Integrand size = 21, antiderivative size = 204 \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=-\frac {231 b^5 \sqrt {b \sqrt {x}+a x}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {231 b^6 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{256 a^{13/2}} \] Output:
-231/256*b^5*(b*x^(1/2)+a*x)^(1/2)/a^6+77/128*b^4*x^(1/2)*(b*x^(1/2)+a*x)^ (1/2)/a^5-77/160*b^3*x*(b*x^(1/2)+a*x)^(1/2)/a^4+33/80*b^2*x^(3/2)*(b*x^(1 /2)+a*x)^(1/2)/a^3-11/30*b*x^2*(b*x^(1/2)+a*x)^(1/2)/a^2+1/3*x^(5/2)*(b*x^ (1/2)+a*x)^(1/2)/a+231/256*b^6*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/ 2))/a^(13/2)
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.62 \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {\sqrt {b \sqrt {x}+a x} \left (-3465 b^5+2310 a b^4 \sqrt {x}-1848 a^2 b^3 x+1584 a^3 b^2 x^{3/2}-1408 a^4 b x^2+1280 a^5 x^{5/2}\right )}{3840 a^6}+\frac {231 b^6 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{256 a^{13/2}} \] Input:
Integrate[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]
Output:
(Sqrt[b*Sqrt[x] + a*x]*(-3465*b^5 + 2310*a*b^4*Sqrt[x] - 1848*a^2*b^3*x + 1584*a^3*b^2*x^(3/2) - 1408*a^4*b*x^2 + 1280*a^5*x^(5/2)))/(3840*a^6) + (2 31*b^6*ArcTanh[(Sqrt[a]*Sqrt[b*Sqrt[x] + a*x])/(b + a*Sqrt[x])])/(256*a^(1 3/2))
Time = 0.40 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1924, 1134, 1134, 1134, 1134, 1134, 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.
| \(\displaystyle \int \frac {x^{5/2}}{\sqrt {a x+b \sqrt {x}}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 2 \int \frac {x^3}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \int \frac {x^{5/2}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{12 a}\right )\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \int \frac {x^2}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \int \frac {x^{3/2}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \left (\frac {x \sqrt {a x+b \sqrt {x}}}{3 a}-\frac {5 b \int \frac {x}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \left (\frac {x \sqrt {a x+b \sqrt {x}}}{3 a}-\frac {5 b \left (\frac {\sqrt {x} \sqrt {a x+b \sqrt {x}}}{2 a}-\frac {3 b \int \frac {\sqrt {x}}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \left (\frac {x \sqrt {a x+b \sqrt {x}}}{3 a}-\frac {5 b \left (\frac {\sqrt {x} \sqrt {a x+b \sqrt {x}}}{2 a}-\frac {3 b \left (\frac {\sqrt {a x+b \sqrt {x}}}{a}-\frac {b \int \frac {1}{\sqrt {\sqrt {x} b+a x}}d\sqrt {x}}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \left (\frac {x \sqrt {a x+b \sqrt {x}}}{3 a}-\frac {5 b \left (\frac {\sqrt {x} \sqrt {a x+b \sqrt {x}}}{2 a}-\frac {3 b \left (\frac {\sqrt {a x+b \sqrt {x}}}{a}-\frac {b \int \frac {1}{1-a x}d\frac {\sqrt {x}}{\sqrt {\sqrt {x} b+a x}}}{a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{6 a}-\frac {11 b \left (\frac {x^2 \sqrt {a x+b \sqrt {x}}}{5 a}-\frac {9 b \left (\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{4 a}-\frac {7 b \left (\frac {x \sqrt {a x+b \sqrt {x}}}{3 a}-\frac {5 b \left (\frac {\sqrt {x} \sqrt {a x+b \sqrt {x}}}{2 a}-\frac {3 b \left (\frac {\sqrt {a x+b \sqrt {x}}}{a}-\frac {b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{a^{3/2}}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )\) |
Input:
Int[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]
Output:
2*((x^(5/2)*Sqrt[b*Sqrt[x] + a*x])/(6*a) - (11*b*((x^2*Sqrt[b*Sqrt[x] + a* x])/(5*a) - (9*b*((x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(4*a) - (7*b*((x*Sqrt[b* Sqrt[x] + a*x])/(3*a) - (5*b*((Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(2*a) - (3*b *(Sqrt[b*Sqrt[x] + a*x]/a - (b*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/a^(3/2)))/(4*a)))/(6*a)))/(8*a)))/(10*a)))/(12*a))
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)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
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 ]
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {5}{2}} \sqrt {b \sqrt {x}+x a}}{3 a}-\frac {11 b \left (\frac {x^{2} \sqrt {b \sqrt {x}+x a}}{5 a}-\frac {9 b \left (\frac {x^{\frac {3}{2}} \sqrt {b \sqrt {x}+x a}}{4 a}-\frac {7 b \left (\frac {x \sqrt {b \sqrt {x}+x a}}{3 a}-\frac {5 b \left (\frac {\sqrt {x}\, \sqrt {b \sqrt {x}+x a}}{2 a}-\frac {3 b \left (\frac {\sqrt {b \sqrt {x}+x a}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+\sqrt {x}\, a}{\sqrt {a}}+\sqrt {b \sqrt {x}+x a}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{6 a}\) | \(177\) |
| default | \(\frac {\sqrt {b \sqrt {x}+x a}\, \left (2560 x^{\frac {3}{2}} \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {11}{2}}+8544 \sqrt {x}\, \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2}-5376 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {9}{2}} b x +16860 \sqrt {x}\, \sqrt {b \sqrt {x}+x a}\, a^{\frac {5}{2}} b^{4}-12240 \left (b \sqrt {x}+x a \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}+8430 \sqrt {b \sqrt {x}+x a}\, a^{\frac {3}{2}} b^{5}-15360 a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, b^{5}+7680 a \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{6}-4215 \ln \left (\frac {2 \sqrt {x}\, a +2 \sqrt {b \sqrt {x}+x a}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{6}\right )}{7680 \sqrt {\sqrt {x}\, \left (\sqrt {x}\, a +b \right )}\, a^{\frac {15}{2}}}\) | \(245\) |
Input:
int(x^(5/2)/(b*x^(1/2)+x*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*x^(5/2)*(b*x^(1/2)+x*a)^(1/2)/a-11/6*b/a*(1/5*x^2*(b*x^(1/2)+x*a)^(1/2 )/a-9/10*b/a*(1/4*x^(3/2)*(b*x^(1/2)+x*a)^(1/2)/a-7/8*b/a*(1/3*x*(b*x^(1/2 )+x*a)^(1/2)/a-5/6*b/a*(1/2*x^(1/2)*(b*x^(1/2)+x*a)^(1/2)/a-3/4*b/a*((b*x^ (1/2)+x*a)^(1/2)/a-1/2*b/a^(3/2)*ln((1/2*b+x^(1/2)*a)/a^(1/2)+(b*x^(1/2)+x *a)^(1/2)))))))
Timed out. \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\text {Timed out} \] Input:
integrate(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")
Output:
Timed out
Time = 1.80 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.98 \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=2 \left (\begin {cases} \sqrt {a x + b \sqrt {x}} \left (\frac {x^{\frac {5}{2}}}{6 a} - \frac {11 b x^{2}}{60 a^{2}} + \frac {33 b^{2} x^{\frac {3}{2}}}{160 a^{3}} - \frac {77 b^{3} x}{320 a^{4}} + \frac {77 b^{4} \sqrt {x}}{256 a^{5}} - \frac {231 b^{5}}{512 a^{6}}\right ) + \frac {231 b^{6} \left (\begin {cases} \frac {\log {\left (2 \sqrt {a} \sqrt {a x + b \sqrt {x}} + 2 a \sqrt {x} + b \right )}}{\sqrt {a}} & \text {for}\: \frac {b^{2}}{a} \neq 0 \\\frac {\left (\sqrt {x} + \frac {b}{2 a}\right ) \log {\left (\sqrt {x} + \frac {b}{2 a} \right )}}{\sqrt {a \left (\sqrt {x} + \frac {b}{2 a}\right )^{2}}} & \text {otherwise} \end {cases}\right )}{1024 a^{6}} & \text {for}\: a \neq 0 \\\frac {2 \left (b \sqrt {x}\right )^{\frac {13}{2}}}{13 b^{7}} & \text {for}\: b \neq 0 \\\tilde {\infty } x^{\frac {7}{2}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate(x**(5/2)/(b*x**(1/2)+a*x)**(1/2),x)
Output:
2*Piecewise((sqrt(a*x + b*sqrt(x))*(x**(5/2)/(6*a) - 11*b*x**2/(60*a**2) + 33*b**2*x**(3/2)/(160*a**3) - 77*b**3*x/(320*a**4) + 77*b**4*sqrt(x)/(256 *a**5) - 231*b**5/(512*a**6)) + 231*b**6*Piecewise((log(2*sqrt(a)*sqrt(a*x + b*sqrt(x)) + 2*a*sqrt(x) + b)/sqrt(a), Ne(b**2/a, 0)), ((sqrt(x) + b/(2 *a))*log(sqrt(x) + b/(2*a))/sqrt(a*(sqrt(x) + b/(2*a))**2), True))/(1024*a **6), Ne(a, 0)), (2*(b*sqrt(x))**(13/2)/(13*b**7), Ne(b, 0)), (zoo*x**(7/2 ), True))
\[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {x^{\frac {5}{2}}}{\sqrt {a x + b \sqrt {x}}} \,d x } \] Input:
integrate(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(5/2)/sqrt(a*x + b*sqrt(x)), x)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.60 \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {1}{3840} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, \sqrt {x} {\left (\frac {10 \, \sqrt {x}}{a} - \frac {11 \, b}{a^{2}}\right )} + \frac {99 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {231 \, b^{3}}{a^{4}}\right )} \sqrt {x} + \frac {1155 \, b^{4}}{a^{5}}\right )} \sqrt {x} - \frac {3465 \, b^{5}}{a^{6}}\right )} - \frac {231 \, b^{6} \log \left ({\left | 2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b \right |}\right )}{512 \, a^{\frac {13}{2}}} \] Input:
integrate(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")
Output:
1/3840*sqrt(a*x + b*sqrt(x))*(2*(4*(2*(8*sqrt(x)*(10*sqrt(x)/a - 11*b/a^2) + 99*b^2/a^3)*sqrt(x) - 231*b^3/a^4)*sqrt(x) + 1155*b^4/a^5)*sqrt(x) - 34 65*b^5/a^6) - 231/512*b^6*log(abs(2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + b))/a^(13/2)
Timed out. \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {x^{5/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \] Input:
int(x^(5/2)/(a*x + b*x^(1/2))^(1/2),x)
Output:
int(x^(5/2)/(a*x + b*x^(1/2))^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.66 \[ \int \frac {x^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {1280 x^{\frac {11}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{6}+1584 x^{\frac {7}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{4} b^{2}+2310 x^{\frac {3}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{2} b^{4}-1408 x^{\frac {9}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{5} b -1848 x^{\frac {5}{4}} \sqrt {\sqrt {x}\, a +b}\, a^{3} b^{3}-3465 x^{\frac {1}{4}} \sqrt {\sqrt {x}\, a +b}\, a \,b^{5}+3465 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {\sqrt {x}\, a +b}+x^{\frac {1}{4}} \sqrt {a}}{\sqrt {b}}\right ) b^{6}}{3840 a^{7}} \] Input:
int(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x)
Output:
(1280*x**(3/4)*sqrt(sqrt(x)*a + b)*a**6*x**2 + 1584*x**(3/4)*sqrt(sqrt(x)* a + b)*a**4*b**2*x + 2310*x**(3/4)*sqrt(sqrt(x)*a + b)*a**2*b**4 - 1408*x* *(1/4)*sqrt(sqrt(x)*a + b)*a**5*b*x**2 - 1848*x**(1/4)*sqrt(sqrt(x)*a + b) *a**3*b**3*x - 3465*x**(1/4)*sqrt(sqrt(x)*a + b)*a*b**5 + 3465*sqrt(a)*log ((sqrt(sqrt(x)*a + b) + x**(1/4)*sqrt(a))/sqrt(b))*b**6)/(3840*a**7)