Integrand size = 15, antiderivative size = 122 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\frac {5 a^3 \sqrt {x} \sqrt {a+b x}}{64 b^3}-\frac {5 a^2 x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {a x^{5/2} \sqrt {a+b x}}{24 b}+\frac {1}{4} x^{7/2} \sqrt {a+b x}-\frac {5 a^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}} \]
-5/64*a^4*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(7/2)-5/96*a^2*x^(3/2)* (b*x+a)^(1/2)/b^2+1/24*a*x^(5/2)*(b*x+a)^(1/2)/b+1/4*x^(7/2)*(b*x+a)^(1/2) +5/64*a^3*x^(1/2)*(b*x+a)^(1/2)/b^3
Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (15 a^3-10 a^2 b x+8 a b^2 x^2+48 b^3 x^3\right )+30 a^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{192 b^{7/2}} \]
(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^3 - 10*a^2*b*x + 8*a*b^2*x^2 + 48*b^3 *x^3) + 30*a^4*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])])/(192* b^(7/2))
Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.
\(\displaystyle \int x^{5/2} \sqrt {a+b x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{8} a \int \frac {x^{5/2}}{\sqrt {a+b x}}dx+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\) |
(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[( Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/(4*b)))/(6*b)))/8
3.5.89.3.1 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 = 98, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {\left (48 b^{3} x^{3}+8 a \,b^{2} x^{2}-10 a^{2} b x +15 a^{3}\right ) \sqrt {x}\, \sqrt {b x +a}}{192 b^{3}}-\frac {5 a^{4} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{128 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(98\) |
default | \(\frac {x^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{2}}}{2 b}-\frac {a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\) | \(128\) |
1/192*(48*b^3*x^3+8*a*b^2*x^2-10*a^2*b*x+15*a^3)*x^(1/2)*(b*x+a)^(1/2)/b^3 -5/128*a^4/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 = 162, normalized size of antiderivative = 1.33 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\left [\frac {15 \, a^{4} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (48 \, b^{4} x^{3} + 8 \, a b^{3} x^{2} - 10 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{4}}, \frac {15 \, a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (48 \, b^{4} x^{3} + 8 \, a b^{3} x^{2} - 10 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{4}}\right ] \]
[1/384*(15*a^4*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(48*b^4*x^3 + 8*a*b^3*x^2 - 10*a^2*b^2*x + 15*a^3*b)*sqrt(b*x + a)*sqrt( x))/b^4, 1/192*(15*a^4*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (48*b^4*x^3 + 8*a*b^3*x^2 - 10*a^2*b^2*x + 15*a^3*b)*sqrt(b*x + a)*sqrt (x))/b^4]
Time = 29.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.25 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\frac {5 a^{\frac {7}{2}} \sqrt {x}}{64 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b \sqrt {1 + \frac {b x}{a}}} + \frac {7 \sqrt {a} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {7}{2}}} + \frac {b x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
5*a**(7/2)*sqrt(x)/(64*b**3*sqrt(1 + b*x/a)) + 5*a**(5/2)*x**(3/2)/(192*b* *2*sqrt(1 + b*x/a)) - a**(3/2)*x**(5/2)/(96*b*sqrt(1 + b*x/a)) + 7*sqrt(a) *x**(7/2)/(24*sqrt(1 + b*x/a)) - 5*a**4*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(64 *b**(7/2)) + b*x**(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a))
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (88) = 176\).
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.46 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\frac {5 \, a^{4} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{128 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {b x + a} a^{4} b^{3}}{\sqrt {x}} + \frac {73 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {55 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{192 \, {\left (b^{7} - \frac {4 \, {\left (b x + a\right )} b^{6}}{x} + \frac {6 \, {\left (b x + a\right )}^{2} b^{5}}{x^{2}} - \frac {4 \, {\left (b x + a\right )}^{3} b^{4}}{x^{3}} + \frac {{\left (b x + a\right )}^{4} b^{3}}{x^{4}}\right )}} \]
5/128*a^4*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/ sqrt(x)))/b^(7/2) + 1/192*(15*sqrt(b*x + a)*a^4*b^3/sqrt(x) + 73*(b*x + a) ^(3/2)*a^4*b^2/x^(3/2) - 55*(b*x + a)^(5/2)*a^4*b/x^(5/2) + 15*(b*x + a)^( 7/2)*a^4/x^(7/2))/(b^7 - 4*(b*x + a)*b^6/x + 6*(b*x + a)^2*b^5/x^2 - 4*(b* x + a)^3*b^4/x^3 + (b*x + a)^4*b^3/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (88) = 176\).
Time = 152.74 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.84 \[ \int x^{5/2} \sqrt {a+b x} \, dx=\frac {\frac {8 \, {\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {3}{2}}} + \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} - \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} a {\left | b \right |}}{b^{2}} - \frac {{\left (\frac {105 \, a^{4} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {5}{2}}} - {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} - \frac {25 \, a}{b^{3}}\right )} + \frac {163 \, a^{2}}{b^{3}}\right )} - \frac {279 \, a^{3}}{b^{3}}\right )} \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}\right )} {\left | b \right |}}{b}}{192 \, b} \]
1/192*(8*(15*a^3*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)) )/b^(3/2) + sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a )/b^2 - 13*a/b^2) + 33*a^2/b^2))*a*abs(b)/b^2 - (105*a^4*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)))/b^(5/2) - (2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 - 25*a/b^3) + 163*a^2/b^3) - 279*a^3/b^3)*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a))*abs(b)/b)/b
Timed out. \[ \int x^{5/2} \sqrt {a+b x} \, dx=\int x^{5/2}\,\sqrt {a+b\,x} \,d x \]