Integrand size = 29, antiderivative size = 119 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}} \] Output:
-64/315*d^2*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-16/63*d*(-c*e^2*x^2 +c*d^2)^(5/2)/c/e/(e*x+d)^(3/2)-2/9*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^( 1/2)
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.52 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (107 d^2+110 d e x+35 e^2 x^2\right )}{315 e \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2),x]
Output:
(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(107*d^2 + 110*d*e*x + 35*e^2*x^ 2))/(315*e*Sqrt[d + e*x])
Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {459, 459, 458}
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.
\(\displaystyle \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {8}{9} d \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}}dx-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {8}{9} d \left (\frac {4}{7} d \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}}dx-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle \frac {8}{9} d \left (-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}}\right )-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\) |
Input:
Int[Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2),x]
Output:
(-2*(c*d^2 - c*e^2*x^2)^(5/2))/(9*c*e*Sqrt[d + e*x]) + (8*d*((-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2 ))/(7*c*e*(d + e*x)^(3/2))))/9
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {2 \left (-e x +d \right ) \left (35 e^{2} x^{2}+110 d e x +107 d^{2}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{315 e \left (e x +d \right )^{\frac {3}{2}}}\) | \(55\) |
orering | \(-\frac {2 \left (-e x +d \right ) \left (35 e^{2} x^{2}+110 d e x +107 d^{2}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{315 e \left (e x +d \right )^{\frac {3}{2}}}\) | \(55\) |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2} \left (35 e^{2} x^{2}+110 d e x +107 d^{2}\right )}{315 \sqrt {e x +d}\, e}\) | \(57\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (35 e^{4} x^{4}+40 d \,e^{3} x^{3}-78 d^{2} e^{2} x^{2}-104 d^{3} e x +107 d^{4}\right ) \left (-e x +d \right )}{315 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(118\) |
Input:
int((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/315*(-e*x+d)*(35*e^2*x^2+110*d*e*x+107*d^2)*(-c*e^2*x^2+c*d^2)^(3/2)/e/ (e*x+d)^(3/2)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c e^{4} x^{4} + 40 \, c d e^{3} x^{3} - 78 \, c d^{2} e^{2} x^{2} - 104 \, c d^{3} e x + 107 \, c d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \] Input:
integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")
Output:
-2/315*(35*c*e^4*x^4 + 40*c*d*e^3*x^3 - 78*c*d^2*e^2*x^2 - 104*c*d^3*e*x + 107*c*d^4)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)
\[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\int \left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \] Input:
integrate((e*x+d)**(1/2)*(-c*e**2*x**2+c*d**2)**(3/2),x)
Output:
Integral((-c*(-d + e*x)*(d + e*x))**(3/2)*sqrt(d + e*x), x)
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c^{\frac {3}{2}} e^{4} x^{4} + 40 \, c^{\frac {3}{2}} d e^{3} x^{3} - 78 \, c^{\frac {3}{2}} d^{2} e^{2} x^{2} - 104 \, c^{\frac {3}{2}} d^{3} e x + 107 \, c^{\frac {3}{2}} d^{4}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \] Input:
integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")
Output:
-2/315*(35*c^(3/2)*e^4*x^4 + 40*c^(3/2)*d*e^3*x^3 - 78*c^(3/2)*d^2*e^2*x^2 - 104*c^(3/2)*d^3*e*x + 107*c^(3/2)*d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (101) = 202\).
Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.79 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (315 \, \sqrt {-c e x + c d} c d^{4} - \frac {42 \, {\left (15 \, \sqrt {-c e x + c d} c^{2} d^{2} - 10 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c d + 3 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d}\right )} d^{2}}{c} + \frac {315 \, \sqrt {-c e x + c d} c^{4} d^{4} - 420 \, {\left (-c e x + c d\right )}^{\frac {3}{2}} c^{3} d^{3} + 378 \, {\left (c e x - c d\right )}^{2} \sqrt {-c e x + c d} c^{2} d^{2} + 180 \, {\left (c e x - c d\right )}^{3} \sqrt {-c e x + c d} c d + 35 \, {\left (c e x - c d\right )}^{4} \sqrt {-c e x + c d}}{c^{3}}\right )}}{315 \, e} \] Input:
integrate((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")
Output:
-2/315*(315*sqrt(-c*e*x + c*d)*c*d^4 - 42*(15*sqrt(-c*e*x + c*d)*c^2*d^2 - 10*(-c*e*x + c*d)^(3/2)*c*d + 3*(c*e*x - c*d)^2*sqrt(-c*e*x + c*d))*d^2/c + (315*sqrt(-c*e*x + c*d)*c^4*d^4 - 420*(-c*e*x + c*d)^(3/2)*c^3*d^3 + 37 8*(c*e*x - c*d)^2*sqrt(-c*e*x + c*d)*c^2*d^2 + 180*(c*e*x - c*d)^3*sqrt(-c *e*x + c*d)*c*d + 35*(c*e*x - c*d)^4*sqrt(-c*e*x + c*d))/c^3)/e
Time = 7.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {214\,c\,d^4\,\sqrt {d+e\,x}}{315\,e^2}-\frac {52\,c\,d^2\,x^2\,\sqrt {d+e\,x}}{105}+\frac {2\,c\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {208\,c\,d^3\,x\,\sqrt {d+e\,x}}{315\,e}+\frac {16\,c\,d\,e\,x^3\,\sqrt {d+e\,x}}{63}\right )}{x+\frac {d}{e}} \] Input:
int((c*d^2 - c*e^2*x^2)^(3/2)*(d + e*x)^(1/2),x)
Output:
-((c*d^2 - c*e^2*x^2)^(1/2)*((214*c*d^4*(d + e*x)^(1/2))/(315*e^2) - (52*c *d^2*x^2*(d + e*x)^(1/2))/105 + (2*c*e^2*x^4*(d + e*x)^(1/2))/9 - (208*c*d ^3*x*(d + e*x)^(1/2))/(315*e) + (16*c*d*e*x^3*(d + e*x)^(1/2))/63))/(x + d /e)
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.47 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {-e x +d}\, c \left (-35 e^{4} x^{4}-40 d \,e^{3} x^{3}+78 d^{2} e^{2} x^{2}+104 d^{3} e x -107 d^{4}\right )}{315 e} \] Input:
int((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x)
Output:
(2*sqrt(c)*sqrt(d - e*x)*c*( - 107*d**4 + 104*d**3*e*x + 78*d**2*e**2*x**2 - 40*d*e**3*x**3 - 35*e**4*x**4))/(315*e)