)

[Out]

(2*x^(1/2)*(45*a^2 + 5*b^2*x^4 + 18*a*b*x^2))/45

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sympy [A]  time = 0.75, size = 32, normalized size = 0.94 \[ 2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9} \]

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*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/2)/9

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