Integrand size = 21, antiderivative size = 113 \[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {15 b \sqrt {b \sqrt {x}+a x}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{a^2}+\frac {15 b^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^{7/2}} \]
15/2*b^2*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/2))/a^(7/2)-4*x^(3/2)/ a/(b*x^(1/2)+a*x)^(1/2)-15/2*b*(b*x^(1/2)+a*x)^(1/2)/a^3+5*x^(1/2)*(b*x^(1 /2)+a*x)^(1/2)/a^2
Time = 0.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {\sqrt {b \sqrt {x}+a x} \left (-15 b^2-5 a b \sqrt {x}+2 a^2 x\right )}{2 a^3 \left (b+a \sqrt {x}\right )}+\frac {15 b^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{2 a^{7/2}} \]
(Sqrt[b*Sqrt[x] + a*x]*(-15*b^2 - 5*a*b*Sqrt[x] + 2*a^2*x))/(2*a^3*(b + a* Sqrt[x])) + (15*b^2*ArcTanh[(Sqrt[a]*Sqrt[b*Sqrt[x] + a*x])/(b + a*Sqrt[x] )])/(2*a^(7/2))
Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1924, 1124, 2192, 27, 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.
| \(\displaystyle \int \frac {x^{3/2}}{\left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 2 \int \frac {x^2}{\left (\sqrt {x} b+a x\right )^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 1124 |
| \(\displaystyle 2 \left (\frac {\int \frac {x a^2-b \sqrt {x} a+b^2}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {a b \left (4 b-7 a \sqrt {x}\right )}{2 \sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{2 a}+\frac {1}{2} a \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {1}{4} b \int \frac {4 b-7 a \sqrt {x}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}+\frac {1}{2} a \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle 2 \left (\frac {\frac {1}{4} b \left (\frac {15}{2} b \int \frac {1}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}-7 \sqrt {a x+b \sqrt {x}}\right )+\frac {1}{2} a \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle 2 \left (\frac {\frac {1}{4} b \left (15 b \int \frac {1}{1-a x}d\frac {\sqrt {x}}{\sqrt {\sqrt {x} b+a x}}-7 \sqrt {a x+b \sqrt {x}}\right )+\frac {1}{2} a \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {\frac {1}{4} b \left (\frac {15 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{\sqrt {a}}-7 \sqrt {a x+b \sqrt {x}}\right )+\frac {1}{2} a \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^3}-\frac {2 b^2 \sqrt {x}}{a^3 \sqrt {a x+b \sqrt {x}}}\right )\) |
2*((-2*b^2*Sqrt[x])/(a^3*Sqrt[b*Sqrt[x] + a*x]) + ((a*Sqrt[x]*Sqrt[b*Sqrt[ x] + a*x])/2 + (b*(-7*Sqrt[b*Sqrt[x] + a*x] + (15*b*ArcTanh[(Sqrt[a]*Sqrt[ x])/Sqrt[b*Sqrt[x] + a*x]])/Sqrt[a]))/4)/a^3)
3.2.25.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 ]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 2.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {3}{2}}}{a \sqrt {b \sqrt {x}+a x}}-\frac {5 b \left (\frac {x}{a \sqrt {b \sqrt {x}+a x}}-\frac {3 b \left (-\frac {\sqrt {x}}{a \sqrt {b \sqrt {x}+a x}}-\frac {b \left (-\frac {1}{a \sqrt {b \sqrt {x}+a x}}+\frac {b +2 a \sqrt {x}}{b a \sqrt {b \sqrt {x}+a x}}\right )}{2 a}+\frac {\ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{2 a}\) | \(148\) |
| default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (4 x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}}+10 x \sqrt {b \sqrt {x}+a x}\, a^{\frac {7}{2}} b -32 x \,a^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b +16 x \,a^{3} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{2}-x \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{3} b^{2}+8 \sqrt {x}\, \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}} b^{2}-64 \sqrt {x}\, a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2}+32 \sqrt {x}\, a^{2} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{3}+16 a^{\frac {5}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} b -2 \sqrt {x}\, \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{2} b^{3}+2 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b^{3}-32 a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{3}+16 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{4}-\ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a \,b^{4}\right )}{4 a^{\frac {9}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \left (a \sqrt {x}+b \right )^{2}}\) | \(440\) |
x^(3/2)/a/(b*x^(1/2)+a*x)^(1/2)-5/2*b/a*(x/a/(b*x^(1/2)+a*x)^(1/2)-3/2*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. \[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {1}{2} \, \sqrt {a x + b \sqrt {x}} {\left (\frac {2 \, \sqrt {x}}{a^{2}} - \frac {7 \, b}{a^{3}}\right )} - \frac {15 \, b^{2} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{4 \, a^{\frac {7}{2}}} - \frac {4 \, b^{3}}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} a^{\frac {7}{2}}} \]
1/2*sqrt(a*x + b*sqrt(x))*(2*sqrt(x)/a^2 - 7*b/a^3) - 15/4*b^2*log(abs(-2* sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^(7/2) - 4*b^3/(( sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + b)*a^(7/2))
Timed out. \[ \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^{3/2}}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]