3.278 \(\int \frac {(a+b x^2)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {2 a^2}{3 x^{3/2}}+4 a b \sqrt {x}+\frac {2}{5} b^2 x^{5/2} \]

[Out]

-2/3*a^2/x^(3/2)+2/5*b^2*x^(5/2)+4*a*b*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} \[ -\frac {2 a^2}{3 x^{3/2}}+4 a b \sqrt {x}+\frac {2}{5} b^2 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) + 4*a*b*Sqrt[x] + (2*b^2*x^(5/2))/5

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.88 \[ \frac {2 \left (-5 a^2+30 a b x^2+3 b^2 x^4\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^(5/2),x]

[Out]

(2*(-5*a^2 + 30*a*b*x^2 + 3*b^2*x^4))/(15*x^(3/2))

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fricas [A]  time = 0.68, size = 26, normalized size = 0.76 \[ \frac {2 \, {\left (3 \, b^{2} x^{4} + 30 \, a b x^{2} - 5 \, a^{2}\right )}}{15 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^2*x^4 + 30*a*b*x^2 - 5*a^2)/x^(3/2)

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giac [A]  time = 0.64, size = 24, normalized size = 0.71 \[ \frac {2}{5} \, b^{2} x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2),x, algorithm="giac")

[Out]

2/5*b^2*x^(5/2) + 4*a*b*sqrt(x) - 2/3*a^2/x^(3/2)

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maple [A]  time = 0.00, size = 27, normalized size = 0.79 \[ -\frac {2 \left (-3 b^{2} x^{4}-30 a b \,x^{2}+5 a^{2}\right )}{15 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(5/2),x)

[Out]

-2/15*(-3*b^2*x^4-30*a*b*x^2+5*a^2)/x^(3/2)

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maxima [A]  time = 1.35, size = 24, normalized size = 0.71 \[ \frac {2}{5} \, b^{2} x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(5/2),x, algorithm="maxima")

[Out]

2/5*b^2*x^(5/2) + 4*a*b*sqrt(x) - 2/3*a^2/x^(3/2)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.76 \[ \frac {-10\,a^2+60\,a\,b\,x^2+6\,b^2\,x^4}{15\,x^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^(5/2),x)

[Out]

(6*b^2*x^4 - 10*a^2 + 60*a*b*x^2)/(15*x^(3/2))

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sympy [A]  time = 1.11, size = 32, normalized size = 0.94 \[ - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(5/2),x)

[Out]

-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5

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