Integrand size = 37, antiderivative size = 83 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{9 e^3}+\frac {2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \] Output:
2/7*(-a*e^2+c*d^2)^2*(e*x+d)^(7/2)/e^3-4/9*c*d*(-a*e^2+c*d^2)*(e*x+d)^(9/2 )/e^3+2/11*c^2*d^2*(e*x+d)^(11/2)/e^3
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (99 a^2 e^4-22 a c d e^2 (2 d-7 e x)+c^2 d^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \] Input:
Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(2*(d + e*x)^(7/2)*(99*a^2*e^4 - 22*a*c*d*e^2*(2*d - 7*e*x) + c^2*d^2*(8*d ^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)
Time = 0.39 (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.
\(\displaystyle \int \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{e^2}+\frac {(d+e x)^{5/2} \left (a e^2-c d^2\right )^2}{e^2}+\frac {c^2 d^2 (d+e x)^{9/2}}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac {2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac {2 c^2 d^2 (d+e x)^{11/2}}{11 e^3}\) |
Input:
Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(2*(c*d^2 - a*e^2)^2*(d + e*x)^(7/2))/(7*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(9/2))/(9*e^3) + (2*c^2*d^2*(d + e*x)^(11/2))/(11*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 = 2.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (a^{2} e^{4}+\frac {14 x a c d \,e^{3}}{9}-\frac {4 \left (-\frac {63 c \,x^{2}}{44}+a \right ) c \,d^{2} e^{2}}{9}-\frac {28 x \,c^{2} d^{3} e}{99}+\frac {8 c^{2} d^{4}}{99}\right )}{7 e^{3}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\) | \(68\) |
default | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\) | \(68\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 x^{2} c^{2} d^{2} e^{2}+154 x a c d \,e^{3}-28 x \,c^{2} d^{3} e +99 a^{2} e^{4}-44 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{693 e^{3}}\) | \(73\) |
orering | \(\frac {2 \left (63 x^{2} c^{2} d^{2} e^{2}+154 x a c d \,e^{3}-28 x \,c^{2} d^{3} e +99 a^{2} e^{4}-44 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{693 e^{3} \left (c d x +a e \right )^{2}}\) | \(110\) |
trager | \(\frac {2 \left (63 c^{2} d^{2} e^{5} x^{5}+154 a c d \,e^{6} x^{4}+161 e^{4} d^{3} c^{2} x^{4}+99 a^{2} e^{7} x^{3}+418 a c \,d^{2} e^{5} x^{3}+113 c^{2} d^{4} e^{3} x^{3}+297 a^{2} d \,e^{6} x^{2}+330 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+297 a^{2} d^{2} e^{5} x +22 a c \,d^{4} e^{3} x -4 c^{2} d^{6} e x +99 a^{2} d^{3} e^{4}-44 a c \,d^{5} e^{2}+8 c^{2} d^{7}\right ) \sqrt {e x +d}}{693 e^{3}}\) | \(192\) |
risch | \(\frac {2 \left (63 c^{2} d^{2} e^{5} x^{5}+154 a c d \,e^{6} x^{4}+161 e^{4} d^{3} c^{2} x^{4}+99 a^{2} e^{7} x^{3}+418 a c \,d^{2} e^{5} x^{3}+113 c^{2} d^{4} e^{3} x^{3}+297 a^{2} d \,e^{6} x^{2}+330 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+297 a^{2} d^{2} e^{5} x +22 a c \,d^{4} e^{3} x -4 c^{2} d^{6} e x +99 a^{2} d^{3} e^{4}-44 a c \,d^{5} e^{2}+8 c^{2} d^{7}\right ) \sqrt {e x +d}}{693 e^{3}}\) | \(192\) |
Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERB OSE)
Output:
2/7*(e*x+d)^(7/2)*(a^2*e^4+14/9*x*a*c*d*e^3-4/9*(-63/44*c*x^2+a)*c*d^2*e^2 -28/99*x*c^2*d^3*e+8/99*c^2*d^4)/e^3
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (71) = 142\).
Time = 0.08 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.22 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (63 \, c^{2} d^{2} e^{5} x^{5} + 8 \, c^{2} d^{7} - 44 \, a c d^{5} e^{2} + 99 \, a^{2} d^{3} e^{4} + 7 \, {\left (23 \, c^{2} d^{3} e^{4} + 22 \, a c d e^{6}\right )} x^{4} + {\left (113 \, c^{2} d^{4} e^{3} + 418 \, a c d^{2} e^{5} + 99 \, a^{2} e^{7}\right )} x^{3} + 3 \, {\left (c^{2} d^{5} e^{2} + 110 \, a c d^{3} e^{4} + 99 \, a^{2} d e^{6}\right )} x^{2} - {\left (4 \, c^{2} d^{6} e - 22 \, a c d^{4} e^{3} - 297 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" fricas")
Output:
2/693*(63*c^2*d^2*e^5*x^5 + 8*c^2*d^7 - 44*a*c*d^5*e^2 + 99*a^2*d^3*e^4 + 7*(23*c^2*d^3*e^4 + 22*a*c*d*e^6)*x^4 + (113*c^2*d^4*e^3 + 418*a*c*d^2*e^5 + 99*a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 + 110*a*c*d^3*e^4 + 99*a^2*d*e^6)*x^2 - (4*c^2*d^6*e - 22*a*c*d^4*e^3 - 297*a^2*d^2*e^5)*x)*sqrt(e*x + d)/e^3
Time = 0.88 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{2}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{9 e^{2}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{7 e^{2}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {9}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
Piecewise((2*(c**2*d**2*(d + e*x)**(11/2)/(11*e**2) + (d + e*x)**(9/2)*(2* a*c*d*e**2 - 2*c**2*d**3)/(9*e**2) + (d + e*x)**(7/2)*(a**2*e**4 - 2*a*c*d **2*e**2 + c**2*d**4)/(7*e**2))/e, Ne(e, 0)), (c**2*d**(9/2)*x**3/3, True) )
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} d^{2} - 154 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" maxima")
Output:
2/693*(63*(e*x + d)^(11/2)*c^2*d^2 - 154*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^( 9/2) + 99*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(7/2))/e^3
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (71) = 142\).
Time = 0.18 (sec) , antiderivative size = 566, normalized size of antiderivative = 6.82 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" giac")
Output:
2/3465*(3465*sqrt(e*x + d)*a^2*d^3*e^2 + 2310*((e*x + d)^(3/2) - 3*sqrt(e* x + d)*d)*a*c*d^4 + 3465*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^2*e^2 + 1386*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)* a*c*d^3 + 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*c^2*d^5/e^2 + 693*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqr t(e*x + d)*d^2)*a^2*d*e^2 + 594*(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^2 + 297*(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^4/e^2 + 99*(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^2*e^2 + 22*(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)*a*c*d + 33*(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^3/e^2 + 5*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 9 90*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d ^4 - 693*sqrt(e*x + d)*d^5)*c^2*d^2/e^2)/e
Time = 5.35 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (99\,a^2\,e^4+99\,c^2\,d^4+63\,c^2\,d^2\,{\left (d+e\,x\right )}^2-154\,c^2\,d^3\,\left (d+e\,x\right )-198\,a\,c\,d^2\,e^2+154\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{693\,e^3} \] Input:
int((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
(2*(d + e*x)^(7/2)*(99*a^2*e^4 + 99*c^2*d^4 + 63*c^2*d^2*(d + e*x)^2 - 154 *c^2*d^3*(d + e*x) - 198*a*c*d^2*e^2 + 154*a*c*d*e^2*(d + e*x)))/(693*e^3)
Time = 0.21 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.29 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (63 c^{2} d^{2} e^{5} x^{5}+154 a c d \,e^{6} x^{4}+161 c^{2} d^{3} e^{4} x^{4}+99 a^{2} e^{7} x^{3}+418 a c \,d^{2} e^{5} x^{3}+113 c^{2} d^{4} e^{3} x^{3}+297 a^{2} d \,e^{6} x^{2}+330 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+297 a^{2} d^{2} e^{5} x +22 a c \,d^{4} e^{3} x -4 c^{2} d^{6} e x +99 a^{2} d^{3} e^{4}-44 a c \,d^{5} e^{2}+8 c^{2} d^{7}\right )}{693 e^{3}} \] Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
(2*sqrt(d + e*x)*(99*a**2*d**3*e**4 + 297*a**2*d**2*e**5*x + 297*a**2*d*e* *6*x**2 + 99*a**2*e**7*x**3 - 44*a*c*d**5*e**2 + 22*a*c*d**4*e**3*x + 330* a*c*d**3*e**4*x**2 + 418*a*c*d**2*e**5*x**3 + 154*a*c*d*e**6*x**4 + 8*c**2 *d**7 - 4*c**2*d**6*e*x + 3*c**2*d**5*e**2*x**2 + 113*c**2*d**4*e**3*x**3 + 161*c**2*d**3*e**4*x**4 + 63*c**2*d**2*e**5*x**5))/(693*e**3)