Integrand size = 37, antiderivative size = 83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^3}+\frac {2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \] Output:
2/5*(-a*e^2+c*d^2)^2*(e*x+d)^(5/2)/e^3-4/7*c*d*(-a*e^2+c*d^2)*(e*x+d)^(7/2 )/e^3+2/9*c^2*d^2*(e*x+d)^(9/2)/e^3
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 (d+e x)^{5/2} \left (63 a^2 e^4+18 a c d e^2 (-2 d+5 e x)+c^2 d^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/Sqrt[d + e*x],x]
Output:
(2*(d + e*x)^(5/2)*(63*a^2*e^4 + 18*a*c*d*e^2*(-2*d + 5*e*x) + c^2*d^2*(8* d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)
Time = 0.37 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {1121, 2009}
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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{e^2}+\frac {(d+e x)^{3/2} \left (a e^2-c d^2\right )^2}{e^2}+\frac {c^2 d^2 (d+e x)^{7/2}}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac {2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/Sqrt[d + e*x],x]
Output:
(2*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(7/2))/(7*e^3) + (2*c^2*d^2*(d + e*x)^(9/2))/(9*e^3)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (a^{2} e^{4}+\frac {10 x a c d \,e^{3}}{7}-\frac {4 \left (-\frac {35 c \,x^{2}}{36}+a \right ) c \,d^{2} e^{2}}{7}-\frac {20 x \,c^{2} d^{3} e}{63}+\frac {8 c^{2} d^{4}}{63}\right )}{5 e^{3}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(68\) |
default | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(68\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 x^{2} c^{2} d^{2} e^{2}+90 x a c d \,e^{3}-20 x \,c^{2} d^{3} e +63 a^{2} e^{4}-36 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{315 e^{3}}\) | \(73\) |
orering | \(\frac {2 \left (35 x^{2} c^{2} d^{2} e^{2}+90 x a c d \,e^{3}-20 x \,c^{2} d^{3} e +63 a^{2} e^{4}-36 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) \sqrt {e x +d}\, {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{315 e^{3} \left (c d x +a e \right )^{2}}\) | \(110\) |
trager | \(\frac {2 \left (35 d^{2} e^{4} c^{2} x^{4}+90 a c d \,e^{5} x^{3}+50 c^{2} d^{3} e^{3} x^{3}+63 a^{2} e^{6} x^{2}+144 a c \,d^{2} e^{4} x^{2}+3 c^{2} d^{4} e^{2} x^{2}+126 a^{2} d \,e^{5} x +18 a c \,d^{3} e^{3} x -4 c^{2} d^{5} e x +63 a^{2} d^{2} e^{4}-36 a \,d^{4} e^{2} c +8 d^{6} c^{2}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(151\) |
risch | \(\frac {2 \left (35 d^{2} e^{4} c^{2} x^{4}+90 a c d \,e^{5} x^{3}+50 c^{2} d^{3} e^{3} x^{3}+63 a^{2} e^{6} x^{2}+144 a c \,d^{2} e^{4} x^{2}+3 c^{2} d^{4} e^{2} x^{2}+126 a^{2} d \,e^{5} x +18 a c \,d^{3} e^{3} x -4 c^{2} d^{5} e x +63 a^{2} d^{2} e^{4}-36 a \,d^{4} e^{2} c +8 d^{6} c^{2}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(151\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2/(e*x+d)^(1/2),x,method=_RETURNVERB OSE)
Output:
2/5*(e*x+d)^(5/2)*(a^2*e^4+10/7*x*a*c*d*e^3-4/7*(-35/36*c*x^2+a)*c*d^2*e^2 -20/63*x*c^2*d^3*e+8/63*c^2*d^4)/e^3
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (71) = 142\).
Time = 0.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, c^{2} d^{2} e^{4} x^{4} + 8 \, c^{2} d^{6} - 36 \, a c d^{4} e^{2} + 63 \, a^{2} d^{2} e^{4} + 10 \, {\left (5 \, c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (c^{2} d^{4} e^{2} + 48 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x^{2} - 2 \, {\left (2 \, c^{2} d^{5} e - 9 \, a c d^{3} e^{3} - 63 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{3}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(1/2),x, algorithm=" fricas")
Output:
2/315*(35*c^2*d^2*e^4*x^4 + 8*c^2*d^6 - 36*a*c*d^4*e^2 + 63*a^2*d^2*e^4 + 10*(5*c^2*d^3*e^3 + 9*a*c*d*e^5)*x^3 + 3*(c^2*d^4*e^2 + 48*a*c*d^2*e^4 + 2 1*a^2*e^6)*x^2 - 2*(2*c^2*d^5*e - 9*a*c*d^3*e^3 - 63*a^2*d*e^5)*x)*sqrt(e* x + d)/e^3
Time = 0.90 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{2}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{5 e^{2}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {7}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**(1/2),x)
Output:
Piecewise((2*(c**2*d**2*(d + e*x)**(9/2)/(9*e**2) + (d + e*x)**(7/2)*(2*a* c*d*e**2 - 2*c**2*d**3)/(7*e**2) + (d + e*x)**(5/2)*(a**2*e**4 - 2*a*c*d** 2*e**2 + c**2*d**4)/(5*e**2))/e, Ne(e, 0)), (c**2*d**(7/2)*x**3/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (71) = 142\).
Time = 0.03 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{2} d^{2} e^{2} + 42 \, {\left (\frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d}{e} + \frac {5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )}}{e}\right )} a d e + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} {\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac {21 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )}}{e^{2}}\right )}}{315 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(1/2),x, algorithm=" maxima")
Output:
2/315*(315*sqrt(e*x + d)*a^2*d^2*e^2 + 42*((3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e + 5*(c*d^2 + a*e^2)*((e*x + d)^(3 /2) - 3*sqrt(e*x + d)*d)/e)*a*d*e + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7 /2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*( e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e^2)*c*d/e^2 + 21*(c *d^2 + a*e^2)^2*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)/e^2)/e
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (71) = 142\).
Time = 0.15 (sec) , antiderivative size = 368, normalized size of antiderivative = 4.43 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{2} d^{2} e^{2} + 210 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a c d^{3} + 210 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} d e^{2} + 84 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c d^{2} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c^{2} d^{4}}{e^{2}} + 21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} e^{2} + 18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a c d + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c^{2} d^{3}}{e^{2}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}}\right )}}{315 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(1/2),x, algorithm=" giac")
Output:
2/315*(315*sqrt(e*x + d)*a^2*d^2*e^2 + 210*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*c*d^3 + 210*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d*e^2 + 84* (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c*d^2 + 21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c^2 *d^4/e^2 + 21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*a^2*e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d )^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*c*d + 18*(5*(e*x + d)^(7/2) - 21*(e* x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*c^2*d^3/e^ 2 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2)/e
Time = 5.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (63\,a^2\,e^4+63\,c^2\,d^4+35\,c^2\,d^2\,{\left (d+e\,x\right )}^2-90\,c^2\,d^3\,\left (d+e\,x\right )-126\,a\,c\,d^2\,e^2+90\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{315\,e^3} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^(1/2),x)
Output:
(2*(d + e*x)^(5/2)*(63*a^2*e^4 + 63*c^2*d^4 + 35*c^2*d^2*(d + e*x)^2 - 90* c^2*d^3*(d + e*x) - 126*a*c*d^2*e^2 + 90*a*c*d*e^2*(d + e*x)))/(315*e^3)
Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (35 c^{2} d^{2} e^{4} x^{4}+90 a c d \,e^{5} x^{3}+50 c^{2} d^{3} e^{3} x^{3}+63 a^{2} e^{6} x^{2}+144 a c \,d^{2} e^{4} x^{2}+3 c^{2} d^{4} e^{2} x^{2}+126 a^{2} d \,e^{5} x +18 a c \,d^{3} e^{3} x -4 c^{2} d^{5} e x +63 a^{2} d^{2} e^{4}-36 a c \,d^{4} e^{2}+8 c^{2} d^{6}\right )}{315 e^{3}} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(1/2),x)
Output:
(2*sqrt(d + e*x)*(63*a**2*d**2*e**4 + 126*a**2*d*e**5*x + 63*a**2*e**6*x** 2 - 36*a*c*d**4*e**2 + 18*a*c*d**3*e**3*x + 144*a*c*d**2*e**4*x**2 + 90*a* c*d*e**5*x**3 + 8*c**2*d**6 - 4*c**2*d**5*e*x + 3*c**2*d**4*e**2*x**2 + 50 *c**2*d**3*e**3*x**3 + 35*c**2*d**2*e**4*x**4))/(315*e**3)