Integrand size = 17, antiderivative size = 96 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=-2 a d \sqrt {c+\frac {d}{x}}+\frac {1}{4} (4 a c+3 b d) \sqrt {c+\frac {d}{x}} x+\frac {1}{2} b \left (c+\frac {d}{x}\right )^{3/2} x^2+\frac {3 d (4 a c+b d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x}}}{\sqrt {c}}\right )}{4 \sqrt {c}} \] Output:
-2*a*d*(c+d/x)^(1/2)+1/4*(4*a*c+3*b*d)*(c+d/x)^(1/2)*x+1/2*b*(c+d/x)^(3/2) *x^2+3/4*d*(4*a*c+b*d)*arctanh((c+d/x)^(1/2)/c^(1/2))/c^(1/2)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\frac {1}{4} \left (\sqrt {c+\frac {d}{x}} (b x (5 d+2 c x)+a (-8 d+4 c x))+\frac {3 d (4 a c+b d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \] Input:
Integrate[(c + d/x)^(3/2)*(a + b*x),x]
Output:
(Sqrt[c + d/x]*(b*x*(5*d + 2*c*x) + a*(-8*d + 4*c*x)) + (3*d*(4*a*c + b*d) *ArcTanh[Sqrt[c + d/x]/Sqrt[c]])/Sqrt[c])/4
Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {941, 948, 87, 51, 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.
\(\displaystyle \int (a+b x) \left (c+\frac {d}{x}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 941 |
\(\displaystyle \int x \left (\frac {a}{x}+b\right ) \left (c+\frac {d}{x}\right )^{3/2}dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\int \left (\frac {a}{x}+b\right ) \left (c+\frac {d}{x}\right )^{3/2} x^3d\frac {1}{x}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {b x^2 \left (c+\frac {d}{x}\right )^{5/2}}{2 c}-\frac {(4 a c+b d) \int \left (c+\frac {d}{x}\right )^{3/2} x^2d\frac {1}{x}}{4 c}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {b x^2 \left (c+\frac {d}{x}\right )^{5/2}}{2 c}-\frac {(4 a c+b d) \left (\frac {3}{2} d \int \sqrt {c+\frac {d}{x}} xd\frac {1}{x}-x \left (c+\frac {d}{x}\right )^{3/2}\right )}{4 c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {b x^2 \left (c+\frac {d}{x}\right )^{5/2}}{2 c}-\frac {(4 a c+b d) \left (\frac {3}{2} d \left (c \int \frac {x}{\sqrt {c+\frac {d}{x}}}d\frac {1}{x}+2 \sqrt {c+\frac {d}{x}}\right )-x \left (c+\frac {d}{x}\right )^{3/2}\right )}{4 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b x^2 \left (c+\frac {d}{x}\right )^{5/2}}{2 c}-\frac {(4 a c+b d) \left (\frac {3}{2} d \left (\frac {2 c \int \frac {1}{\frac {1}{d x^2}-\frac {c}{d}}d\sqrt {c+\frac {d}{x}}}{d}+2 \sqrt {c+\frac {d}{x}}\right )-x \left (c+\frac {d}{x}\right )^{3/2}\right )}{4 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {b x^2 \left (c+\frac {d}{x}\right )^{5/2}}{2 c}-\frac {(4 a c+b d) \left (\frac {3}{2} d \left (2 \sqrt {c+\frac {d}{x}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x}}}{\sqrt {c}}\right )\right )-x \left (c+\frac {d}{x}\right )^{3/2}\right )}{4 c}\) |
Input:
Int[(c + d/x)^(3/2)*(a + b*x),x]
Output:
(b*(c + d/x)^(5/2)*x^2)/(2*c) - ((4*a*c + b*d)*(-((c + d/x)^(3/2)*x) + (3* d*(2*Sqrt[c + d/x] - 2*Sqrt[c]*ArcTanh[Sqrt[c + d/x]/Sqrt[c]]))/2))/(4*c)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !IntegerQ[p])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[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]]
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {\left (2 b c \,x^{2}+4 a c x +5 b d x -8 a d \right ) \sqrt {\frac {c x +d}{x}}}{4}+\frac {3 \left (4 a c +b d \right ) d \ln \left (\frac {\frac {d}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+d x}\right ) \sqrt {\frac {c x +d}{x}}\, \sqrt {\left (c x +d \right ) x}}{8 \sqrt {c}\, \left (c x +d \right )}\) | \(102\) |
default | \(-\frac {\sqrt {\frac {c x +d}{x}}\, \left (-4 \sqrt {c \,x^{2}+d x}\, c^{\frac {5}{2}} b \,x^{3}-24 \sqrt {c \,x^{2}+d x}\, c^{\frac {5}{2}} a \,x^{2}-10 \sqrt {c \,x^{2}+d x}\, c^{\frac {3}{2}} b d \,x^{2}-12 c^{2} \ln \left (\frac {2 \sqrt {c \,x^{2}+d x}\, \sqrt {c}+2 c x +d}{2 \sqrt {c}}\right ) a d \,x^{2}-3 c \ln \left (\frac {2 \sqrt {c \,x^{2}+d x}\, \sqrt {c}+2 c x +d}{2 \sqrt {c}}\right ) b \,d^{2} x^{2}+16 \left (c \,x^{2}+d x \right )^{\frac {3}{2}} c^{\frac {3}{2}} a \right )}{8 x \sqrt {\left (c x +d \right ) x}\, c^{\frac {3}{2}}}\) | \(184\) |
Input:
int((c+d/x)^(3/2)*(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/4*(2*b*c*x^2+4*a*c*x+5*b*d*x-8*a*d)*((c*x+d)/x)^(1/2)+3/8*(4*a*c+b*d)*d* ln((1/2*d+c*x)/c^(1/2)+(c*x^2+d*x)^(1/2))/c^(1/2)/(c*x+d)*((c*x+d)/x)^(1/2 )*((c*x+d)*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.89 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\left [\frac {3 \, {\left (4 \, a c d + b d^{2}\right )} \sqrt {c} \log \left (2 \, c x + 2 \, \sqrt {c} x \sqrt {\frac {c x + d}{x}} + d\right ) + 2 \, {\left (2 \, b c^{2} x^{2} - 8 \, a c d + {\left (4 \, a c^{2} + 5 \, b c d\right )} x\right )} \sqrt {\frac {c x + d}{x}}}{8 \, c}, -\frac {3 \, {\left (4 \, a c d + b d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x \sqrt {\frac {c x + d}{x}}}{c x + d}\right ) - {\left (2 \, b c^{2} x^{2} - 8 \, a c d + {\left (4 \, a c^{2} + 5 \, b c d\right )} x\right )} \sqrt {\frac {c x + d}{x}}}{4 \, c}\right ] \] Input:
integrate((c+d/x)^(3/2)*(b*x+a),x, algorithm="fricas")
Output:
[1/8*(3*(4*a*c*d + b*d^2)*sqrt(c)*log(2*c*x + 2*sqrt(c)*x*sqrt((c*x + d)/x ) + d) + 2*(2*b*c^2*x^2 - 8*a*c*d + (4*a*c^2 + 5*b*c*d)*x)*sqrt((c*x + d)/ x))/c, -1/4*(3*(4*a*c*d + b*d^2)*sqrt(-c)*arctan(sqrt(-c)*x*sqrt((c*x + d) /x)/(c*x + d)) - (2*b*c^2*x^2 - 8*a*c*d + (4*a*c^2 + 5*b*c*d)*x)*sqrt((c*x + d)/x))/c]
Time = 4.15 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.40 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=a \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {d}} \right )} + a c \sqrt {d} \sqrt {x} \sqrt {\frac {c x}{d} + 1} - a d \left (\begin {cases} \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c + \frac {d}{x}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {c + \frac {d}{x}} & \text {for}\: d \neq 0 \\- \sqrt {c} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + \frac {b c^{2} x^{\frac {5}{2}}}{2 \sqrt {d} \sqrt {\frac {c x}{d} + 1}} + \frac {3 b c \sqrt {d} x^{\frac {3}{2}}}{4 \sqrt {\frac {c x}{d} + 1}} + b d^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {c x}{d} + 1} + \frac {b d^{\frac {3}{2}} \sqrt {x}}{4 \sqrt {\frac {c x}{d} + 1}} + \frac {3 b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {d}} \right )}}{4 \sqrt {c}} \] Input:
integrate((c+d/x)**(3/2)*(b*x+a),x)
Output:
a*sqrt(c)*d*asinh(sqrt(c)*sqrt(x)/sqrt(d)) + a*c*sqrt(d)*sqrt(x)*sqrt(c*x/ d + 1) - a*d*Piecewise((2*c*atan(sqrt(c + d/x)/sqrt(-c))/sqrt(-c) + 2*sqrt (c + d/x), Ne(d, 0)), (-sqrt(c)*log(x), True)) + b*c**2*x**(5/2)/(2*sqrt(d )*sqrt(c*x/d + 1)) + 3*b*c*sqrt(d)*x**(3/2)/(4*sqrt(c*x/d + 1)) + b*d**(3/ 2)*sqrt(x)*sqrt(c*x/d + 1) + b*d**(3/2)*sqrt(x)/(4*sqrt(c*x/d + 1)) + 3*b* d**2*asinh(sqrt(c)*sqrt(x)/sqrt(d))/(4*sqrt(c))
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (78) = 156\).
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.76 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\frac {1}{2} \, {\left (2 \, \sqrt {c + \frac {d}{x}} c x - 3 \, \sqrt {c} d \log \left (\frac {\sqrt {c + \frac {d}{x}} - \sqrt {c}}{\sqrt {c + \frac {d}{x}} + \sqrt {c}}\right ) - 4 \, \sqrt {c + \frac {d}{x}} d\right )} a - \frac {1}{8} \, {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x}} - \sqrt {c}}{\sqrt {c + \frac {d}{x}} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x}\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {c + \frac {d}{x}} c d^{2}\right )}}{{\left (c + \frac {d}{x}\right )}^{2} - 2 \, {\left (c + \frac {d}{x}\right )} c + c^{2}}\right )} b \] Input:
integrate((c+d/x)^(3/2)*(b*x+a),x, algorithm="maxima")
Output:
1/2*(2*sqrt(c + d/x)*c*x - 3*sqrt(c)*d*log((sqrt(c + d/x) - sqrt(c))/(sqrt (c + d/x) + sqrt(c))) - 4*sqrt(c + d/x)*d)*a - 1/8*(3*d^2*log((sqrt(c + d/ x) - sqrt(c))/(sqrt(c + d/x) + sqrt(c)))/sqrt(c) - 2*(5*(c + d/x)^(3/2)*d^ 2 - 3*sqrt(c + d/x)*c*d^2)/((c + d/x)^2 - 2*(c + d/x)*c + c^2))*b
Exception generated. \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c+d/x)^(3/2)*(b*x+a),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 = 6.94 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\frac {5\,b\,x^2\,{\left (c+\frac {d}{x}\right )}^{3/2}}{4}+\frac {3\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x}}}{\sqrt {c}}\right )}{4\,\sqrt {c}}-\frac {3\,b\,c\,x^2\,\sqrt {c+\frac {d}{x}}}{4}-\frac {2\,a\,x\,{\left (c+\frac {d}{x}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {c\,x}{d}\right )}{{\left (\frac {c\,x}{d}+1\right )}^{3/2}} \] Input:
int((c + d/x)^(3/2)*(a + b*x),x)
Output:
(5*b*x^2*(c + d/x)^(3/2))/4 + (3*b*d^2*atanh((c + d/x)^(1/2)/c^(1/2)))/(4* c^(1/2)) - (3*b*c*x^2*(c + d/x)^(1/2))/4 - (2*a*x*(c + d/x)^(3/2)*hypergeo m([-3/2, -1/2], 1/2, -(c*x)/d))/((c*x)/d + 1)^(3/2)
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \left (c+\frac {d}{x}\right )^{3/2} (a+b x) \, dx=\frac {4 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{2} x -8 \sqrt {x}\, \sqrt {c x +d}\, a c d +2 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{2} x^{2}+5 \sqrt {x}\, \sqrt {c x +d}\, b c d x +12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +d}+\sqrt {x}\, \sqrt {c}}{\sqrt {d}}\right ) a c d x +3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +d}+\sqrt {x}\, \sqrt {c}}{\sqrt {d}}\right ) b \,d^{2} x -9 \sqrt {c}\, a c d x -\sqrt {c}\, b \,d^{2} x}{4 c x} \] Input:
int((c+d/x)^(3/2)*(b*x+a),x)
Output:
(4*sqrt(x)*sqrt(c*x + d)*a*c**2*x - 8*sqrt(x)*sqrt(c*x + d)*a*c*d + 2*sqrt (x)*sqrt(c*x + d)*b*c**2*x**2 + 5*sqrt(x)*sqrt(c*x + d)*b*c*d*x + 12*sqrt( c)*log((sqrt(c*x + d) + sqrt(x)*sqrt(c))/sqrt(d))*a*c*d*x + 3*sqrt(c)*log( (sqrt(c*x + d) + sqrt(x)*sqrt(c))/sqrt(d))*b*d**2*x - 9*sqrt(c)*a*c*d*x - sqrt(c)*b*d**2*x)/(4*c*x)