3.251 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} (c+\frac{d}{x})^3} \, dx\)
Optimal. Leaf size=250 \[ -\frac{d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}-\frac{(6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^4}+\frac{d \sqrt{a+\frac{b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{d \sqrt{a+\frac{b}{x}} (2 b c-3 a d)}{2 a c^2 \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )^2} \]
[Out]
(d*(2*b*c - 3*a*d)*Sqrt[a + b/x])/(2*a*c^2*(b*c - a*d)*(c + d/x)^2) + (d*(b*c - 4*a*d)*(4*b*c - 3*a*d)*Sqrt[a
+ b/x])/(4*a*c^3*(b*c - a*d)^2*(c + d/x)) + (Sqrt[a + b/x]*x)/(a*c*(c + d/x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*a*
b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c - a*d)^(5/2)) - ((b*c + 6*a*d
)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(3/2)*c^4)
________________________________________________________________________________________
Rubi [A] time = 0.400077, antiderivative size = 250, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{375, 103, 151, 156, 63, 208, 205} \[ -\frac{d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}-\frac{(6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^4}+\frac{d \sqrt{a+\frac{b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{d \sqrt{a+\frac{b}{x}} (2 b c-3 a d)}{2 a c^2 \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[a + b/x]*(c + d/x)^3),x]
[Out]
(d*(2*b*c - 3*a*d)*Sqrt[a + b/x])/(2*a*c^2*(b*c - a*d)*(c + d/x)^2) + (d*(b*c - 4*a*d)*(4*b*c - 3*a*d)*Sqrt[a
+ b/x])/(4*a*c^3*(b*c - a*d)^2*(c + d/x)) + (Sqrt[a + b/x]*x)/(a*c*(c + d/x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*a*
b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c - a*d)^(5/2)) - ((b*c + 6*a*d
)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(3/2)*c^4)
Rule 375
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Rule 103
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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
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 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x} (c+d x)^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (b c+6 a d)+\frac{5 b d x}{2}}{x \sqrt{a+b x} (c+d x)^3} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{d (2 b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac{d}{x}\right )^2}+\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-(b c-a d) (b c+6 a d)-\frac{3}{2} b d (2 b c-3 a d) x}{x \sqrt{a+b x} (c+d x)^2} \, dx,x,\frac{1}{x}\right )}{2 a c^2 (b c-a d)}\\ &=\frac{d (2 b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac{d}{x}\right )^2}+\frac{d (b c-4 a d) (4 b c-3 a d) \sqrt{a+\frac{b}{x}}}{4 a c^3 (b c-a d)^2 \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{(b c-a d)^2 (b c+6 a d)+\frac{1}{4} b d (b c-4 a d) (4 b c-3 a d) x}{x \sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{2 a c^3 (b c-a d)^2}\\ &=\frac{d (2 b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac{d}{x}\right )^2}+\frac{d (b c-4 a d) (4 b c-3 a d) \sqrt{a+\frac{b}{x}}}{4 a c^3 (b c-a d)^2 \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}+\frac{(b c+6 a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a c^4}-\frac{\left (d^2 \left (35 b^2 c^2-56 a b c d+24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{8 c^4 (b c-a d)^2}\\ &=\frac{d (2 b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac{d}{x}\right )^2}+\frac{d (b c-4 a d) (4 b c-3 a d) \sqrt{a+\frac{b}{x}}}{4 a c^3 (b c-a d)^2 \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}+\frac{(b c+6 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a b c^4}-\frac{\left (d^2 \left (35 b^2 c^2-56 a b c d+24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{a d}{b}+\frac{d x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 b c^4 (b c-a d)^2}\\ &=\frac{d (2 b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 a c^2 (b c-a d) \left (c+\frac{d}{x}\right )^2}+\frac{d (b c-4 a d) (4 b c-3 a d) \sqrt{a+\frac{b}{x}}}{4 a c^3 (b c-a d)^2 \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} x}{a c \left (c+\frac{d}{x}\right )^2}-\frac{d^{3/2} \left (35 b^2 c^2-56 a b c d+24 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}-\frac{(b c+6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^4}\\ \end{align*}
Mathematica [A] time = 1.40395, size = 216, normalized size = 0.86 \[ \frac{\frac{c x \sqrt{a+\frac{b}{x}} \left (2 a^2 d^2 \left (2 c^2 x^2+9 c d x+6 d^2\right )-a b c d \left (8 c^2 x^2+29 c d x+19 d^2\right )+4 b^2 c^2 (c x+d)^2\right )}{(c x+d)^2 (b c-a d)^2}-\frac{a d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}-\frac{4 (6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}}{4 a c^4} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[a + b/x]*(c + d/x)^3),x]
[Out]
((c*Sqrt[a + b/x]*x*(4*b^2*c^2*(d + c*x)^2 + 2*a^2*d^2*(6*d^2 + 9*c*d*x + 2*c^2*x^2) - a*b*c*d*(19*d^2 + 29*c*
d*x + 8*c^2*x^2)))/((b*c - a*d)^2*(d + c*x)^2) - (a*d^(3/2)*(35*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqr
t[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2) - (4*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqr
t[a])/(4*a*c^4)
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Maple [B] time = 0.014, size = 2269, normalized size = 9.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(c+d/x)^3/(a+b/x)^(1/2),x)
[Out]
-1/8*((a*x+b)/x)^(1/2)*x*(-70*a^(5/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*
x+d))*x*b^3*c^4*d^3+60*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x^2*b
^2*c^5*d^2-12*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*c^6*d+
22*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*x*b*c^6*d+18*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^
(1/2)*x^2*b*c^5*d^2-8*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^
6*d+16*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b^3*c^6*d+24*a^(11/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)
*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*d^7-35*a^(5/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2
)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b^3*c^5*d^2+48*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*(
(a*d-b*c)*d/c^2)^(1/2)*x*c^2*d^5+18*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*b*c^5*d^2-36*a^(9/2)*((a
*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*c^3*d^4-160*a^(9/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c
-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*b*c^2*d^5+182*a^(7/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x
+b*c*x-b*d)/(c*x+d))*x*b^2*c^3*d^4+8*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^2*b^3*c^7-4*ln(1/2*(2
*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*b^4*c^5*d^2+8*a^(3/2)*((a*d-b*c)*d/c^2)
^(1/2)*((a*x+b)*x)^(1/2)*b^3*c^5*d^2-4*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^
2)^(1/2)*x^2*b^4*c^7+24*a^(11/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))
*x^2*c^2*d^5-8*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*c^4*d^3+48*a^(11/2)*ln((2*((a*d-b*c)*d/c^2)^(
1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*c*d^6+24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*c*d^6-24*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c^2*d^5-80*a^(9
/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b*c*d^6+91*a^(7/2)*ln((2*((a
*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^2*c^2*d^5-35*a^(5/2)*ln((2*((a*d-b*c)*d
/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^3*c^3*d^4+120*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^4*d^3-24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/
a^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^5*d^2+102*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b*c
^4*d^3-92*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b^2*c^5*d^2-46*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*(
(a*x+b)*x)^(1/2)*x^2*b^2*c^6*d-136*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)
^(1/2)*x*b*c^3*d^4-22*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^3*b*c^6*d-68*ln(1/2*(2*((a*x+b)*x)^(
1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^4*d^3-68*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*b*c^2*d^5+60*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1
/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^3*d^4-12*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*((a*
d-b*c)*d/c^2)^(1/2)*b^3*c^4*d^3+62*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*b*c^3*d^4-46*a^(5/2)*((a*
d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*b^2*c^4*d^3+12*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^3*c^5
*d^2+24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^3*d^4-12*a^(7/
2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*x*c^5*d^2-80*a^(9/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1
/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b*c^3*d^4+91*a^(7/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2
*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b^2*c^4*d^3)/c^5/((a*x+b)*x)^(1/2)/(a*d-b*c)^3/(c*x+d)^2/a^(5/2)/((a*d-b*c)*d/c
^2)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x}}{\left (c + \frac{d}{x}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d/x)^3/(a+b/x)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(a + b/x)*(c + d/x)^3), x)
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Fricas [B] time = 3.7945, size = 4749, normalized size = 19. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d/x)^3/(a+b/x)^(1/2),x, algorithm="fricas")
[Out]
[1/8*(4*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*
d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*sqrt(a)*log(2*a
*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (35*a^2*b^2*c^2*d^3 - 56*a^3*b*c*d^4 + 24*a^4*d^5 + (35*a^2*b^2*c^4*
d - 56*a^3*b*c^3*d^2 + 24*a^4*c^2*d^3)*x^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*d^4)*x)*sqrt(
-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d
)) + 2*(4*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2)*x^3 + (8*a*b^2*c^4*d - 29*a^2*b*c^3*d^2 + 18*a^3*c^2*d^3)*
x^2 + (4*a*b^2*c^3*d^2 - 19*a^2*b*c^2*d^3 + 12*a^3*c*d^4)*x)*sqrt((a*x + b)/x))/(a^2*b^2*c^6*d^2 - 2*a^3*b*c^5
*d^3 + a^4*c^4*d^4 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^
4*c^5*d^3)*x), 1/8*(8*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d -
11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*s
qrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (35*a^2*b^2*c^2*d^3 - 56*a^3*b*c*d^4 + 24*a^4*d^5 + (35*a^2*b^2
*c^4*d - 56*a^3*b*c^3*d^2 + 24*a^4*c^2*d^3)*x^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*d^4)*x)*
sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*
x + d)) + 2*(4*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2)*x^3 + (8*a*b^2*c^4*d - 29*a^2*b*c^3*d^2 + 18*a^3*c^2*
d^3)*x^2 + (4*a*b^2*c^3*d^2 - 19*a^2*b*c^2*d^3 + 12*a^3*c*d^4)*x)*sqrt((a*x + b)/x))/(a^2*b^2*c^6*d^2 - 2*a^3*
b*c^5*d^3 + a^4*c^4*d^4 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2
+ a^4*c^5*d^3)*x), -1/4*((35*a^2*b^2*c^2*d^3 - 56*a^3*b*c*d^4 + 24*a^4*d^5 + (35*a^2*b^2*c^4*d - 56*a^3*b*c^3
*d^2 + 24*a^4*c^2*d^3)*x^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*d^4)*x)*sqrt(d/(b*c - a*d))*a
rctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 2*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 -
11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d
+ 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b)
- (4*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2)*x^3 + (8*a*b^2*c^4*d - 29*a^2*b*c^3*d^2 + 18*a^3*c^2*d^3)*x^2 +
(4*a*b^2*c^3*d^2 - 19*a^2*b*c^2*d^3 + 12*a^3*c*d^4)*x)*sqrt((a*x + b)/x))/(a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3
+ a^4*c^4*d^4 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5
*d^3)*x), -1/4*((35*a^2*b^2*c^2*d^3 - 56*a^3*b*c*d^4 + 24*a^4*d^5 + (35*a^2*b^2*c^4*d - 56*a^3*b*c^3*d^2 + 24*
a^4*c^2*d^3)*x^2 + 2*(35*a^2*b^2*c^3*d^2 - 56*a^3*b*c^2*d^3 + 24*a^4*c*d^4)*x)*sqrt(d/(b*c - a*d))*arctan(-(b*
c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 4*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*
c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*
c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (4*(a*b^2*c^5 - 2
*a^2*b*c^4*d + a^3*c^3*d^2)*x^3 + (8*a*b^2*c^4*d - 29*a^2*b*c^3*d^2 + 18*a^3*c^2*d^3)*x^2 + (4*a*b^2*c^3*d^2 -
19*a^2*b*c^2*d^3 + 12*a^3*c*d^4)*x)*sqrt((a*x + b)/x))/(a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4 + (a^
2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x)]
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Sympy [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(1/(c+d/x)**3/(a+b/x)**(1/2),x)
[Out]
Exception raised: ValueError
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Giac [A] time = 1.22351, size = 459, normalized size = 1.84 \begin{align*} -\frac{1}{4} \, b{\left (\frac{{\left (35 \, b^{2} c^{2} d^{2} - 56 \, a b c d^{3} + 24 \, a^{2} d^{4}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{3} c^{6} - 2 \, a b^{2} c^{5} d + a^{2} b c^{4} d^{2}\right )} \sqrt{b c d - a d^{2}}} + \frac{13 \, b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}} - 21 \, a b c d^{3} \sqrt{\frac{a x + b}{x}} + 8 \, a^{2} d^{4} \sqrt{\frac{a x + b}{x}} + \frac{11 \,{\left (a x + b\right )} b c d^{3} \sqrt{\frac{a x + b}{x}}}{x} - \frac{8 \,{\left (a x + b\right )} a d^{4} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}{\left (b c - a d + \frac{{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac{4 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a c^{3}} - \frac{4 \,{\left (b c + 6 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d/x)^3/(a+b/x)^(1/2),x, algorithm="giac")
[Out]
-1/4*b*((35*b^2*c^2*d^2 - 56*a*b*c*d^3 + 24*a^2*d^4)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^3*c^6
- 2*a*b^2*c^5*d + a^2*b*c^4*d^2)*sqrt(b*c*d - a*d^2)) + (13*b^2*c^2*d^2*sqrt((a*x + b)/x) - 21*a*b*c*d^3*sqrt
((a*x + b)/x) + 8*a^2*d^4*sqrt((a*x + b)/x) + 11*(a*x + b)*b*c*d^3*sqrt((a*x + b)/x)/x - 8*(a*x + b)*a*d^4*sqr
t((a*x + b)/x)/x)/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*(b*c - a*d + (a*x + b)*d/x)^2) + 4*sqrt((a*x + b)/x)/
((a - (a*x + b)/x)*a*c^3) - 4*(b*c + 6*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b*c^4))