Integrand size = 15, antiderivative size = 164 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {5 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}} \]
1/12*a*x^(7/2)*(b*x+a)^(3/2)+1/6*x^(7/2)*(b*x+a)^(5/2)-5/512*a^6*arctanh(b ^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(7/2)-5/768*a^4*x^(3/2)*(b*x+a)^(1/2)/b^2+ 1/192*a^3*x^(5/2)*(b*x+a)^(1/2)/b+1/32*a^2*x^(7/2)*(b*x+a)^(1/2)+5/512*a^5 *x^(1/2)*(b*x+a)^(1/2)/b^3
Time = 0.50 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )+30 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{1536 b^{7/2}} \]
(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^5 - 10*a^4*b*x + 8*a^3*b^2*x^2 + 432* a^2*b^3*x^3 + 640*a*b^4*x^4 + 256*b^5*x^5) + 30*a^6*ArcTanh[(Sqrt[b]*Sqrt[ x])/(Sqrt[a] - Sqrt[a + b*x])])/(1536*b^(7/2))
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {60, 60, 60, 60, 60, 60, 65, 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.
| \(\displaystyle \int x^{5/2} (a+b x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \int x^{5/2} (a+b x)^{3/2}dx+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \int x^{5/2} \sqrt {a+b x}dx+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \int \frac {x^{5/2}}{\sqrt {a+b x}}dx+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a+b x)^{5/2}\) |
(x^(7/2)*(a + b*x)^(5/2))/6 + (5*a*((x^(7/2)*(a + b*x)^(3/2))/5 + (3*a*((x ^(7/2)*Sqrt[a + b*x])/4 + (a*((x^(5/2)*Sqrt[a + b*x])/(3*b) - (5*a*((x^(3/ 2)*Sqrt[a + b*x])/(2*b) - (3*a*((Sqrt[x]*Sqrt[a + b*x])/b - (a*ArcTanh[(Sq rt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/(4*b)))/(6*b)))/8))/10))/12
3.6.45.3.1 Defintions of rubi rules used
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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
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])
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {\left (256 b^{5} x^{5}+640 a \,b^{4} x^{4}+432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}-10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {b x +a}}{1536 b^{3}}-\frac {5 a^{6} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(120\) |
| default | \(\frac {x^{\frac {5}{2}} \left (b x +a \right )^{\frac {7}{2}}}{6 b}-\frac {5 a \left (\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {7}{2}}}{5 b}-\frac {3 a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {7}{2}}}{4 b}-\frac {a \left (\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\) | \(160\) |
1/1536*(256*b^5*x^5+640*a*b^4*x^4+432*a^2*b^3*x^3+8*a^3*b^2*x^2-10*a^4*b*x +15*a^5)*x^(1/2)*(b*x+a)^(1/2)/b^3-5/1024*a^6/b^(7/2)*ln((1/2*a+b*x)/b^(1/ 2)+(b*x^2+a*x)^(1/2))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)
Time = 0.24 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.26 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\left [\frac {15 \, a^{6} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{3072 \, b^{4}}, \frac {15 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt {b x + a} \sqrt {x}}{1536 \, b^{4}}\right ] \]
[1/3072*(15*a^6*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(256*b^6*x^5 + 640*a*b^5*x^4 + 432*a^2*b^4*x^3 + 8*a^3*b^3*x^2 - 10*a^4 *b^2*x + 15*a^5*b)*sqrt(b*x + a)*sqrt(x))/b^4, 1/1536*(15*a^6*sqrt(-b)*arc tan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (256*b^6*x^5 + 640*a*b^5*x^4 + 4 32*a^2*b^4*x^3 + 8*a^3*b^3*x^2 - 10*a^4*b^2*x + 15*a^5*b)*sqrt(b*x + a)*sq rt(x))/b^4]
Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (118) = 236\).
Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.49 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {5 \, a^{6} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {b x + a} a^{6} b^{5}}{\sqrt {x}} - \frac {85 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} + \frac {198 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x + a\right )} b^{8}}{x} + \frac {15 \, {\left (b x + a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x + a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x + a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x + a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x + a\right )}^{6} b^{3}}{x^{6}}\right )}} \]
5/1024*a^6*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a) /sqrt(x)))/b^(7/2) + 1/1536*(15*sqrt(b*x + a)*a^6*b^5/sqrt(x) - 85*(b*x + a)^(3/2)*a^6*b^4/x^(3/2) + 198*(b*x + a)^(5/2)*a^6*b^3/x^(5/2) + 198*(b*x + a)^(7/2)*a^6*b^2/x^(7/2) - 85*(b*x + a)^(9/2)*a^6*b/x^(9/2) + 15*(b*x + a)^(11/2)*a^6/x^(11/2))/(b^9 - 6*(b*x + a)*b^8/x + 15*(b*x + a)^2*b^7/x^2 - 20*(b*x + a)^3*b^6/x^3 + 15*(b*x + a)^4*b^5/x^4 - 6*(b*x + a)^5*b^4/x^5 + (b*x + a)^6*b^3/x^6)
Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\int x^{5/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \]
Time = 0.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (a+b x)^{5/2} \, dx=\frac {15 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b -10 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} x^{2}+432 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} x^{3}+640 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} x^{4}+256 \sqrt {x}\, \sqrt {b x +a}\, b^{6} x^{5}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6}}{1536 b^{4}} \]
(15*sqrt(x)*sqrt(a + b*x)*a**5*b - 10*sqrt(x)*sqrt(a + b*x)*a**4*b**2*x + 8*sqrt(x)*sqrt(a + b*x)*a**3*b**3*x**2 + 432*sqrt(x)*sqrt(a + b*x)*a**2*b* *4*x**3 + 640*sqrt(x)*sqrt(a + b*x)*a*b**5*x**4 + 256*sqrt(x)*sqrt(a + b*x )*b**6*x**5 - 15*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a* *6)/(1536*b**4)