3.930 \(\int (a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}} x^3 \, dx\)

Optimal. Leaf size=90 \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]

[Out]

((4*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(8*c) + (a*(c + d/x^2)^(3/2)*x^4)/(4*c) + (d*(4*b*c - a*d)*ArcTanh[Sqrt[c
+ d/x^2]/Sqrt[c]])/(8*c^(3/2))

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Rubi [A]  time = 0.0678375, antiderivative size = 90, 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 = {446, 78, 47, 63, 208} \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(8*c) + (a*(c + d/x^2)^(3/2)*x^4)/(4*c) + (d*(4*b*c - a*d)*ArcTanh[Sqrt[c
+ d/x^2]/Sqrt[c]])/(8*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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^3 \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) \sqrt{c+d x}}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{\left (2 b c-\frac{a d}{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^2} \, dx,x,\frac{1}{x^2}\right )}{4 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{(d (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{16 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{(4 b c-a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+\frac{d}{x^2}}\right )}{8 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}+\frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.104588, size = 100, normalized size = 1.11 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left (a \left (2 c x^2+d\right )+4 b c\right )+\sqrt{d} (4 b c-a d) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )\right )}{8 c^{3/2} \sqrt{\frac{c x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/d]*(4*b*c + a*(d + 2*c*x^2)) + Sqrt[d]*(4*b*c - a*d)*ArcSinh[(S
qrt[c]*x)/Sqrt[d]]))/(8*c^(3/2)*Sqrt[1 + (c*x^2)/d])

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Maple [A]  time = 0.007, size = 122, normalized size = 1.4 \begin{align*}{\frac{x}{8}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{3/2}xa-\sqrt{c}\sqrt{c{x}^{2}+d}xad+4\,{c}^{3/2}\sqrt{c{x}^{2}+d}xb-\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) a{d}^{2}+4\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) bcd \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{c}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((c*x^2+d)/x^2)^(1/2)*x*(2*c^(1/2)*(c*x^2+d)^(3/2)*x*a-c^(1/2)*(c*x^2+d)^(1/2)*x*a*d+4*c^(3/2)*(c*x^2+d)^(
1/2)*x*b-ln(c^(1/2)*x+(c*x^2+d)^(1/2))*a*d^2+4*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*b*c*d)/(c*x^2+d)^(1/2)/c^(3/2)

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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^3*(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37069, size = 431, normalized size = 4.79 \begin{align*} \left [-\frac{{\left (4 \, b c d - a d^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c^{2}}, -\frac{{\left (4 \, b c d - a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) -{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^3*(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*((4*b*c*d - a*d^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - 2*(2*a*c^2*x^4 + (
4*b*c^2 + a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^2, -1/8*((4*b*c*d - a*d^2)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c
*x^2 + d)/x^2)/(c*x^2 + d)) - (2*a*c^2*x^4 + (4*b*c^2 + a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^2]

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Sympy [A]  time = 28.8253, size = 144, normalized size = 1.6 \begin{align*} \frac{a c x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{3}{2}} x}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{3}{2}}} + \frac{b \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**3*(c+d/x**2)**(1/2),x)

[Out]

a*c*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*a*sqrt(d)*x**3/(8*sqrt(c*x**2/d + 1)) + a*d**(3/2)*x/(8*c*sqrt(c*x
**2/d + 1)) - a*d**2*asinh(sqrt(c)*x/sqrt(d))/(8*c**(3/2)) + b*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 + b*d*asinh(sqrt
(c)*x/sqrt(d))/(2*sqrt(c))

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Giac [A]  time = 1.14636, size = 142, normalized size = 1.58 \begin{align*} \frac{1}{8} \,{\left (2 \, a x^{2} \mathrm{sgn}\left (x\right ) + \frac{4 \, b c^{2} \mathrm{sgn}\left (x\right ) + a c d \mathrm{sgn}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + d} x - \frac{{\left (4 \, b c d \mathrm{sgn}\left (x\right ) - a d^{2} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{8 \, c^{\frac{3}{2}}} + \frac{{\left (4 \, b c d \log \left ({\left | d \right |}\right ) - a d^{2} \log \left ({\left | d \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{16 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^3*(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

1/8*(2*a*x^2*sgn(x) + (4*b*c^2*sgn(x) + a*c*d*sgn(x))/c^2)*sqrt(c*x^2 + d)*x - 1/8*(4*b*c*d*sgn(x) - a*d^2*sgn
(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^(3/2) + 1/16*(4*b*c*d*log(abs(d)) - a*d^2*log(abs(d)))*sgn(x)/c^
(3/2)