3.251 \(\int \frac {1}{\sqrt {a+\frac {b}{x}} (c+\frac {d}{x})^3} \, dx\)

Optimal. Leaf size=250 \[ -\frac {(6 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^4}-\frac {d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}+\frac {d \sqrt {a+\frac {b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left (c+\frac {d}{x}\right ) (b c-a d)^2}+\frac {d \sqrt {a+\frac {b}{x}} (2 b c-3 a d)}{2 a c^2 \left (c+\frac {d}{x}\right )^2 (b c-a d)}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )^2} \]

[Out]

-1/4*d^(3/2)*(24*a^2*d^2-56*a*b*c*d+35*b^2*c^2)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^4/(-a*d+b*c)^
(5/2)-(6*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(3/2)/c^4+1/2*d*(-3*a*d+2*b*c)*(a+b/x)^(1/2)/a/c^2/(-a*d+b*
c)/(c+d/x)^2+1/4*d*(-4*a*d+b*c)*(-3*a*d+4*b*c)*(a+b/x)^(1/2)/a/c^3/(-a*d+b*c)^2/(c+d/x)+x*(a+b/x)^(1/2)/a/c/(c
+d/x)^2

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Rubi [A]  time = 0.40, antiderivative size = 250, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 1.75, 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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fricas [B]  time = 1.94, size = 2307, normalized size = 9.23 \[ \text {result too large to display} \]

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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giac [A]  time = 0.29, size = 352, normalized size = 1.41 \[ -\frac {1}{4} \, b^{4} {\left (\frac {{\left (35 \, b^{2} c^{2} d^{2} - 56 \, a b c d^{3} + 24 \, a^{2} d^{4}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{6} c^{6} - 2 \, a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2}\right )} \sqrt {b c d - a d^{2}}} + \frac {13 \, b^{2} c^{2} d^{2} \sqrt {\frac {a x + b}{x}} - 21 \, a b c d^{3} \sqrt {\frac {a x + b}{x}} + 8 \, a^{2} d^{4} \sqrt {\frac {a x + b}{x}} + \frac {11 \, {\left (a x + b\right )} b c d^{3} \sqrt {\frac {a x + b}{x}}}{x} - \frac {8 \, {\left (a x + b\right )} a d^{4} \sqrt {\frac {a x + b}{x}}}{x}}{{\left (b^{5} c^{5} - 2 \, a b^{4} c^{4} d + a^{2} b^{3} c^{3} d^{2}\right )} {\left (b c - a d + \frac {{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac {4 \, \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a b^{3} c^{3}} - \frac {4 \, {\left (b c + 6 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a b^{4} c^{4}}\right )} \]

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^4*((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^6*c
^6 - 2*a*b^5*c^5*d + a^2*b^4*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
*sqrt((a*x + b)/x)/x)/((b^5*c^5 - 2*a*b^4*c^4*d + a^2*b^3*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*b^3*c^3) - 4*(b*c + 6*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b^4*c
^4))

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maple [B]  time = 0.07, size = 2269, normalized size = 9.08 \[ \text {result too large to display} \]

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*(60*a^3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)
*b^2*c^3*d^4-12*a^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*b^3*c^4*d^3+
12*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*c^5*d^2-12*a^(7/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d
)^(1/2)*x*c^5*d^2-80*a^(9/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x^
2*b*c^3*d^4+91*a^(7/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x^2*b^2*
c^4*d^3-35*a^(5/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x^2*b^3*c^5*
d^2+24*a^5*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x^2*c^3*d^4+18*a^(5/2
)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*b*c^5*d^2-36*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*c
^3*d^4-160*a^(9/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x*b*c^2*d^5-
4*a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^4*c^7+8*a^(3/2)*((a*x+
b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b^3*c^5*d^2-4*a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*
d-b*c)/c^2*d)^(1/2)*b^4*c^5*d^2+182*a^(7/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)
*c)/(c*x+d))*x*b^2*c^3*d^4-70*a^(5/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c
*x+d))*x*b^3*c^4*d^3+48*a^5*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*c^
2*d^5+62*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b*c^3*d^4-46*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c
^2*d)^(1/2)*b^2*c^4*d^3-68*a^4*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*b
*c^2*d^5+8*a^(3/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^3*c^7+24*a^(11/2)*ln((-2*a*d*x+b*c*x-b*d+2*
((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*d^7-92*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)
*x*b^2*c^5*d^2-136*a^4*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b*c^3*d
^4+120*a^3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b^2*c^4*d^3-24*a^2*
ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b^3*c^5*d^2+60*a^3*ln(1/2*(2*a
*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^2*c^5*d^2-12*a^2*ln(1/2*(2*a*x+b+2*((
a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^3*c^6*d+102*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*
c)/c^2*d)^(1/2)*x*b*c^4*d^3+22*a^(5/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b*c^6*d+18*a^(7/2)*((a*x+b)
*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b*c^5*d^2-46*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^2*c
^6*d-68*a^4*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x^2*b*c^4*d^3-22*a^(
7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*b*c^6*d-8*a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^
(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b^4*c^6*d+16*a^(3/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^3*c^6*d-24
*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*c^2*d^5-80*a^(9/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*
d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*b*c*d^6+91*a^(7/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a
*x+b)*x)^(1/2)*c)/(c*x+d))*b^2*c^2*d^5-35*a^(5/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)
^(1/2)*c)/(c*x+d))*b^3*c^3*d^4+24*a^5*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^
(1/2)*c*d^6+24*a^(11/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x^2*c^2
*d^5-8*a^(7/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*c^4*d^3+48*a^(11/2)*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*
c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))*x*c*d^6)/c^5/((a*x+b)*x)^(1/2)/(a*d-b*c)^3/(c*x+d)^2/a^(5/2)/((a
*d-b*c)/c^2*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x}} {\left (c + \frac {d}{x}\right )}^{3}}\,{d x} \]

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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mupad [B]  time = 5.48, size = 2890, normalized size = 11.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((d^3*(a*d - b*c)^5)^(1/2)*(a + b/x)^(1/2) - a^3*d^4 + b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3)*(d^3*(
a*d - b*c)^5)^(1/2)*(3*a^2*d^2 + (35*b^2*c^2)/8 - 7*a*b*c*d))/(b^5*c^9 - a^5*c^4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^
2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 - 5*a*b^4*c^8*d) - ((b*(a + b/x)^(5/2)*(12*a^2*d^4 + 4*b^2*c^2*d^2 - 19*a*b
*c*d^3))/(4*a*c^3*(a*d - b*c)^2) - ((a + b/x)^(1/2)*(4*b^4*c^3 - 12*a^3*b*d^3 + 25*a^2*b^2*c*d^2 - 12*a*b^3*c^
2*d))/(4*a*c^3*(a*d - b*c)) + (d*(a + b/x)^(3/2)*(8*b^4*c^3 - 24*a^3*b*d^3 + 56*a^2*b^2*c*d^2 - 37*a*b^3*c^2*d
))/(4*c^3*(a^2*d - a*b*c)*(a*d - b*c)))/((a + b/x)^2*(3*a*d^2 - 2*b*c*d) - (a + b/x)*(3*a^2*d^2 + b^2*c^2 - 4*
a*b*c*d) - d^2*(a + b/x)^3 + a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) - (log((d^3*(a*d - b*c)^5)^(1/2)*(a + b/x)^(1/
2) + a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)*(d^3*(a*d - b*c)^5)^(1/2)*(24*a^2*d^2 + 35*b^2*c^2
- 56*a*b*c*d))/(8*(b^5*c^9 - a^5*c^4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 - 5*a*b^
4*c^8*d)) - (atan((((((a + b/x)^(1/2)*(1152*a^6*b^2*d^9 + 16*b^8*c^6*d^3 + 128*a*b^7*c^5*d^4 - 4800*a^5*b^3*c*
d^8 + 1129*a^2*b^6*c^4*d^5 - 5136*a^3*b^5*c^3*d^6 + 7520*a^4*b^4*c^2*d^7))/(8*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*
a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)) - (((4*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 - 45*a^3*b^6*
c^11*d^4 + 74*a^4*b^5*c^10*d^5 - 49*a^5*b^4*c^9*d^6 + 12*a^6*b^3*c^8*d^7)/(a^2*b^4*c^13 + a^6*c^9*d^4 - 4*a^3*
b^3*c^12*d - 4*a^5*b*c^10*d^3 + 6*a^4*b^2*c^11*d^2) - ((a + b/x)^(1/2)*(6*a*d + b*c)*(64*a^2*b^7*c^13*d^2 - 38
4*a^3*b^6*c^12*d^3 + 896*a^4*b^5*c^11*d^4 - 1024*a^5*b^4*c^10*d^5 + 576*a^6*b^3*c^9*d^6 - 128*a^7*b^2*c^8*d^7)
)/(16*c^4*(a^3)^(1/2)*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)))*(
6*a*d + b*c))/(2*c^4*(a^3)^(1/2)))*(6*a*d + b*c)*1i)/(2*c^4*(a^3)^(1/2)) + ((((a + b/x)^(1/2)*(1152*a^6*b^2*d^
9 + 16*b^8*c^6*d^3 + 128*a*b^7*c^5*d^4 - 4800*a^5*b^3*c*d^8 + 1129*a^2*b^6*c^4*d^5 - 5136*a^3*b^5*c^3*d^6 + 75
20*a^4*b^4*c^2*d^7))/(8*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2))
+ (((4*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 - 45*a^3*b^6*c^11*d^4 + 74*a^4*b^5*c^10*d^5 - 49*a^5*b^4*c^9*d^6 +
12*a^6*b^3*c^8*d^7)/(a^2*b^4*c^13 + a^6*c^9*d^4 - 4*a^3*b^3*c^12*d - 4*a^5*b*c^10*d^3 + 6*a^4*b^2*c^11*d^2) +
((a + b/x)^(1/2)*(6*a*d + b*c)*(64*a^2*b^7*c^13*d^2 - 384*a^3*b^6*c^12*d^3 + 896*a^4*b^5*c^11*d^4 - 1024*a^5*b
^4*c^10*d^5 + 576*a^6*b^3*c^9*d^6 - 128*a^7*b^2*c^8*d^7))/(16*c^4*(a^3)^(1/2)*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*
a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)))*(6*a*d + b*c))/(2*c^4*(a^3)^(1/2)))*(6*a*d + b*c)*1i)/(
2*c^4*(a^3)^(1/2)))/((216*a^5*b^3*d^9 + (35*b^8*c^5*d^4)/2 - (49*a*b^7*c^4*d^5)/8 - 810*a^4*b^4*c*d^8 - (1877*
a^2*b^6*c^3*d^6)/4 + 1044*a^3*b^5*c^2*d^7)/(a^2*b^4*c^13 + a^6*c^9*d^4 - 4*a^3*b^3*c^12*d - 4*a^5*b*c^10*d^3 +
6*a^4*b^2*c^11*d^2) + ((((a + b/x)^(1/2)*(1152*a^6*b^2*d^9 + 16*b^8*c^6*d^3 + 128*a*b^7*c^5*d^4 - 4800*a^5*b^
3*c*d^8 + 1129*a^2*b^6*c^4*d^5 - 5136*a^3*b^5*c^3*d^6 + 7520*a^4*b^4*c^2*d^7))/(8*(a^2*b^4*c^10 + a^6*c^6*d^4
- 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)) - (((4*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 - 45*a^3*
b^6*c^11*d^4 + 74*a^4*b^5*c^10*d^5 - 49*a^5*b^4*c^9*d^6 + 12*a^6*b^3*c^8*d^7)/(a^2*b^4*c^13 + a^6*c^9*d^4 - 4*
a^3*b^3*c^12*d - 4*a^5*b*c^10*d^3 + 6*a^4*b^2*c^11*d^2) - ((a + b/x)^(1/2)*(6*a*d + b*c)*(64*a^2*b^7*c^13*d^2
- 384*a^3*b^6*c^12*d^3 + 896*a^4*b^5*c^11*d^4 - 1024*a^5*b^4*c^10*d^5 + 576*a^6*b^3*c^9*d^6 - 128*a^7*b^2*c^8*
d^7))/(16*c^4*(a^3)^(1/2)*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)
))*(6*a*d + b*c))/(2*c^4*(a^3)^(1/2)))*(6*a*d + b*c))/(2*c^4*(a^3)^(1/2)) - ((((a + b/x)^(1/2)*(1152*a^6*b^2*d
^9 + 16*b^8*c^6*d^3 + 128*a*b^7*c^5*d^4 - 4800*a^5*b^3*c*d^8 + 1129*a^2*b^6*c^4*d^5 - 5136*a^3*b^5*c^3*d^6 + 7
520*a^4*b^4*c^2*d^7))/(8*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2))
+ (((4*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 - 45*a^3*b^6*c^11*d^4 + 74*a^4*b^5*c^10*d^5 - 49*a^5*b^4*c^9*d^6 +
12*a^6*b^3*c^8*d^7)/(a^2*b^4*c^13 + a^6*c^9*d^4 - 4*a^3*b^3*c^12*d - 4*a^5*b*c^10*d^3 + 6*a^4*b^2*c^11*d^2) +
((a + b/x)^(1/2)*(6*a*d + b*c)*(64*a^2*b^7*c^13*d^2 - 384*a^3*b^6*c^12*d^3 + 896*a^4*b^5*c^11*d^4 - 1024*a^5*
b^4*c^10*d^5 + 576*a^6*b^3*c^9*d^6 - 128*a^7*b^2*c^8*d^7))/(16*c^4*(a^3)^(1/2)*(a^2*b^4*c^10 + a^6*c^6*d^4 - 4
*a^3*b^3*c^9*d - 4*a^5*b*c^7*d^3 + 6*a^4*b^2*c^8*d^2)))*(6*a*d + b*c))/(2*c^4*(a^3)^(1/2)))*(6*a*d + b*c))/(2*
c^4*(a^3)^(1/2))))*(6*a*d + b*c)*1i)/(c^4*(a^3)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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