3.324 \(\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^7} \, dx\)

Optimal. Leaf size=265 \[ -\frac{d^2 \left (8 a^2 c^2+20 a b c+11 b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 c^3 x^2 (a c+b)^2}+\frac{b d^3 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right )}{16 c^{7/2} (a c+b)^{5/2}}+\frac{d (4 a c+3 b) \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c^3 x^4 (a c+b)}-\frac{\left (c+d x^2\right )^3 \left (\frac{a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{6 c^2 x^6 (a c+b)} \]

[Out]

-((11*b^2 + 20*a*b*c + 8*a^2*c^2)*d^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(16*c^3*(b + a*c)^2*x
^2) + ((3*b + 4*a*c)*d*(c + d*x^2)^2*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(8*c^3*(b + a*c)*x^4) - ((c + d*x^
2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(3/2))/(6*c^2*(b + a*c)*x^6) + (b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*d^3*Ar
cTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(16*c^(7/2)*(b + a*c)^(5/2))

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Rubi [A]  time = 0.606903, antiderivative size = 271, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {6722, 1975, 446, 99, 151, 12, 93, 208} \[ \frac{b d^3 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a \left (c+d x^2\right )+b}}\right )}{16 c^{7/2} (a c+b)^{5/2} \sqrt{a \left (c+d x^2\right )+b}}-\frac{d^2 (2 a c+5 b) (4 a c+3 b) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{48 c^3 x^2 (a c+b)^2}+\frac{d (4 a c+5 b) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 x^4 (a c+b)}-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{6 c x^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/(c + d*x^2)]/x^7,x]

[Out]

-((c + d*x^2)*Sqrt[a + b/(c + d*x^2)])/(6*c*x^6) + ((5*b + 4*a*c)*d*(c + d*x^2)*Sqrt[a + b/(c + d*x^2)])/(24*c
^2*(b + a*c)*x^4) - ((5*b + 2*a*c)*(3*b + 4*a*c)*d^2*(c + d*x^2)*Sqrt[a + b/(c + d*x^2)])/(48*c^3*(b + a*c)^2*
x^2) + (b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*d^3*Sqrt[c + d*x^2]*Sqrt[a + b/(c + d*x^2)]*ArcTanh[(Sqrt[b + a*c]*Sq
rt[c + d*x^2])/(Sqrt[c]*Sqrt[b + a*(c + d*x^2)])])/(16*c^(7/2)*(b + a*c)^(5/2)*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 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^7} \, 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^7 \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^7 \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^4 \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}}}{6 c x^6}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2} (5 b+4 a c) d-2 a d^2 x}{x^3 \sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{6 c \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{6 c x^6}+\frac{(5 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{4} (5 b+2 a c) (3 b+4 a c) d^2-\frac{1}{2} a (5 b+4 a c) d^3 x}{x^2 \sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{12 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{6 c x^6}+\frac{(5 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac{(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int -\frac{3 b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3}{8 x \sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{12 c^3 (b+a c)^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}}}{6 c x^6}+\frac{(5 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac{(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}-\frac{\left (b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \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 )}{32 c^3 (b+a c)^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}}}{6 c x^6}+\frac{(5 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac{(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}-\frac{\left (b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \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 )}{16 c^3 (b+a c)^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}}}{6 c x^6}+\frac{(5 b+4 a c) d \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{24 c^2 (b+a c) x^4}-\frac{(5 b+2 a c) (3 b+4 a c) d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{48 c^3 (b+a c)^2 x^2}+\frac{b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \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 )}{16 c^{7/2} (b+a c)^{5/2} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}

Mathematica [A]  time = 0.459011, size = 259, normalized size = 0.98 \[ \frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (3 b d^3 x^6 \left (8 a^2 c^2+12 a b c+5 b^2\right ) \sqrt{c+d x^2} \tanh ^{-1}\left (\frac{\sqrt{a c+b} \sqrt{c+d x^2}}{\sqrt{c} \sqrt{a c+a d x^2+b}}\right )-\sqrt{c} \sqrt{a c+b} \left (c+d x^2\right ) \sqrt{a \left (c+d x^2\right )+b} \left (8 a^2 c^2 \left (c^2-c d x^2+d^2 x^4\right )+2 a b c \left (8 c^2-9 c d x^2+13 d^2 x^4\right )+b^2 \left (8 c^2-10 c d x^2+15 d^2 x^4\right )\right )\right )}{48 c^{7/2} x^6 (a c+b)^{5/2} \sqrt{a \left (c+d x^2\right )+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/(c + d*x^2)]/x^7,x]

[Out]

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-(Sqrt[c]*Sqrt[b + a*c]*(c + d*x^2)*Sqrt[b + a*(c + d*x^2)]*(8*a^2*c^2
*(c^2 - c*d*x^2 + d^2*x^4) + 2*a*b*c*(8*c^2 - 9*c*d*x^2 + 13*d^2*x^4) + b^2*(8*c^2 - 10*c*d*x^2 + 15*d^2*x^4))
) + 3*b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*d^3*x^6*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])]))/(48*c^(7/2)*(b + a*c)^(5/2)*x^6*Sqrt[b + a*(c + d*x^2)])

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Maple [B]  time = 0.028, size = 1518, normalized size = 5.7 \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^7,x)

[Out]

-1/96*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(-24*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^6*a^5*b*c^8*d^3-96*a^3*d^4*(a*d^2*x^4+2*a*c*d*x^2+b
*d*x^2+a*c^2+b*c)^(1/2)*x^8*c^2*(a*c^2+b*c)^(5/2)-108*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^6*a^4*b^2*c^7*d^3-156*a^2*d^4*(a*d^2*x^4+2*a*c*d*x^2
+b*d*x^2+a*c^2+b*c)^(1/2)*x^8*c*b*(a*c^2+b*c)^(5/2)-195*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^6*a^3*b^3*c^6*d^3-66*a*d^4*(a*d^2*x^4+2*a*c*d*x^2+
b*d*x^2+a*c^2+b*c)^(1/2)*x^8*b^2*(a*c^2+b*c)^(5/2)-144*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a^3*c^3
*d^3*(a*c^2+b*c)^(5/2)*x^6-177*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^6*a^2*b^4*c^5*d^3-324*(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^3*b*(a*c^2+b*c)^(5/2)*x^6-81*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^6*a*b^5*c^4*d^3-252*(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^3*b^2*(a*c^2+b*c)^(5/2)*x^6-15*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^6*b^6*c^3*d^3+96*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*a^2*c^2*
d^2*(a*c^2+b*c)^(5/2)*x^4-66*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*b^3*d^3*(a*c^2+b*c)^(5/2)*x^6+156
*(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^2*b*(a*c^2+b*c)^(5/2)*x^4+66*(a*d^2*x^4+2*a*c*d*x^2+b*d
*x^2+a*c^2+b*c)^(3/2)*b^2*d^2*(a*c^2+b*c)^(5/2)*x^4-48*(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)^(5/2)*x^2*a^2*c^3*d-84*(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)^(5/2)*x^2*a*b*c^2*d-36
*(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)^(5/2)*x^2*b^2*c*d+16*(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)^(5/2)*a^2*c^4+32*(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)^(
5/2)*a*b*c^3+16*(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)^(5/2)*b^2*c^2)/((d*x^2+c)*(a*d*x^2
+a*c+b))^(1/2)/c^4/(a*c+b)^3/x^6/(a*c^2+b*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/(d*x^2+c))^(1/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.93521, size = 1601, normalized size = 6.04 \begin{align*} \left [\frac{3 \,{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt{a c^{2} + b c} d^{3} x^{6} \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 (8 \, a^{3} c^{7} +{\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} +{\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \,{\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{192 \,{\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}, -\frac{3 \,{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt{-a c^{2} - b c} d^{3} x^{6} \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 (8 \, a^{3} c^{7} +{\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} +{\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \,{\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{96 \,{\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^7,x, algorithm="fricas")

[Out]

[1/192*(3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*sqrt(a*c^2 + b*c)*d^3*x^6*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*(
8*a^3*c^7 + (8*a^3*c^4 + 34*a^2*b*c^3 + 41*a*b^2*c^2 + 15*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^3
*c^4 + (8*a^2*b*c^4 + 13*a*b^2*c^3 + 5*b^3*c^2)*d^2*x^4 - 2*(a^2*b*c^5 + 2*a*b^2*c^4 + b^3*c^3)*d*x^2)*sqrt((a
*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^7 + 3*a^2*b*c^6 + 3*a*b^2*c^5 + b^3*c^4)*x^6), -1/96*(3*(8*a^2*b*c^2 +
12*a*b^2*c + 5*b^3)*sqrt(-a*c^2 - b*c)*d^3*x^6*arctan(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*(8*a^
3*c^7 + (8*a^3*c^4 + 34*a^2*b*c^3 + 41*a*b^2*c^2 + 15*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^3*c^4
+ (8*a^2*b*c^4 + 13*a*b^2*c^3 + 5*b^3*c^2)*d^2*x^4 - 2*(a^2*b*c^5 + 2*a*b^2*c^4 + b^3*c^3)*d*x^2)*sqrt((a*d*x
^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^7 + 3*a^2*b*c^6 + 3*a*b^2*c^5 + b^3*c^4)*x^6)]

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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^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(1/2)/x**7,x)

[Out]

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**7, 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^{7}}\,{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^7,x, algorithm="giac")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))/x^7, x)