Integrand size = 15, antiderivative size = 98 \[ \int x^{3/2} \sqrt {a+b x} \, dx=-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}} \] Output:
-1/8*a^2*x^(1/2)*(b*x+a)^(1/2)/b^2+1/12*a*x^(3/2)*(b*x+a)^(1/2)/b+1/3*x^(5 /2)*(b*x+a)^(1/2)+1/8*a^3*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(5/2)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int x^{3/2} \sqrt {a+b x} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-3 a^2+2 a b x+8 b^2 x^2\right )+6 a^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{24 b^{5/2}} \] Input:
Integrate[x^(3/2)*Sqrt[a + b*x],x]
Output:
(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-3*a^2 + 2*a*b*x + 8*b^2*x^2) + 6*a^3*ArcT anh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/(24*b^(5/2))
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
\(\displaystyle \int x^{3/2} \sqrt {a+b x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{6} a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx+\frac {1}{3} x^{5/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\) |
Input:
Int[x^(3/2)*Sqrt[a + b*x],x]
Output:
(x^(5/2)*Sqrt[a + b*x])/3 + (a*((x^(3/2)*Sqrt[a + b*x])/(2*b) - (3*a*((Sqr t[x]*Sqrt[a + b*x])/b - (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/ 2)))/(4*b)))/6
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 = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (-8 b^{2} x^{2}-2 a b x +3 a^{2}\right ) \sqrt {x}\, \sqrt {b x +a}}{24 b^{2}}+\frac {a^{3} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{16 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(87\) |
default | \(\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{2}}}{2 b}-\frac {a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4 b}\right )}{2 b}\) | \(106\) |
Input:
int(x^(3/2)*(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/24*(-8*b^2*x^2-2*a*b*x+3*a^2)*x^(1/2)*(b*x+a)^(1/2)/b^2+1/16*a^3/b^(5/2 )*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 = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41 \[ \int x^{3/2} \sqrt {a+b x} \, dx=\left [\frac {3 \, a^{3} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{3}}, -\frac {3 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) - {\left (8 \, b^{3} x^{2} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{3}}\right ] \] Input:
integrate(x^(3/2)*(b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/48*(3*a^3*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2* (8*b^3*x^2 + 2*a*b^2*x - 3*a^2*b)*sqrt(b*x + a)*sqrt(x))/b^3, -1/24*(3*a^3 *sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/sqrt(b*x + a)) - (8*b^3*x^2 + 2*a*b^2*x - 3*a^2*b)*sqrt(b*x + a)*sqrt(x))/b^3]
Time = 4.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int x^{3/2} \sqrt {a+b x} \, dx=- \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {5 \sqrt {a} x^{\frac {5}{2}}}{12 \sqrt {1 + \frac {b x}{a}}} + \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {b x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \] Input:
integrate(x**(3/2)*(b*x+a)**(1/2),x)
Output:
-a**(5/2)*sqrt(x)/(8*b**2*sqrt(1 + b*x/a)) - a**(3/2)*x**(3/2)/(24*b*sqrt( 1 + b*x/a)) + 5*sqrt(a)*x**(5/2)/(12*sqrt(1 + b*x/a)) + a**3*asinh(sqrt(b) *sqrt(x)/sqrt(a))/(8*b**(5/2)) + b*x**(7/2)/(3*sqrt(a)*sqrt(1 + b*x/a))
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (70) = 140\).
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.49 \[ \int x^{3/2} \sqrt {a+b x} \, dx=-\frac {a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {\frac {3 \, \sqrt {b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {8 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{5} - \frac {3 \, {\left (b x + a\right )} b^{4}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x + a\right )}^{3} b^{2}}{x^{3}}\right )}} \] Input:
integrate(x^(3/2)*(b*x+a)^(1/2),x, algorithm="maxima")
Output:
-1/16*a^3*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/ sqrt(x)))/b^(5/2) - 1/24*(3*sqrt(b*x + a)*a^3*b^2/sqrt(x) + 8*(b*x + a)^(3 /2)*a^3*b/x^(3/2) - 3*(b*x + a)^(5/2)*a^3/x^(5/2))/(b^5 - 3*(b*x + a)*b^4/ x + 3*(b*x + a)^2*b^3/x^2 - (b*x + a)^3*b^2/x^3)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (70) = 140\).
Time = 148.62 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int x^{3/2} \sqrt {a+b x} \, dx=-\frac {\frac {6 \, {\left (3 \, a^{2} \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) - \sqrt {{\left (b x + a\right )} b - a b} {\left (2 \, b x - 3 \, a\right )} \sqrt {b x + a}\right )} a {\left | b \right |}}{b^{3}} - \frac {{\left (15 \, a^{3} \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) + {\left (2 \, {\left (4 \, b x - 9 \, a\right )} {\left (b x + a\right )} + 33 \, a^{2}\right )} \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}\right )} {\left | b \right |}}{b^{3}}}{24 \, b} \] Input:
integrate(x^(3/2)*(b*x+a)^(1/2),x, algorithm="giac")
Output:
-1/24*(6*(3*a^2*sqrt(b)*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b))) - sqrt((b*x + a)*b - a*b)*(2*b*x - 3*a)*sqrt(b*x + a))*a*abs(b)/b ^3 - (15*a^3*sqrt(b)*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a *b))) + (2*(4*b*x - 9*a)*(b*x + a) + 33*a^2)*sqrt((b*x + a)*b - a*b)*sqrt( b*x + a))*abs(b)/b^3)/b
Timed out. \[ \int x^{3/2} \sqrt {a+b x} \, dx=\int x^{3/2}\,\sqrt {a+b\,x} \,d x \] Input:
int(x^(3/2)*(a + b*x)^(1/2),x)
Output:
int(x^(3/2)*(a + b*x)^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int x^{3/2} \sqrt {a+b x} \, dx=\frac {-3 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b +2 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, b^{3} x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3}}{24 b^{3}} \] Input:
int(x^(3/2)*(b*x+a)^(1/2),x)
Output:
( - 3*sqrt(x)*sqrt(a + b*x)*a**2*b + 2*sqrt(x)*sqrt(a + b*x)*a*b**2*x + 8* sqrt(x)*sqrt(a + b*x)*b**3*x**2 + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*s qrt(b))/sqrt(a))*a**3)/(24*b**3)