Integrand size = 17, antiderivative size = 77 \[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 x}{a \sqrt {b \sqrt {x}+a x}}+\frac {6 \sqrt {b \sqrt {x}+a x}}{a^2}-\frac {6 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{a^{5/2}} \]
-6*b*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/2))/a^(5/2)-4*x/a/(b*x^(1/ 2)+a*x)^(1/2)+6*(b*x^(1/2)+a*x)^(1/2)/a^2
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {2 \left (3 b+a \sqrt {x}\right ) \sqrt {b \sqrt {x}+a x}}{a^2 \left (b+a \sqrt {x}\right )}-\frac {6 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{a^{5/2}} \]
(2*(3*b + a*Sqrt[x])*Sqrt[b*Sqrt[x] + a*x])/(a^2*(b + a*Sqrt[x])) - (6*b*A rcTanh[(Sqrt[a]*Sqrt[b*Sqrt[x] + a*x])/(b + a*Sqrt[x])])/a^(5/2)
Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1924, 1124, 25, 1160, 1091, 219}
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.
\(\displaystyle \int \frac {x}{\left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1924 |
\(\displaystyle 2 \int \frac {x^{3/2}}{\left (\sqrt {x} b+a x\right )^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 1124 |
\(\displaystyle 2 \left (\frac {\int -\frac {b-a \sqrt {x}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{a^2}+\frac {2 b \sqrt {x}}{a^2 \sqrt {a x+b \sqrt {x}}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (\frac {2 b \sqrt {x}}{a^2 \sqrt {a x+b \sqrt {x}}}-\frac {\int \frac {b-a \sqrt {x}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{a^2}\right )\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 2 \left (\frac {2 b \sqrt {x}}{a^2 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {3}{2} b \int \frac {1}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}-\sqrt {a x+b \sqrt {x}}}{a^2}\right )\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle 2 \left (\frac {2 b \sqrt {x}}{a^2 \sqrt {a x+b \sqrt {x}}}-\frac {3 b \int \frac {1}{1-a x}d\frac {\sqrt {x}}{\sqrt {\sqrt {x} b+a x}}-\sqrt {a x+b \sqrt {x}}}{a^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {2 b \sqrt {x}}{a^2 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {3 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{\sqrt {a}}-\sqrt {a x+b \sqrt {x}}}{a^2}\right )\) |
2*((2*b*Sqrt[x])/(a^2*Sqrt[b*Sqrt[x] + a*x]) - (-Sqrt[b*Sqrt[x] + a*x] + ( 3*b*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/Sqrt[a])/a^2)
3.2.12.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + b*x + c*x^2])), x] + Simp[e^2/c^(m - 1) Int[(1/Sqrt[a + b*x + c*x^2])*Exp andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp [1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j ] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1 ]
Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(59)=118\).
Time = 2.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.61
method | result | size |
derivativedivides | \(\frac {2 x}{a \sqrt {b \sqrt {x}+a x}}-\frac {3 b \left (-\frac {\sqrt {x}}{a \sqrt {b \sqrt {x}+a x}}-\frac {b \left (-\frac {1}{a \sqrt {b \sqrt {x}+a x}}+\frac {b +2 a \sqrt {x}}{b a \sqrt {b \sqrt {x}+a x}}\right )}{2 a}+\frac {\ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{a^{\frac {3}{2}}}\right )}{a}\) | \(124\) |
default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (6 x \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {5}{2}}-3 x \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b +12 \sqrt {x}\, \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {3}{2}} b -6 \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{2}-4 a^{\frac {3}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}}+6 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}\, b^{2}-3 \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{3}\right )}{a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \left (a \sqrt {x}+b \right )^{2}}\) | \(236\) |
2*x/a/(b*x^(1/2)+a*x)^(1/2)-3*b/a*(-x^(1/2)/a/(b*x^(1/2)+a*x)^(1/2)-1/2*b/ a*(-1/a/(b*x^(1/2)+a*x)^(1/2)+1/b/a*(b+2*a*x^(1/2))/(b*x^(1/2)+a*x)^(1/2)) +1/a^(3/2)*ln((1/2*b+a*x^(1/2))/a^(1/2)+(b*x^(1/2)+a*x)^(1/2)))
Timed out. \[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {x}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {3 \, b \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{a^{\frac {5}{2}}} + \frac {2 \, \sqrt {a x + b \sqrt {x}}}{a^{2}} + \frac {4 \, b^{2}}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} a^{\frac {5}{2}}} \]
3*b*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^( 5/2) + 2*sqrt(a*x + b*sqrt(x))/a^2 + 4*b^2/((sqrt(a)*(sqrt(a)*sqrt(x) - sq rt(a*x + b*sqrt(x))) + b)*a^(5/2))
Timed out. \[ \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]