3.323 \(\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^5} \, dx\)

Optimal. Leaf size=174 \[ -\frac{b d^2 (4 a c+3 b) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right )}{8 c^{5/2} (a c+b)^{3/2}}+\frac{d (4 a c+5 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac{\left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 c^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.507267, antiderivative size = 218, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6722, 1975, 446, 96, 94, 93, 208} \[ -\frac{b d^2 (4 a c+3 b) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a \left (c+d x^2\right )+b}}\right )}{8 c^{5/2} (a c+b)^{3/2} \sqrt{a \left (c+d x^2\right )+b}}+\frac{d (4 a c+3 b) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{4 c x^4 (a c+b)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/(c + d*x^2)]/x^5,x]

[Out]

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

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{\sqrt{a+\frac{b}{c+d x^2}}}{x^5} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a \left (c+d x^2\right )}}{x^5 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{x^5 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x}}{x^3 \sqrt{c+d x}} \, dx,x,x^2\right )}{2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac{\left ((3 b+4 a c) d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x}}{x^2 \sqrt{c+d x}} \, dx,x,x^2\right )}{8 c (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{(3 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac{\left (b (3 b+4 a c) d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{16 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{(3 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac{\left (b (3 b+4 a c) d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{-c-(-b-a c) x^2} \, dx,x,\frac{\sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{8 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{(3 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac{b (3 b+4 a c) d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{b+a c} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{b+a \left (c+d x^2\right )}}\right )}{8 c^{5/2} (b+a c)^{3/2} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}

Mathematica [A]  time = 0.277231, size = 193, normalized size = 1.11 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (b d^2 x^4 (4 a c+3 b) \sqrt{c+d x^2} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a c+a d x^2+b}}\right )+\sqrt{c} \sqrt{a c+b} \left (c+d x^2\right ) \left (2 a c \left (c-d x^2\right )+b \left (2 c-3 d x^2\right )\right ) \sqrt{a \left (c+d x^2\right )+b}\right )}{8 c^{5/2} x^4 (a c+b)^{3/2} \sqrt{a \left (c+d x^2\right )+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/(c + d*x^2)]/x^5,x]

[Out]

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

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Maple [B]  time = 0.025, size = 923, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/(d*x^2+c))^(1/2)/x^5,x)

[Out]

1/16*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(-12*a^2*d^3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*
x^6*c*(a*c^2+b*c)^(3/2)-4*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a
*c^2+b*c)^(1/2)+2*b*c)/x^2)*x^4*a^3*b*c^5*d^2-10*a*d^3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*x^6*b*(
a*c^2+b*c)^(3/2)-11*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b
*c)^(1/2)+2*b*c)/x^2)*x^4*a^2*b^2*c^4*d^2-20*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a^2*c^2*d^2*x^4*(
a*c^2+b*c)^(3/2)-10*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b
*c)^(1/2)+2*b*c)/x^2)*x^4*a*b^3*c^3*d^2-28*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a*c*d^2*b*x^4*(a*c^
2+b*c)^(3/2)-3*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(
1/2)+2*b*c)/x^2)*x^4*b^4*c^2*d^2-10*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*b^2*d^2*x^4*(a*c^2+b*c)^(3
/2)+12*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*a*c*d*x^2*(a*c^2+b*c)^(3/2)+10*(a*d^2*x^4+2*a*c*d*x^2+b
*d*x^2+a*c^2+b*c)^(3/2)*b*d*x^2*(a*c^2+b*c)^(3/2)-4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c
)^(3/2)*a*c^2-4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(3/2)*b*c)/((d*x^2+c)*(a*d*x^2+a*c
+b))^(1/2)/c^3/(a*c+b)^2/x^4/(a*c^2+b*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/(d*x^2+c))^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.841, size = 1218, normalized size = 7. \begin{align*} \left [\frac{{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt{a c^{2} + b c} d^{2} x^{4} \log \left (\frac{{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \,{\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \,{\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} +{\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt{a c^{2} + b c} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \,{\left (2 \, a^{2} c^{5} -{\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} -{\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{32 \,{\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}, \frac{{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt{-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac{{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt{-a c^{2} - b c} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} c^{3} + 2 \, a b c^{2} +{\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) - 2 \,{\left (2 \, a^{2} c^{5} -{\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} -{\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{16 \,{\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/32*((4*a*b*c + 3*b^2)*sqrt(a*c^2 + b*c)*d^2*x^4*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a
*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c + b)*d^2*x^4 + 2*a*c^3 + (4*a*c^2 + 3
*b*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/x^4) - 4*(2*a^2*c^5 - (2*a^2*c
^3 + 5*a*b*c^2 + 3*b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b
)/(d*x^2 + c)))/((a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*x^4), 1/16*((4*a*b*c + 3*b^2)*sqrt(-a*c^2 - b*c)*d^2*x^4*arct
an(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt(-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*c^3
+ 2*a*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) - 2*(2*a^2*c^5 - (2*a^2*c^3 + 5*a*b*c^2 + 3*b^2*c)*d^2*x^4 +
4*a*b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^2*c^5 + 2*a*b*c^
4 + b^2*c^3)*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a c + a d x^{2} + b}{c + d x^{2}}}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(1/2)/x**5,x)

[Out]

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{d x^{2} + c}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))/x^5, x)