Integrand size = 19, antiderivative size = 85 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=-2 (b c+a d) \sqrt {a+\frac {b}{x}}-\frac {2}{3} d \left (a+\frac {b}{x}\right )^{3/2}+a c \sqrt {a+\frac {b}{x}} x+\sqrt {a} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Output:
-2*(a*d+b*c)*(a+b/x)^(1/2)-2/3*d*(a+b/x)^(3/2)+a*c*(a+b/x)^(1/2)*x+a^(1/2) *(2*a*d+3*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {\sqrt {a+\frac {b}{x}} (a x (-8 d+3 c x)-2 b (d+3 c x))}{3 x}+\sqrt {a} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b/x)^(3/2)*(c + d/x),x]
Output:
(Sqrt[a + b/x]*(a*x*(-8*d + 3*c*x) - 2*b*(d + 3*c*x)))/(3*x) + Sqrt[a]*(3* b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]
Time = 0.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {899, 87, 60, 60, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx\) |
\(\Big \downarrow \) 899 |
\(\displaystyle -\int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {(2 a d+3 b c) \int \left (a+\frac {b}{x}\right )^{3/2} xd\frac {1}{x}}{2 a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {(2 a d+3 b c) \left (a \int \sqrt {a+\frac {b}{x}} xd\frac {1}{x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )}{2 a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {(2 a d+3 b c) \left (a \left (a \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )}{2 a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {(2 a d+3 b c) \left (a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b}+2 \sqrt {a+\frac {b}{x}}\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right )}{2 a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {c x \left (a+\frac {b}{x}\right )^{5/2}}{a}-\frac {\left (a \left (2 \sqrt {a+\frac {b}{x}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+\frac {b}{x}\right )^{3/2}\right ) (2 a d+3 b c)}{2 a}\) |
Input:
Int[(a + b/x)^(3/2)*(c + d/x),x]
Output:
(c*(a + b/x)^(5/2)*x)/a - ((3*b*c + 2*a*d)*((2*(a + b/x)^(3/2))/3 + a*(2*S qrt[a + b/x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])))/(2*a)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
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]
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {\left (3 a c \,x^{2}-8 a d x -6 b c x -2 b d \right ) \sqrt {\frac {a x +b}{x}}}{3 x}+\frac {\left (2 a d +3 b c \right ) \sqrt {a}\, \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{2 a x +2 b}\) | \(105\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-12 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} d \,x^{3}-18 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b c \,x^{3}-6 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b d \,x^{3}-9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} c \,x^{3}+12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} d x +12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b c x +4 d \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \sqrt {a}\right )}{6 x^{2} \sqrt {x \left (a x +b \right )}\, b \sqrt {a}}\) | \(205\) |
Input:
int((a+b/x)^(3/2)*(c+1/x*d),x,method=_RETURNVERBOSE)
Output:
1/3*(3*a*c*x^2-8*a*d*x-6*b*c*x-2*b*d)/x*((a*x+b)/x)^(1/2)+1/2*(2*a*d+3*b*c )*a^(1/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))*((a*x+b)/x)^(1/2)*(x*( a*x+b))^(1/2)/(a*x+b)
Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.99 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\left [\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {a} x \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (3 \, a c x^{2} - 2 \, b d - 2 \, {\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, x}, -\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (3 \, a c x^{2} - 2 \, b d - 2 \, {\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, x}\right ] \] Input:
integrate((a+b/x)^(3/2)*(c+d/x),x, algorithm="fricas")
Output:
[1/6*(3*(3*b*c + 2*a*d)*sqrt(a)*x*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x ) + b) + 2*(3*a*c*x^2 - 2*b*d - 2*(3*b*c + 4*a*d)*x)*sqrt((a*x + b)/x))/x, -1/3*(3*(3*b*c + 2*a*d)*sqrt(-a)*x*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a *x + b)) - (3*a*c*x^2 - 2*b*d - 2*(3*b*c + 4*a*d)*x)*sqrt((a*x + b)/x))/x]
Time = 19.05 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\sqrt {a} b c \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} + a \sqrt {b} c \sqrt {x} \sqrt {\frac {a x}{b} + 1} - a d \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + \frac {b}{x}} & \text {for}\: b \neq 0 \\- \sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b/x)**(3/2)*(c+d/x),x)
Output:
sqrt(a)*b*c*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + a*sqrt(b)*c*sqrt(x)*sqrt(a*x/ b + 1) - a*d*Piecewise((2*a*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + 2*sqrt (a + b/x), Ne(b, 0)), (-sqrt(a)*log(x), True)) - b*c*Piecewise((2*a*atan(s qrt(a + b/x)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b/x), Ne(b, 0)), (-sqrt(a)*lo g(x), True)) + b*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/ (3*b), True))
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c - \frac {1}{3} \, {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} d \] Input:
integrate((a+b/x)^(3/2)*(c+d/x),x, algorithm="maxima")
Output:
1/2*(2*sqrt(a + b/x)*a*x - 3*sqrt(a)*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt (a + b/x) + sqrt(a))) - 4*sqrt(a + b/x)*b)*c - 1/3*(3*a^(3/2)*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))) + 2*(a + b/x)^(3/2) + 6*sqrt( a + b/x)*a)*d
Exception generated. \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b/x)^(3/2)*(c+d/x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 2.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=2\,a^{3/2}\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )-\frac {2\,d\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}-2\,a\,d\,\sqrt {a+\frac {b}{x}}-\frac {2\,c\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,x}{b}\right )}{{\left (\frac {a\,x}{b}+1\right )}^{3/2}} \] Input:
int((a + b/x)^(3/2)*(c + d/x),x)
Output:
2*a^(3/2)*d*atanh((a + b/x)^(1/2)/a^(1/2)) - (2*d*(a + b/x)^(3/2))/3 - 2*a *d*(a + b/x)^(1/2) - (2*c*x*(a + b/x)^(3/2)*hypergeom([-3/2, -1/2], 1/2, - (a*x)/b))/((a*x)/b + 1)^(3/2)
Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44 \[ \int \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) \, dx=\frac {6 \sqrt {x}\, \sqrt {a x +b}\, a c \,x^{2}-16 \sqrt {x}\, \sqrt {a x +b}\, a d x -12 \sqrt {x}\, \sqrt {a x +b}\, b c x -4 \sqrt {x}\, \sqrt {a x +b}\, b d +12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a d \,x^{2}+18 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b c \,x^{2}+5 \sqrt {a}\, b c \,x^{2}}{6 x^{2}} \] Input:
int((a+b/x)^(3/2)*(c+d/x),x)
Output:
(6*sqrt(x)*sqrt(a*x + b)*a*c*x**2 - 16*sqrt(x)*sqrt(a*x + b)*a*d*x - 12*sq rt(x)*sqrt(a*x + b)*b*c*x - 4*sqrt(x)*sqrt(a*x + b)*b*d + 12*sqrt(a)*log(( sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*a*d*x**2 + 18*sqrt(a)*log((sqrt( a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b*c*x**2 + 5*sqrt(a)*b*c*x**2)/(6*x** 2)