\(\int \frac {\sqrt {a+\frac {b}{x}}}{(c+\frac {d}{x})^3} \, dx\) [230]

Optimal result

Integrand size = 21, antiderivative size = 213 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {(b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4} \]

[Out]

(-6*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/c^4/a^(1/2)+1/4*(24*a^2*d^2-40*a*b*c*d+15*b^2*c^2)*arctan(d^(1/2)*
(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))*d^(1/2)/c^4/(-a*d+b*c)^(3/2)+3/2*d*(a+b/x)^(1/2)/c^2/(c+d/x)^2+1/4*d*(-12*a*d+
11*b*c)*(a+b/x)^(1/2)/c^3/(-a*d+b*c)/(c+d/x)+x*(a+b/x)^(1/2)/c/(c+d/x)^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {382, 101, 156, 162, 65, 214, 211} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (b c-6 a d)}{\sqrt {a} c^4}+\frac {d \sqrt {a+\frac {b}{x}} (11 b c-12 a d)}{4 c^3 \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {x \sqrt {a+\frac {b}{x}}}{c \left (c+\frac {d}{x}\right )^2} \]

[In]

Int[Sqrt[a + b/x]/(c + d/x)^3,x]

[Out]

(3*d*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) + (d*(11*b*c - 12*a*d)*Sqrt[a + b/x])/(4*c^3*(b*c - a*d)*(c + d/x)) +
(Sqrt[a + b/x]*x)/(c*(c + d/x)^2) + (Sqrt[d]*(15*b^2*c^2 - 40*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)^(3/2)) + ((b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(Sqrt[a]*c^4
)

Rule 65

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 101

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 156

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] && ILtQ[m, -1]

Rule 162

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 382

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*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\text {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 )}{c} \\ & = \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\text {Subst}\left (\int \frac {-((b c-6 a d) (b c-a d))+\frac {9}{2} b d (b c-a d) x}{x \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{2 c^2 (b c-a d)} \\ & = \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\text {Subst}\left (\int \frac {(b c-6 a d) (b c-a d)^2-\frac {1}{4} b d (11 b c-12 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3 (b c-a d)^2} \\ & = \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {(b c-6 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \text {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)} \\ & = \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {(b c-6 a d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \text {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)} \\ & = \frac {3 d \sqrt {a+\frac {b}{x}}}{2 c^2 \left (c+\frac {d}{x}\right )^2}+\frac {d (11 b c-12 a d) \sqrt {a+\frac {b}{x}}}{4 c^3 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 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)^{3/2}}+\frac {(b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (-2 a d \left (6 d^2+9 c d x+2 c^2 x^2\right )+b c \left (11 d^2+17 c d x+4 c^2 x^2\right )\right )}{(b c-a d) (d+c x)^2}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {4 (b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{4 c^4} \]

[In]

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

[Out]

((c*Sqrt[a + b/x]*x*(-2*a*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + b*c*(11*d^2 + 17*c*d*x + 4*c^2*x^2)))/((b*c - a*d)
*(d + c*x)^2) + (Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]
])/(b*c - a*d)^(3/2) + (4*(b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/(4*c^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(968\) vs. \(2(185)=370\).

Time = 0.26 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.55

method	result	size
risch	\(\frac {x \sqrt {\frac {a x +b}{x}}}{c^{3}}-\frac {\left (\frac {\left (6 a d -b c \right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{c \sqrt {a}}+\frac {2 d^{2} \left (4 a d -3 b c \right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{3}}-\frac {2 d^{3} \left (a d -b c \right ) \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{2 \left (a d -b c \right ) d \left (x +\frac {d}{c}\right )^{2}}+\frac {3 \left (2 a d -b c \right ) c \left (-\frac {c^{2} \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{\left (a d -b c \right ) d \left (x +\frac {d}{c}\right )}-\frac {\left (2 a d -b c \right ) c \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{4 \left (a d -b c \right ) d}+\frac {a \,c^{2} \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{2 \left (a d -b c \right ) d \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right )}{c^{4}}+\frac {6 d \left (2 a d -b c \right ) \ln \left (\frac {\frac {2 \left (a d -b c \right ) d}{c^{2}}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a \left (x +\frac {d}{c}\right )^{2}-\frac {\left (2 a d -b c \right ) \left (x +\frac {d}{c}\right )}{c}+\frac {\left (a d -b c \right ) d}{c^{2}}}}{x +\frac {d}{c}}\right )}{c^{2} \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 c^{3} \left (a x +b \right )}\)	\(969\)
default	\(\text {Expression too large to display}\)	\(1972\)

[In]

int((a+b/x)^(1/2)/(c+d/x)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^3*x*((a*x+b)/x)^(1/2)-1/2/c^3*((6*a*d-b*c)/c*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+2/c^3*d^2*(
4*a*d-3*b*c)*(-1/(a*d-b*c)/d*c^2/(x+d/c)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)-1/2*(2*a*d-
b*c)*c/(a*d-b*c)/d/((a*d-b*c)*d/c^2)^(1/2)*ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*d-b*c)*d/c^2)^(1/
2)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c)))-2*d^3*(a*d-b*c)/c^4*(-1/2/(a*d-b*c)/d*
c^2/(x+d/c)^2*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)+3/4*(2*a*d-b*c)*c/(a*d-b*c)/d*(-1/(a*d
-b*c)/d*c^2/(x+d/c)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2)-1/2*(2*a*d-b*c)*c/(a*d-b*c)/d/((
a*d-b*c)*d/c^2)^(1/2)*ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x+d/c)^2-(2*a*
d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c)))+1/2*a/(a*d-b*c)/d*c^2/((a*d-b*c)*d/c^2)^(1/2)*ln((2*(a*d-b*
c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*d-b*c)*d/c^2)^(1/2)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(
1/2))/(x+d/c)))+6/c^2*d*(2*a*d-b*c)/((a*d-b*c)*d/c^2)^(1/2)*ln((2*(a*d-b*c)*d/c^2-(2*a*d-b*c)/c*(x+d/c)+2*((a*
d-b*c)*d/c^2)^(1/2)*(a*(x+d/c)^2-(2*a*d-b*c)/c*(x+d/c)+(a*d-b*c)*d/c^2)^(1/2))/(x+d/c)))*((a*x+b)/x)^(1/2)*(x*
(a*x+b))^(1/2)/(a*x+b)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (185) = 370\).

Time = 0.41 (sec) , antiderivative size = 1749, normalized size of antiderivative = 8.21 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((a+b/x)^(1/2)/(c+d/x)^3,x, algorithm="fricas")

[Out]

[-1/8*(4*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d -
7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (15*a*b^2*c^2*d^2 -
40*a^2*b*c*d^3 + 24*a^3*d^4 + (15*a*b^2*c^4 - 40*a^2*b*c^3*d + 24*a^3*c^2*d^2)*x^2 + 2*(15*a*b^2*c^3*d - 40*a^
2*b*c^2*d^2 + 24*a^3*c*d^3)*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*c^4 - a^2*c^3*d)*x^3 + (17*a*b*c^3*d - 18*a^2*c^2*d^2)*x^2
+ (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^4*d^3 + (a*b*c^7 - a^2*c^6*d)*x^2
+ 2*(a*b*c^6*d - a^2*c^5*d^2)*x), 1/4*((15*a*b^2*c^2*d^2 - 40*a^2*b*c*d^3 + 24*a^3*d^4 + (15*a*b^2*c^4 - 40*a
^2*b*c^3*d + 24*a^3*c^2*d^2)*x^2 + 2*(15*a*b^2*c^3*d - 40*a^2*b*c^2*d^2 + 24*a^3*c*d^3)*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)) - 2*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6
*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt(a
)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*(a*b*c^4 - a^2*c^3*d)*x^3 + (17*a*b*c^3*d - 18*a^2*c^2*d
^2)*x^2 + (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^4*d^3 + (a*b*c^7 - a^2*c^
6*d)*x^2 + 2*(a*b*c^6*d - a^2*c^5*d^2)*x), -1/8*(8*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c
^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x
+ b)/x)/a) + (15*a*b^2*c^2*d^2 - 40*a^2*b*c*d^3 + 24*a^3*d^4 + (15*a*b^2*c^4 - 40*a^2*b*c^3*d + 24*a^3*c^2*d^2
)*x^2 + 2*(15*a*b^2*c^3*d - 40*a^2*b*c^2*d^2 + 24*a^3*c*d^3)*x)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqr
t(-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*c^4 - a^2*c^3*d)*x^3 + (17
*a*b*c^3*d - 18*a^2*c^2*d^2)*x^2 + (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^
4*d^3 + (a*b*c^7 - a^2*c^6*d)*x^2 + 2*(a*b*c^6*d - a^2*c^5*d^2)*x), 1/4*((15*a*b^2*c^2*d^2 - 40*a^2*b*c*d^3 +
24*a^3*d^4 + (15*a*b^2*c^4 - 40*a^2*b*c^3*d + 24*a^3*c^2*d^2)*x^2 + 2*(15*a*b^2*c^3*d - 40*a^2*b*c^2*d^2 + 24*
a^3*c*d^3)*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^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b
*c^2*d^2 + 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (4*(a*b*c^4 - a^2*c^3*d)*x^3 + (17*
a*b*c^3*d - 18*a^2*c^2*d^2)*x^2 + (11*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(a*b*c^5*d^2 - a^2*c^4
*d^3 + (a*b*c^7 - a^2*c^6*d)*x^2 + 2*(a*b*c^6*d - a^2*c^5*d^2)*x)]

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int \frac {x^{3} \sqrt {a + \frac {b}{x}}}{\left (c x + d\right )^{3}}\, dx \]

[In]

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

[Out]

Integral(x**3*sqrt(a + b/x)/(c*x + d)**3, x)

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int { \frac {\sqrt {a + \frac {b}{x}}}{{\left (c + \frac {d}{x}\right )}^{3}} \,d x } \]

[In]

integrate((a+b/x)^(1/2)/(c+d/x)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/x)/(c + d/x)^3, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (185) = 370\).

Time = 0.34 (sec) , antiderivative size = 809, normalized size of antiderivative = 3.80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=-\frac {{\left (15 \, \sqrt {a} b^{2} c^{2} d \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 40 \, a^{\frac {3}{2}} b c d^{2} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) + 24 \, a^{\frac {5}{2}} d^{3} \arctan \left (\frac {\sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right ) - 2 \, \sqrt {b c d - a d^{2}} b^{2} c^{2} \log \left ({\left | b \right |}\right ) + 14 \, \sqrt {b c d - a d^{2}} a b c d \log \left ({\left | b \right |}\right ) - 12 \, \sqrt {b c d - a d^{2}} a^{2} d^{2} \log \left ({\left | b \right |}\right ) + 9 \, \sqrt {b c d - a d^{2}} a b c d - 10 \, \sqrt {b c d - a d^{2}} a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{4 \, {\left (\sqrt {b c d - a d^{2}} \sqrt {a} b c^{5} - \sqrt {b c d - a d^{2}} a^{\frac {3}{2}} c^{4} d\right )}} - \frac {{\left (15 \, b^{2} c^{2} d \mathrm {sgn}\left (x\right ) - 40 \, a b c d^{2} \mathrm {sgn}\left (x\right ) + 24 \, a^{2} d^{3} \mathrm {sgn}\left (x\right )\right )} \arctan \left (-\frac {{\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} c + \sqrt {a} d}{\sqrt {b c d - a d^{2}}}\right )}{4 \, {\left (b c^{5} - a c^{4} d\right )} \sqrt {b c d - a d^{2}}} + \frac {\sqrt {a x^{2} + b x} \mathrm {sgn}\left (x\right )}{c^{3}} - \frac {9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} b^{2} c^{3} d \mathrm {sgn}\left (x\right ) - 32 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a b c^{2} d^{2} \mathrm {sgn}\left (x\right ) + 24 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{2} c d^{3} \mathrm {sgn}\left (x\right ) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} \sqrt {a} b^{2} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {3}{2}} b c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{\frac {5}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} b^{3} c^{2} d^{2} \mathrm {sgn}\left (x\right ) - 44 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a b^{2} c d^{3} \mathrm {sgn}\left (x\right ) + 40 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{2} b d^{4} \mathrm {sgn}\left (x\right ) - 9 \, \sqrt {a} b^{3} c d^{3} \mathrm {sgn}\left (x\right ) + 10 \, a^{\frac {3}{2}} b^{2} d^{4} \mathrm {sgn}\left (x\right )}{4 \, {\left (b c^{5} - a c^{4} d\right )} {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} c + 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} d + b d\right )}^{2}} - \frac {{\left (b c \mathrm {sgn}\left (x\right ) - 6 \, a d \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, \sqrt {a} c^{4}} \]

[In]

integrate((a+b/x)^(1/2)/(c+d/x)^3,x, algorithm="giac")

[Out]

-1/4*(15*sqrt(a)*b^2*c^2*d*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 40*a^(3/2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*
c*d - a*d^2)) + 24*a^(5/2)*d^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 2*sqrt(b*c*d - a*d^2)*b^2*c^2*log(abs(b
)) + 14*sqrt(b*c*d - a*d^2)*a*b*c*d*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^2*d^2*log(abs(b)) + 9*sqrt(b*c*d -
a*d^2)*a*b*c*d - 10*sqrt(b*c*d - a*d^2)*a^2*d^2)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*b*c^5 - sqrt(b*c*d - a*d^
2)*a^(3/2)*c^4*d) - 1/4*(15*b^2*c^2*d*sgn(x) - 40*a*b*c*d^2*sgn(x) + 24*a^2*d^3*sgn(x))*arctan(-((sqrt(a)*x -
sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/((b*c^5 - a*c^4*d)*sqrt(b*c*d - a*d^2)) + sqrt(a*x^2 +
b*x)*sgn(x)/c^3 - 1/4*(9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*b^2*c^3*d*sgn(x) - 32*(sqrt(a)*x - sqrt(a*x^2 + b*x
))^3*a*b*c^2*d^2*sgn(x) + 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^2*c*d^3*sgn(x) + 3*(sqrt(a)*x - sqrt(a*x^2 +
b*x))^2*sqrt(a)*b^2*c^2*d^2*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^(3/2)*b*c*d^3*sgn(x) + 40*(sqrt(a)
*x - sqrt(a*x^2 + b*x))^2*a^(5/2)*d^4*sgn(x) + 7*(sqrt(a)*x - sqrt(a*x^2 + b*x))*b^3*c^2*d^2*sgn(x) - 44*(sqrt
(a)*x - sqrt(a*x^2 + b*x))*a*b^2*c*d^3*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^2*b*d^4*sgn(x) - 9*sqrt(a
)*b^3*c*d^3*sgn(x) + 10*a^(3/2)*b^2*d^4*sgn(x))/((b*c^5 - a*c^4*d)*((sqrt(a)*x - sqrt(a*x^2 + b*x))^2*c + 2*(s
qrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*d + b*d)^2) - 1/2*(b*c*sgn(x) - 6*a*d*sgn(x))*log(abs(2*(sqrt(a)*x - sqr
t(a*x^2 + b*x))*sqrt(a) + b))/(sqrt(a)*c^4)

Mupad [B] (verification not implemented)

Time = 7.94 (sec) , antiderivative size = 1895, normalized size of antiderivative = 8.90 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \]

[In]

int((a + b/x)^(1/2)/(c + d/x)^3,x)

[Out]

(log((a + b/x)^(1/2)*(d*(a*d - b*c)^3)^(1/2) - a^2*d^2 - b^2*c^2 + 2*a*b*c*d)*(d*(a*d - b*c)^3)^(1/2)*(3*a^2*d
^2 + (15*b^2*c^2)/8 - 5*a*b*c*d))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) - ((b*(a + b/x)^(1
/2)*(12*a^2*d^2 + 4*b^2*c^2 - 17*a*b*c*d))/(4*c^3) + (b*(a + b/x)^(5/2)*(12*a*d^3 - 11*b*c*d^2))/(4*c^3*(a*d -
b*c)) - (d*(a + b/x)^(3/2)*(17*b^3*c^2 + 24*a^2*b*d^2 - 40*a*b^2*c*d))/(4*c^3*(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((a + b/x)^(1/2)*(d*(a*d - b*c)^3)^(1/2) + a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(d*(a*d - b*c)^3)^(1/2)
*(24*a^2*d^2 + 15*b^2*c^2 - 40*a*b*c*d))/(8*(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d)) - (atan
((((((a + b/x)^(1/2)*(1152*a^4*b^2*d^7 + 241*b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^2*
b^4*c^2*d^5))/(8*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d)) - ((6*a*d - b*c)*((4*b^6*c^11*d^2 - 21*a*b^5*c^10*d^3
+ 29*a^2*b^4*c^9*d^4 - 12*a^3*b^3*c^8*d^5)/(b^2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) - ((a + b/x)^(1/2)*(6*a*d -
b*c)*(64*b^5*c^11*d^2 - 256*a*b^4*c^10*d^3 + 320*a^2*b^3*c^9*d^4 - 128*a^3*b^2*c^8*d^5))/(16*a^(1/2)*c^4*(b^2
*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d))))/(2*a^(1/2)*c^4))*(6*a*d - b*c)*1i)/(2*a^(1/2)*c^4) + ((((a + b/x)^(1/2)*(
1152*a^4*b^2*d^7 + 241*b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^2*b^4*c^2*d^5))/(8*(b^2*
c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d)) + ((6*a*d - b*c)*((4*b^6*c^11*d^2 - 21*a*b^5*c^10*d^3 + 29*a^2*b^4*c^9*d^4 -
12*a^3*b^3*c^8*d^5)/(b^2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) + ((a + b/x)^(1/2)*(6*a*d - b*c)*(64*b^5*c^11*d^2
- 256*a*b^4*c^10*d^3 + 320*a^2*b^3*c^9*d^4 - 128*a^3*b^2*c^8*d^5))/(16*a^(1/2)*c^4*(b^2*c^8 + a^2*c^6*d^2 - 2
*a*b*c^7*d))))/(2*a^(1/2)*c^4))*(6*a*d - b*c)*1i)/(2*a^(1/2)*c^4))/((216*a^4*b^3*d^7 + (165*b^7*c^4*d^3)/8 - (
805*a*b^6*c^3*d^4)/4 - 594*a^3*b^4*c*d^6 + 558*a^2*b^5*c^2*d^5)/(b^2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) - ((((
a + b/x)^(1/2)*(1152*a^4*b^2*d^7 + 241*b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^2*b^4*c^
2*d^5))/(8*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*d)) - ((6*a*d - b*c)*((4*b^6*c^11*d^2 - 21*a*b^5*c^10*d^3 + 29*a
^2*b^4*c^9*d^4 - 12*a^3*b^3*c^8*d^5)/(b^2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) - ((a + b/x)^(1/2)*(6*a*d - b*c)*
(64*b^5*c^11*d^2 - 256*a*b^4*c^10*d^3 + 320*a^2*b^3*c^9*d^4 - 128*a^3*b^2*c^8*d^5))/(16*a^(1/2)*c^4*(b^2*c^8 +
a^2*c^6*d^2 - 2*a*b*c^7*d))))/(2*a^(1/2)*c^4))*(6*a*d - b*c))/(2*a^(1/2)*c^4) + ((((a + b/x)^(1/2)*(1152*a^4*
b^2*d^7 + 241*b^6*c^4*d^3 - 1424*a*b^5*c^3*d^4 - 3264*a^3*b^3*c*d^6 + 3296*a^2*b^4*c^2*d^5))/(8*(b^2*c^8 + a^2
*c^6*d^2 - 2*a*b*c^7*d)) + ((6*a*d - b*c)*((4*b^6*c^11*d^2 - 21*a*b^5*c^10*d^3 + 29*a^2*b^4*c^9*d^4 - 12*a^3*b
^3*c^8*d^5)/(b^2*c^11 + a^2*c^9*d^2 - 2*a*b*c^10*d) + ((a + b/x)^(1/2)*(6*a*d - b*c)*(64*b^5*c^11*d^2 - 256*a*
b^4*c^10*d^3 + 320*a^2*b^3*c^9*d^4 - 128*a^3*b^2*c^8*d^5))/(16*a^(1/2)*c^4*(b^2*c^8 + a^2*c^6*d^2 - 2*a*b*c^7*
d))))/(2*a^(1/2)*c^4))*(6*a*d - b*c))/(2*a^(1/2)*c^4)))*(6*a*d - b*c)*1i)/(a^(1/2)*c^4)