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

Optimal. Leaf size=59 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0402656, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 78, 63, 208} \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0367489, size = 79, normalized size = 1.34 \[ \frac{\sqrt{c x^2+d} (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right )+a \sqrt{c} x \left (c x^2+d\right )}{2 c^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 90, normalized size = 1.5 \begin{align*}{\frac{1}{2\,x}\sqrt{c{x}^{2}+d} \left ({c}^{{\frac{3}{2}}}\sqrt{c{x}^{2}+d}xa+2\,b\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) acd \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{c}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2+d)^(1/2)*(c^(3/2)*(c*x^2+d)^(1/2)*x*a+2*b*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^2-ln(c^(1/2)*x+(c*x^2+d)^
(1/2))*a*c*d)/((c*x^2+d)/x^2)^(1/2)/x/c^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.42454, size = 335, normalized size = 5.68 \begin{align*} \left [\frac{2 \, a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right )}{4 \, c^{2}}, \frac{a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{2 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*c*x^2*sqrt((c*x^2 + d)/x^2) - (2*b*c - a*d)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^
2) - d))/c^2, 1/2*(a*c*x^2*sqrt((c*x^2 + d)/x^2) - (2*b*c - a*d)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)
/x^2)/(c*x^2 + d)))/c^2]

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Sympy [A]  time = 38.1443, size = 66, normalized size = 1.12 \begin{align*} \frac{a \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2 c} - \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17194, size = 107, normalized size = 1.81 \begin{align*} -\frac{1}{2} \, d{\left (\frac{a \sqrt{\frac{c x^{2} + d}{x^{2}}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )} c} + \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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