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

Optimal. Leaf size=90 \[ \frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{8 c^2}-\frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{5/2}}+\frac{a x^4 \sqrt{c+\frac{d}{x^2}}}{4 c} \]

[Out]

((4*b*c - 3*a*d)*Sqrt[c + d/x^2]*x^2)/(8*c^2) + (a*Sqrt[c + d/x^2]*x^4)/(4*c) - (d*(4*b*c - 3*a*d)*ArcTanh[Sqr
t[c + d/x^2]/Sqrt[c]])/(8*c^(5/2))

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Rubi [A]  time = 0.0673245, 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, 51, 63, 208} \[ \frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{8 c^2}-\frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{5/2}}+\frac{a x^4 \sqrt{c+\frac{d}{x^2}}}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*c - 3*a*d)*Sqrt[c + d/x^2]*x^2)/(8*c^2) + (a*Sqrt[c + d/x^2]*x^4)/(4*c) - (d*(4*b*c - 3*a*d)*ArcTanh[Sqr
t[c + d/x^2]/Sqrt[c]])/(8*c^(5/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 \frac{\left (a+\frac{b}{x^2}\right ) x^3}{\sqrt{c+\frac{d}{x^2}}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b x}{x^3 \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \sqrt{c+\frac{d}{x^2}} x^4}{4 c}-\frac{\left (2 b c-\frac{3 a d}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{4 c}\\ &=\frac{(4 b c-3 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^4}{4 c}+\frac{(d (4 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{16 c^2}\\ &=\frac{(4 b c-3 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^4}{4 c}+\frac{(4 b c-3 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^2}\\ &=\frac{(4 b c-3 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^4}{4 c}-\frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0791951, size = 95, normalized size = 1.06 \[ \frac{\sqrt{c} x \left (c x^2+d\right ) \left (2 a c x^2-3 a d+4 b c\right )+d \sqrt{c x^2+d} (3 a d-4 b c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right )}{8 c^{5/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*x*(d + c*x^2)*(4*b*c - 3*a*d + 2*a*c*x^2) + d*(-4*b*c + 3*a*d)*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sq
rt[d + c*x^2]])/(8*c^(5/2)*Sqrt[c + d/x^2]*x)

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Maple [A]  time = 0.01, size = 129, normalized size = 1.4 \begin{align*}{\frac{1}{8\,x}\sqrt{c{x}^{2}+d} \left ( 2\,{c}^{5/2}\sqrt{c{x}^{2}+d}{x}^{3}a-3\,{c}^{3/2}\sqrt{c{x}^{2}+d}xad+4\,{c}^{5/2}\sqrt{c{x}^{2}+d}xb+3\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) ac{d}^{2}-4\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) b{c}^{2}d \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{c}^{-{\frac{7}{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)^(1/2)*(2*c^(5/2)*(c*x^2+d)^(1/2)*x^3*a-3*c^(3/2)*(c*x^2+d)^(1/2)*x*a*d+4*c^(5/2)*(c*x^2+d)^(1/2)
*x*b+3*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*a*c*d^2-4*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*b*c^2*d)/((c*x^2+d)/x^2)^(1/2)/x/
c^(7/2)

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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 = 1.3239, size = 440, normalized size = 4.89 \begin{align*} \left [-\frac{{\left (4 \, b c d - 3 \, 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} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c^{3}}, \frac{{\left (4 \, b c d - 3 \, 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} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c^{3}}\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 - 3*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 - 3*a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^3, 1/8*((4*b*c*d - 3*a*d^2)*sqrt(-c)*arctan(sqrt(-c)*x^2*sq
rt((c*x^2 + d)/x^2)/(c*x^2 + d)) + (2*a*c^2*x^4 + (4*b*c^2 - 3*a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^3]

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Sympy [A]  time = 32.2973, size = 150, normalized size = 1.67 \begin{align*} \frac{a x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a \sqrt{d} x^{3}}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 a d^{\frac{3}{2}} x}{8 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{5}{2}}} + \frac{b \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2 c} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 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)

[Out]

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

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Giac [B]  time = 1.17982, size = 220, normalized size = 2.44 \begin{align*} \frac{1}{8} \, d^{2}{\left (\frac{{\left (4 \, b c - 3 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2} d} - \frac{4 \, b c^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - 5 \, a c d \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{4 \,{\left (c x^{2} + d\right )} b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}} + \frac{3 \,{\left (c x^{2} + d\right )} a d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )}^{2} c^{2} d}\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="giac")

[Out]

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