Integrand size = 19, antiderivative size = 144 \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4 b^3 \sqrt {x}}{a^4 \sqrt {b \sqrt {x}+a x}}+\frac {19 b^2 \sqrt {b \sqrt {x}+a x}}{4 a^4}-\frac {11 b \sqrt {x} \sqrt {b \sqrt {x}+a x}}{6 a^3}+\frac {2 x \sqrt {b \sqrt {x}+a x}}{3 a^2}-\frac {35 b^3 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{4 a^{9/2}} \] Output:
4*b^3*x^(1/2)/a^4/(b*x^(1/2)+a*x)^(1/2)+19/4*b^2*(b*x^(1/2)+a*x)^(1/2)/a^4 -11/6*b*x^(1/2)*(b*x^(1/2)+a*x)^(1/2)/a^3+2/3*x*(b*x^(1/2)+a*x)^(1/2)/a^2- 35/4*b^3*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/2))/a^(9/2)
Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {\sqrt {b \sqrt {x}+a x} \left (105 b^3+35 a b^2 \sqrt {x}-14 a^2 b x+8 a^3 x^{3/2}\right )}{12 a^4 \left (b+a \sqrt {x}\right )}-\frac {35 b^3 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{4 a^{9/2}} \] Input:
Integrate[x^2/(b*Sqrt[x] + a*x)^(3/2),x]
Output:
(Sqrt[b*Sqrt[x] + a*x]*(105*b^3 + 35*a*b^2*Sqrt[x] - 14*a^2*b*x + 8*a^3*x^ (3/2)))/(12*a^4*(b + a*Sqrt[x])) - (35*b^3*ArcTanh[(Sqrt[a]*Sqrt[b*Sqrt[x] + a*x])/(b + a*Sqrt[x])])/(4*a^(9/2))
Time = 0.43 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1924, 1124, 25, 2192, 27, 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.
| \(\displaystyle \int \frac {x^2}{\left (a x+b \sqrt {x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 2 \int \frac {x^{5/2}}{\left (\sqrt {x} b+a x\right )^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 1124 |
| \(\displaystyle 2 \left (\frac {\int -\frac {-x^{3/2} a^3+b x a^2-b^2 \sqrt {x} a+b^3}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{a^4}+\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\int \frac {-x^{3/2} a^3+b x a^2-b^2 \sqrt {x} a+b^3}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{a^4}\right )\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\int \frac {11 b x a^3-6 b^2 \sqrt {x} a^2+6 b^3 a}{2 \sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{3 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\int \frac {11 b x a^3-6 b^2 \sqrt {x} a^2+6 b^3 a}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\frac {\int \frac {3 a^2 b^2 \left (8 b-19 a \sqrt {x}\right )}{2 \sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{2 a}+\frac {11}{2} a^2 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\frac {3}{4} a b^2 \int \frac {8 b-19 a \sqrt {x}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}+\frac {11}{2} a^2 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\frac {3}{4} a b^2 \left (\frac {35}{2} b \int \frac {1}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}-19 \sqrt {a x+b \sqrt {x}}\right )+\frac {11}{2} a^2 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\frac {3}{4} a b^2 \left (35 b \int \frac {1}{1-a x}d\frac {\sqrt {x}}{\sqrt {\sqrt {x} b+a x}}-19 \sqrt {a x+b \sqrt {x}}\right )+\frac {11}{2} a^2 b \sqrt {x} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {2 b^3 \sqrt {x}}{a^4 \sqrt {a x+b \sqrt {x}}}-\frac {\frac {\frac {11}{2} a^2 b \sqrt {x} \sqrt {a x+b \sqrt {x}}+\frac {3}{4} a b^2 \left (\frac {35 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{\sqrt {a}}-19 \sqrt {a x+b \sqrt {x}}\right )}{6 a}-\frac {1}{3} a^2 x \sqrt {a x+b \sqrt {x}}}{a^4}\right )\) |
Input:
Int[x^2/(b*Sqrt[x] + a*x)^(3/2),x]
Output:
2*((2*b^3*Sqrt[x])/(a^4*Sqrt[b*Sqrt[x] + a*x]) - (-1/3*(a^2*x*Sqrt[b*Sqrt[ x] + a*x]) + ((11*a^2*b*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/2 + (3*a*b^2*(-19*S qrt[b*Sqrt[x] + a*x] + (35*b*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a* x]])/Sqrt[a]))/4)/(6*a))/a^4)
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 = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {2 x^{2}}{3 a \sqrt {b \sqrt {x}+x a}}-\frac {7 b \left (\frac {x^{\frac {3}{2}}}{2 a \sqrt {b \sqrt {x}+x a}}-\frac {5 b \left (\frac {x}{a \sqrt {b \sqrt {x}+x a}}-\frac {3 b \left (-\frac {\sqrt {x}}{a \sqrt {b \sqrt {x}+x a}}-\frac {b \left (-\frac {1}{a \sqrt {b \sqrt {x}+x a}}+\frac {b +2 \sqrt {x}\, a}{b a \sqrt {b \sqrt {x}+x a}}\right )}{2 a}+\frac {\ln \left (\frac {\frac {b}{2}+\sqrt {x}\, a}{\sqrt {a}}+\sqrt {b \sqrt {x}+x a}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{3 a}\) | \(175\) |
| default | \(\frac {\sqrt {b \sqrt {x}+x a}\, \left (16 x \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {9}{2}}-60 \sqrt {b \sqrt {x}+x a}\, x^{\frac {3}{2}} a^{\frac {9}{2}} b +32 \sqrt {x}\, a^{\frac {7}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} b -150 \sqrt {b \sqrt {x}+x a}\, x \,a^{\frac {7}{2}} b^{2}+240 x \,a^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{2}-120 x \,a^{3} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{3}+16 a^{\frac {5}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} b^{2}-120 \sqrt {x}\, a^{\frac {5}{2}} \sqrt {b \sqrt {x}+x a}\, b^{3}+15 x \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}+480 \sqrt {x}\, a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{3}-240 \sqrt {x}\, a^{2} \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{4}-96 a^{\frac {5}{2}} \left (\sqrt {x}\, \left (\sqrt {x}\, a +b \right )\right )^{\frac {3}{2}} b^{2}-30 a^{\frac {3}{2}} \sqrt {b \sqrt {x}+x a}\, b^{4}+30 \sqrt {x}\, \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b^{4}+240 a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{4}-120 \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{5}+15 \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{5}\right )}{24 a^{\frac {11}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \left (\sqrt {x}\, a +b \right )^{2}}\) | \(503\) |
Input:
int(x^2/(b*x^(1/2)+x*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3*x^2/a/(b*x^(1/2)+x*a)^(1/2)-7/3*b/a*(1/2*x^(3/2)/a/(b*x^(1/2)+x*a)^(1/ 2)-5/4*b/a*(x/a/(b*x^(1/2)+x*a)^(1/2)-3/2*b/a*(-x^(1/2)/a/(b*x^(1/2)+x*a)^ (1/2)-1/2*b/a*(-1/a/(b*x^(1/2)+x*a)^(1/2)+1/b/a*(b+2*x^(1/2)*a)/(b*x^(1/2) +x*a)^(1/2))+1/a^(3/2)*ln((1/2*b+x^(1/2)*a)/a^(1/2)+(b*x^(1/2)+x*a)^(1/2)) )))
Timed out. \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^2/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2/(b*x**(1/2)+a*x)**(3/2),x)
Output:
Integral(x**2/(a*x + b*sqrt(x))**(3/2), x)
\[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2/(a*x + b*sqrt(x))^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {1}{12} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, \sqrt {x} {\left (\frac {4 \, \sqrt {x}}{a^{2}} - \frac {11 \, b}{a^{3}}\right )} + \frac {57 \, b^{2}}{a^{4}}\right )} + \frac {35 \, b^{3} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{8 \, a^{\frac {9}{2}}} + \frac {4 \, b^{4}}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} a^{\frac {9}{2}}} \] Input:
integrate(x^2/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Output:
1/12*sqrt(a*x + b*sqrt(x))*(2*sqrt(x)*(4*sqrt(x)/a^2 - 11*b/a^3) + 57*b^2/ a^4) + 35/8*b^3*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x) )) - b))/a^(9/2) + 4*b^4/((sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x) )) + b)*a^(9/2))
Timed out. \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \] Input:
int(x^2/(a*x + b*x^(1/2))^(3/2),x)
Output:
int(x^2/(a*x + b*x^(1/2))^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {64 x^{\frac {7}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{4}+280 x^{\frac {3}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{2} b^{2}-112 x^{\frac {5}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{3} b +840 x^{\frac {1}{4}} \sqrt {\sqrt {x}\, a +b}\, a \,b^{3}-840 \sqrt {x}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {\sqrt {x}\, a +b}+x^{\frac {1}{4}} \sqrt {a}}{\sqrt {b}}\right ) a \,b^{3}+525 \sqrt {x}\, \sqrt {a}\, a \,b^{3}-840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {\sqrt {x}\, a +b}+x^{\frac {1}{4}} \sqrt {a}}{\sqrt {b}}\right ) b^{4}+525 \sqrt {a}\, b^{4}}{96 a^{5} \left (\sqrt {x}\, a +b \right )} \] Input:
int(x^2/(b*x^(1/2)+a*x)^(3/2),x)
Output:
(64*x**(3/4)*sqrt(sqrt(x)*a + b)*a**4*x + 280*x**(3/4)*sqrt(sqrt(x)*a + b) *a**2*b**2 - 112*x**(1/4)*sqrt(sqrt(x)*a + b)*a**3*b*x + 840*x**(1/4)*sqrt (sqrt(x)*a + b)*a*b**3 - 840*sqrt(x)*sqrt(a)*log((sqrt(sqrt(x)*a + b) + x* *(1/4)*sqrt(a))/sqrt(b))*a*b**3 + 525*sqrt(x)*sqrt(a)*a*b**3 - 840*sqrt(a) *log((sqrt(sqrt(x)*a + b) + x**(1/4)*sqrt(a))/sqrt(b))*b**4 + 525*sqrt(a)* b**4)/(96*a**5*(sqrt(x)*a + b))