3.277 \(\int \frac {(a+b x^2)^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} b^2 x^{7/2} \]

[Out]

4/3*a*b*x^(3/2)+2/7*b^2*x^(7/2)-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} \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} b^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*b^2*x^(7/2))/7

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.88 \[ \frac {2 \left (-21 a^2+14 a b x^2+3 b^2 x^4\right )}{21 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^(3/2),x]

[Out]

(2*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*Sqrt[x])

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fricas [A]  time = 0.83, size = 26, normalized size = 0.76 \[ \frac {2 \, {\left (3 \, b^{2} x^{4} + 14 \, a b x^{2} - 21 \, a^{2}\right )}}{21 \, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*b^2*x^4 + 14*a*b*x^2 - 21*a^2)/sqrt(x)

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giac [A]  time = 0.63, size = 24, normalized size = 0.71 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.79 \[ -\frac {2 \left (-3 b^{2} x^{4}-14 a b \,x^{2}+21 a^{2}\right )}{21 \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(3/2),x)

[Out]

-2/21*(-3*b^2*x^4-14*a*b*x^2+21*a^2)/x^(1/2)

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maxima [A]  time = 1.33, size = 24, normalized size = 0.71 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 26, normalized size = 0.76 \[ \frac {-42\,a^2+28\,a\,b\,x^2+6\,b^2\,x^4}{21\,\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^(3/2),x)

[Out]

(6*b^2*x^4 - 42*a^2 + 28*a*b*x^2)/(21*x^(1/2))

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sympy [A]  time = 0.94, size = 32, normalized size = 0.94 \[ - \frac {2 a^{2}}{\sqrt {x}} + \frac {4 a b x^{\frac {3}{2}}}{3} + \frac {2 b^{2} x^{\frac {7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(3/2),x)

[Out]

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

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