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

Optimal. Leaf size=66 \[ \frac{a x^3 \left (c+\frac{d}{x^2}\right )^{3/2}}{3 c}+b x \sqrt{c+\frac{d}{x^2}}-b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0366468, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {451, 242, 277, 217, 206} \[ \frac{a x^3 \left (c+\frac{d}{x^2}\right )^{3/2}}{3 c}+b x \sqrt{c+\frac{d}{x^2}}-b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 451

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[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0470653, size = 84, normalized size = 1.27 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c x^2+d} \left (a \left (c x^2+d\right )+3 b c\right )-3 b c \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right )\right )}{3 c \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 83, normalized size = 1.3 \begin{align*} -{\frac{x}{3\,c}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 3\,\sqrt{d}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) bc-a \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c{x}^{2}+d}bc \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}} \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*((c*x^2+d)/x^2)^(1/2)*x*(3*d^(1/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*b*c-a*(c*x^2+d)^(3/2)-3*(c*x^2+d)^
(1/2)*b*c)/(c*x^2+d)^(1/2)/c

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.35655, size = 365, normalized size = 5.53 \begin{align*} \left [\frac{3 \, b c \sqrt{d} \log \left (-\frac{c x^{2} - 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \,{\left (a c x^{3} +{\left (3 \, b c + a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, c}, \frac{3 \, b c \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) +{\left (a c x^{3} +{\left (3 \, b c + a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, c}\right ] \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]

[1/6*(3*b*c*sqrt(d)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2) + 2*(a*c*x^3 + (3*b*c + a*d)*x
)*sqrt((c*x^2 + d)/x^2))/c, 1/3*(3*b*c*sqrt(-d)*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + (a*c*x^
3 + (3*b*c + a*d)*x)*sqrt((c*x^2 + d)/x^2))/c]

________________________________________________________________________________________

Sympy [A]  time = 3.04817, size = 107, normalized size = 1.62 \begin{align*} \frac{a \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3} + \frac{a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} + \frac{b \sqrt{c} x}{\sqrt{1 + \frac{d}{c x^{2}}}} - b \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )} + \frac{b d}{\sqrt{c} x \sqrt{1 + \frac{d}{c x^{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)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.16584, size = 157, normalized size = 2.38 \begin{align*} \frac{b d \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-d}} - \frac{{\left (3 \, b c d \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + 3 \, b c \sqrt{-d} \sqrt{d} + a \sqrt{-d} d^{\frac{3}{2}}\right )} \mathrm{sgn}\left (x\right )}{3 \, c \sqrt{-d}} + \frac{{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{2} \mathrm{sgn}\left (x\right ) + 3 \, \sqrt{c x^{2} + d} b c^{3} \mathrm{sgn}\left (x\right )}{3 \, c^{3}} \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]

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