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

Optimal. Leaf size=51 \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]

[Out]

((3*b*c - 2*a*d)*Sqrt[c + d/x^2]*x)/(3*c^2) + (a*Sqrt[c + d/x^2]*x^3)/(3*c)

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Rubi [A]  time = 0.0202861, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 191} \[ \frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]

[Out]

((3*b*c - 2*a*d)*Sqrt[c + d/x^2]*x)/(3*c^2) + (a*Sqrt[c + d/x^2]*x^3)/(3*c)

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) x^2}{\sqrt{c+\frac{d}{x^2}}} \, dx &=\frac{a \sqrt{c+\frac{d}{x^2}} x^3}{3 c}+\frac{(3 b c-2 a d) \int \frac{1}{\sqrt{c+\frac{d}{x^2}}} \, dx}{3 c}\\ &=\frac{(3 b c-2 a d) \sqrt{c+\frac{d}{x^2}} x}{3 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^3}{3 c}\\ \end{align*}

Mathematica [A]  time = 0.0278078, size = 34, normalized size = 0.67 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (a c x^2-2 a d+3 b c\right )}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]

[Out]

(Sqrt[c + d/x^2]*x*(3*b*c - 2*a*d + a*c*x^2))/(3*c^2)

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Maple [A]  time = 0.003, size = 44, normalized size = 0.9 \begin{align*}{\frac{ \left ( a{x}^{2}c-2\,ad+3\,bc \right ) \left ( c{x}^{2}+d \right ) }{3\,x{c}^{2}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/x*(a*c*x^2-2*a*d+3*b*c)*(c*x^2+d)/((c*x^2+d)/x^2)^(1/2)/c^2

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Maxima [A]  time = 0.9383, size = 66, normalized size = 1.29 \begin{align*} \frac{b \sqrt{c + \frac{d}{x^{2}}} x}{c} + \frac{{\left ({\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 3 \, \sqrt{c + \frac{d}{x^{2}}} d x\right )} a}{3 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*sqrt(c + d/x^2)*x/c + 1/3*((c + d/x^2)^(3/2)*x^3 - 3*sqrt(c + d/x^2)*d*x)*a/c^2

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Fricas [A]  time = 1.28396, size = 82, normalized size = 1.61 \begin{align*} \frac{{\left (a c x^{3} +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*c*x^3 + (3*b*c - 2*a*d)*x)*sqrt((c*x^2 + d)/x^2)/c^2

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Sympy [A]  time = 2.37479, size = 70, normalized size = 1.37 \begin{align*} \frac{a \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} - \frac{2 a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{2}} + \frac{b \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/(3*c) - 2*a*d**(3/2)*sqrt(c*x**2/d + 1)/(3*c**2) + b*sqrt(d)*sqrt(c*x**2/d +
1)/c

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{2}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^2)*x^2/sqrt(c + d/x^2), x)