Integrand size = 20, antiderivative size = 96 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {3}{4} a^{3/2} c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right ) \]
1/4*x*(a*x+a)^(3/2)*(-c*x+c)^(3/2)+3/4*a^(3/2)*c^(3/2)*arctan(c^(1/2)*(a*x +a)^(1/2)/a^(1/2)/(-c*x+c)^(1/2))+3/8*a*c*x*(a*x+a)^(1/2)*(-c*x+c)^(1/2)
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {\sqrt {c} (a (1+x))^{3/2} \sqrt {c-c x} \left (\sqrt {c} x \sqrt {1+x} \left (-5+5 x+2 x^2-2 x^3\right )+6 \sqrt {c-c x} \arcsin \left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{8 (-1+x) (1+x)^{3/2}} \]
(Sqrt[c]*(a*(1 + x))^(3/2)*Sqrt[c - c*x]*(Sqrt[c]*x*Sqrt[1 + x]*(-5 + 5*x + 2*x^2 - 2*x^3) + 6*Sqrt[c - c*x]*ArcSin[Sqrt[c - c*x]/(Sqrt[2]*Sqrt[c])] ))/(8*(-1 + x)*(1 + x)^(3/2))
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {40, 40, 45, 218}
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.
\(\displaystyle \int (a x+a)^{3/2} (c-c x)^{3/2} \, dx\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {3}{4} a c \int \sqrt {x a+a} \sqrt {c-c x}dx+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2}\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {3}{4} a c \left (\frac {1}{2} a c \int \frac {1}{\sqrt {x a+a} \sqrt {c-c x}}dx+\frac {1}{2} x \sqrt {a x+a} \sqrt {c-c x}\right )+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {3}{4} a c \left (a c \int \frac {1}{a+\frac {c (x a+a)}{c-c x}}d\frac {\sqrt {x a+a}}{\sqrt {c-c x}}+\frac {1}{2} x \sqrt {a x+a} \sqrt {c-c x}\right )+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3}{4} a c \left (\sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {1}{2} x \sqrt {a x+a} \sqrt {c-c x}\right )+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2}\) |
(x*(a + a*x)^(3/2)*(c - c*x)^(3/2))/4 + (3*a*c*((x*Sqrt[a + a*x]*Sqrt[c - c*x])/2 + Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])]))/4
3.12.38.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* (a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1)) Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ b*c + a*d, 0] && IGtQ[m + 1/2, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x \left (2 x^{2}-5\right ) \left (-1+x \right ) \left (1+x \right ) a^{2} c^{2}}{8 \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}+\frac {3 \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right ) a^{2} c^{2} \sqrt {-a \left (1+x \right ) c \left (-1+x \right )}}{8 \sqrt {a c}\, \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}\) | \(100\) |
default | \(-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}{4 c}+\frac {3 a \left (-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}}}{3 c}+\frac {a \left (\frac {\left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}}{2 a}+\frac {3 c \left (\frac {\sqrt {-c x +c}\, \sqrt {a x +a}}{a}+\frac {c \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}\right )}{2}\right )}{3}\right )}{4}\) | \(150\) |
1/8*x*(2*x^2-5)*(-1+x)*(1+x)*a^2*c^2/(a*(1+x))^(1/2)/(-c*(-1+x))^(1/2)+3/8 /(a*c)^(1/2)*arctan((a*c)^(1/2)*x/(-a*c*x^2+a*c)^(1/2))*a^2*c^2*(-a*(1+x)* c*(-1+x))^(1/2)/(a*(1+x))^(1/2)/(-c*(-1+x))^(1/2)
Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.61 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\left [\frac {3}{16} \, \sqrt {-a c} a c \log \left (2 \, a c x^{2} + 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}, -\frac {3}{8} \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}\right ] \]
[3/16*sqrt(-a*c)*a*c*log(2*a*c*x^2 + 2*sqrt(-a*c)*sqrt(a*x + a)*sqrt(-c*x + c)*x - a*c) - 1/8*(2*a*c*x^3 - 5*a*c*x)*sqrt(a*x + a)*sqrt(-c*x + c), -3 /8*sqrt(a*c)*a*c*arctan(sqrt(a*c)*sqrt(a*x + a)*sqrt(-c*x + c)*x/(a*c*x^2 - a*c)) - 1/8*(2*a*c*x^3 - 5*a*c*x)*sqrt(a*x + a)*sqrt(-c*x + c)]
\[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\int \left (a \left (x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (x - 1\right )\right )^{\frac {3}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.52 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\frac {3 \, a^{2} c^{2} \arcsin \left (x\right )}{8 \, \sqrt {a c}} + \frac {3}{8} \, \sqrt {-a c x^{2} + a c} a c x + \frac {1}{4} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} x \]
3/8*a^2*c^2*arcsin(x)/sqrt(a*c) + 3/8*sqrt(-a*c*x^2 + a*c)*a*c*x + 1/4*(-a *c*x^2 + a*c)^(3/2)*x
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (70) = 140\).
Time = 0.45 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.20 \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=-\frac {{\left (\frac {18 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (a x + a\right )} {\left (2 \, {\left (a x + a\right )} {\left (\frac {3 \, {\left (a x + a\right )}}{a^{3}} - \frac {13}{a^{2}}\right )} + \frac {43}{a}\right )} - 39\right )} \sqrt {a x + a}\right )} c {\left | a \right |}}{24 \, a} + \frac {{\left (\frac {6 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {2 \, {\left (a x + a\right )}}{a^{2}} - \frac {7}{a}\right )} + 9\right )}\right )} c {\left | a \right |}}{6 \, a} - \frac {{\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a}\right )} c {\left | a \right |}}{a} + \frac {{\left (\frac {2 \, a^{3} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (a x - 2 \, a\right )}\right )} c {\left | a \right |}}{2 \, a^{2}} \]
-1/24*(18*a^2*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) + sqrt(-(a*x + a)*a*c + 2*a^2*c)*((a*x + a)*(2*(a*x + a)*(3*(a*x + a)/a^3 - 13/a^2) + 43/a) - 39)*sqrt(a*x + a))*c*abs(a)/a + 1/6*(6*a^2*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a ^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*((a*x + a)*(2*(a*x + a)/a^2 - 7/a) + 9))*c*abs(a)/a - (2*a^2*c*log(abs(-sqrt(-a*c) *sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a))*c*abs(a)/a + 1/2*(2*a^3*c*log(abs(-sqrt (-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) + sqrt( -(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*(a*x - 2*a))*c*abs(a)/a^2
Timed out. \[ \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx=\int {\left (a+a\,x\right )}^{3/2}\,{\left (c-c\,x\right )}^{3/2} \,d x \]