Integrand size = 21, antiderivative size = 115 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\frac {c^2 x \sqrt {c-d x} \sqrt {c+d x}}{8 d}-\frac {c (c-d x)^{3/2} (c+d x)^{3/2}}{12 d^2}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}+\frac {c^4 \arctan \left (\frac {\sqrt {c+d x}}{\sqrt {c-d x}}\right )}{4 d^2} \] Output:
1/8*c^2*x*(-d*x+c)^(1/2)*(d*x+c)^(1/2)/d-1/12*c*(-d*x+c)^(3/2)*(d*x+c)^(3/ 2)/d^2-1/4*(-d*x+c)^(3/2)*(d*x+c)^(5/2)/d^2+1/4*c^4*arctan((d*x+c)^(1/2)/( -d*x+c)^(1/2))/d^2
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\frac {\frac {\sqrt {c-d x} \left (-8 c^4-11 c^3 d x+5 c^2 d^2 x^2+14 c d^3 x^3+6 d^4 x^4\right )}{\sqrt {c+d x}}-6 c^4 \arctan \left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{24 d^2} \] Input:
Integrate[x*Sqrt[c - d*x]*(c + d*x)^(3/2),x]
Output:
((Sqrt[c - d*x]*(-8*c^4 - 11*c^3*d*x + 5*c^2*d^2*x^2 + 14*c*d^3*x^3 + 6*d^ 4*x^4))/Sqrt[c + d*x] - 6*c^4*ArcTan[Sqrt[c - d*x]/Sqrt[c + d*x]])/(24*d^2 )
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {90, 59, 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.
\(\displaystyle \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {c \int \sqrt {c-d x} (c+d x)^{3/2}dx}{4 d}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}\) |
\(\Big \downarrow \) 59 |
\(\displaystyle \frac {c \left (c \int \sqrt {c-d x} \sqrt {c+d x}dx-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{3 d}\right )}{4 d}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {c \left (c \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x}}dx+\frac {1}{2} x \sqrt {c-d x} \sqrt {c+d x}\right )-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{3 d}\right )}{4 d}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {c \left (c \left (c^2 \int \frac {1}{-\frac {(c-d x) d}{c+d x}-d}d\frac {\sqrt {c-d x}}{\sqrt {c+d x}}+\frac {1}{2} x \sqrt {c-d x} \sqrt {c+d x}\right )-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{3 d}\right )}{4 d}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {c \left (c \left (\frac {1}{2} x \sqrt {c-d x} \sqrt {c+d x}-\frac {c^2 \arctan \left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{d}\right )-\frac {(c-d x)^{3/2} (c+d x)^{3/2}}{3 d}\right )}{4 d}-\frac {(c-d x)^{3/2} (c+d x)^{5/2}}{4 d^2}\) |
Input:
Int[x*Sqrt[c - d*x]*(c + d*x)^(3/2),x]
Output:
-1/4*((c - d*x)^(3/2)*(c + d*x)^(5/2))/d^2 + (c*(-1/3*((c - d*x)^(3/2)*(c + d*x)^(3/2))/d + c*((x*Sqrt[c - d*x]*Sqrt[c + d*x])/2 - (c^2*ArcTan[Sqrt[ c - d*x]/Sqrt[c + d*x]])/d)))/(4*d)
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_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[2*c*(n/(m + n + 1) ) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 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 = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {\left (-6 x^{3} d^{3}-8 c \,d^{2} x^{2}+3 c^{2} d x +8 c^{3}\right ) \sqrt {-x d +c}\, \sqrt {x d +c}}{24 d^{2}}+\frac {c^{4} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right ) \sqrt {\left (-x d +c \right ) \left (x d +c \right )}}{8 d \sqrt {d^{2}}\, \sqrt {-x d +c}\, \sqrt {x d +c}}\) | \(116\) |
default | \(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-6 \,\operatorname {csgn}\left (d \right ) d^{3} x^{3} \sqrt {-d^{2} x^{2}+c^{2}}-8 \,\operatorname {csgn}\left (d \right ) c \,d^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}+3 \sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (d \right ) c^{2} d x +8 \,\operatorname {csgn}\left (d \right ) c^{3} \sqrt {-d^{2} x^{2}+c^{2}}-3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right ) c^{4}\right ) \operatorname {csgn}\left (d \right )}{24 d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}\) | \(156\) |
Input:
int(x*(-d*x+c)^(1/2)*(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/24*(-6*d^3*x^3-8*c*d^2*x^2+3*c^2*d*x+8*c^3)/d^2*(-d*x+c)^(1/2)*(d*x+c)^ (1/2)+1/8*c^4/d/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))*((- d*x+c)*(d*x+c))^(1/2)/(-d*x+c)^(1/2)/(d*x+c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=-\frac {6 \, c^{4} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-d x + c} - c}{d x}\right ) - {\left (6 \, d^{3} x^{3} + 8 \, c d^{2} x^{2} - 3 \, c^{2} d x - 8 \, c^{3}\right )} \sqrt {d x + c} \sqrt {-d x + c}}{24 \, d^{2}} \] Input:
integrate(x*(-d*x+c)^(1/2)*(d*x+c)^(3/2),x, algorithm="fricas")
Output:
-1/24*(6*c^4*arctan((sqrt(d*x + c)*sqrt(-d*x + c) - c)/(d*x)) - (6*d^3*x^3 + 8*c*d^2*x^2 - 3*c^2*d*x - 8*c^3)*sqrt(d*x + c)*sqrt(-d*x + c))/d^2
\[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\int x \sqrt {c - d x} \left (c + d x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x*(-d*x+c)**(1/2)*(d*x+c)**(3/2),x)
Output:
Integral(x*sqrt(c - d*x)*(c + d*x)**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\frac {c^{4} \arcsin \left (\frac {d x}{c}\right )}{8 \, d^{2}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} c^{2} x}{8 \, d} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} x}{4 \, d} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} c}{3 \, d^{2}} \] Input:
integrate(x*(-d*x+c)^(1/2)*(d*x+c)^(3/2),x, algorithm="maxima")
Output:
1/8*c^4*arcsin(d*x/c)/d^2 + 1/8*sqrt(-d^2*x^2 + c^2)*c^2*x/d - 1/4*(-d^2*x ^2 + c^2)^(3/2)*x/d - 1/3*(-d^2*x^2 + c^2)^(3/2)*c/d^2
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (91) = 182\).
Time = 0.23 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.64 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=-\frac {18 \, c^{4} \arcsin \left (\frac {\sqrt {2} \sqrt {d x + c}}{2 \, \sqrt {c}}\right ) + 12 \, {\left (2 \, c^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {d x + c}}{2 \, \sqrt {c}}\right ) - \sqrt {d x + c} {\left (d x - 2 \, c\right )} \sqrt {-d x + c}\right )} c^{2} + {\left (39 \, c^{3} - {\left (2 \, {\left (3 \, d x - 10 \, c\right )} {\left (d x + c\right )} + 43 \, c^{2}\right )} {\left (d x + c\right )}\right )} \sqrt {d x + c} \sqrt {-d x + c} - 8 \, {\left (6 \, c^{3} \arcsin \left (\frac {\sqrt {2} \sqrt {d x + c}}{2 \, \sqrt {c}}\right ) + {\left ({\left (2 \, d x - 5 \, c\right )} {\left (d x + c\right )} + 9 \, c^{2}\right )} \sqrt {d x + c} \sqrt {-d x + c}\right )} c}{24 \, d^{2}} \] Input:
integrate(x*(-d*x+c)^(1/2)*(d*x+c)^(3/2),x, algorithm="giac")
Output:
-1/24*(18*c^4*arcsin(1/2*sqrt(2)*sqrt(d*x + c)/sqrt(c)) + 12*(2*c^2*arcsin (1/2*sqrt(2)*sqrt(d*x + c)/sqrt(c)) - sqrt(d*x + c)*(d*x - 2*c)*sqrt(-d*x + c))*c^2 + (39*c^3 - (2*(3*d*x - 10*c)*(d*x + c) + 43*c^2)*(d*x + c))*sqr t(d*x + c)*sqrt(-d*x + c) - 8*(6*c^3*arcsin(1/2*sqrt(2)*sqrt(d*x + c)/sqrt (c)) + ((2*d*x - 5*c)*(d*x + c) + 9*c^2)*sqrt(d*x + c)*sqrt(-d*x + c))*c)/ d^2
Timed out. \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\int x\,{\left (c+d\,x\right )}^{3/2}\,\sqrt {c-d\,x} \,d x \] Input:
int(x*(c + d*x)^(3/2)*(c - d*x)^(1/2),x)
Output:
int(x*(c + d*x)^(3/2)*(c - d*x)^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int x \sqrt {c-d x} (c+d x)^{3/2} \, dx=\frac {-6 \mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right ) c^{4}-8 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{3}-3 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{2} d x +8 \sqrt {d x +c}\, \sqrt {-d x +c}\, c \,d^{2} x^{2}+6 \sqrt {d x +c}\, \sqrt {-d x +c}\, d^{3} x^{3}}{24 d^{2}} \] Input:
int(x*(-d*x+c)^(1/2)*(d*x+c)^(3/2),x)
Output:
( - 6*asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))*c**4 - 8*sqrt(c + d*x)*sqrt(c - d*x)*c**3 - 3*sqrt(c + d*x)*sqrt(c - d*x)*c**2*d*x + 8*sqrt(c + d*x)*sqr t(c - d*x)*c*d**2*x**2 + 6*sqrt(c + d*x)*sqrt(c - d*x)*d**3*x**3)/(24*d**2 )