Integrand size = 37, antiderivative size = 79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3} \] Output:
-2/3*(-a*e^2+c*d^2)^2/e^3/(e*x+d)^(3/2)+4*c*d*(-a*e^2+c*d^2)/e^3/(e*x+d)^( 1/2)+2*c^2*d^2*(e*x+d)^(1/2)/e^3
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {-2 a^2 e^4-4 a c d e^2 (2 d+3 e x)+2 c^2 d^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]
Output:
(-2*a^2*e^4 - 4*a*c*d*e^2*(2*d + 3*e*x) + 2*c^2*d^2*(8*d^2 + 12*d*e*x + 3* e^2*x^2))/(3*e^3*(d + e*x)^(3/2))
Time = 0.36 (sec) , antiderivative size = 79, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{3/2}}+\frac {\left (a e^2-c d^2\right )^2}{e^2 (d+e x)^{5/2}}+\frac {c^2 d^2}{e^2 \sqrt {d+e x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]
Output:
(-2*(c*d^2 - a*e^2)^2)/(3*e^3*(d + e*x)^(3/2)) + (4*c*d*(c*d^2 - a*e^2))/( e^3*Sqrt[d + e*x]) + (2*c^2*d^2*Sqrt[d + e*x])/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 = 1.46 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}}{e^{3}}-\frac {2 \left (6 c d x e +a \,e^{2}+5 c \,d^{2}\right ) \left (a \,e^{2}-c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(62\) |
pseudoelliptic | \(-\frac {2 \left (a^{2} e^{4}+6 x a c d \,e^{3}+4 \left (-\frac {3 c \,x^{2}}{4}+a \right ) c \,d^{2} e^{2}-12 x \,c^{2} d^{3} e -8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(65\) |
gosper | \(-\frac {2 \left (-3 x^{2} c^{2} d^{2} e^{2}+6 x a c d \,e^{3}-12 x \,c^{2} d^{3} e +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(72\) |
trager | \(-\frac {2 \left (-3 x^{2} c^{2} d^{2} e^{2}+6 x a c d \,e^{3}-12 x \,c^{2} d^{3} e +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\) | \(72\) |
derivativedivides | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(78\) |
default | \(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {4 c d \left (a \,e^{2}-c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\) | \(78\) |
orering | \(-\frac {2 \left (-3 x^{2} c^{2} d^{2} e^{2}+6 x a c d \,e^{3}-12 x \,c^{2} d^{3} e +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{3 e^{3} \left (c d x +a e \right )^{2} \left (e x +d \right )^{\frac {7}{2}}}\) | \(109\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2/(e*x+d)^(9/2),x,method=_RETURNVERB OSE)
Output:
2*c^2*d^2*(e*x+d)^(1/2)/e^3-2/3*(6*c*d*e*x+a*e^2+5*c*d^2)*(a*e^2-c*d^2)/e^ 3/(e*x+d)^(3/2)
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} - a^{2} e^{4} + 6 \, {\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(9/2),x, algorithm=" fricas")
Output:
2/3*(3*c^2*d^2*e^2*x^2 + 8*c^2*d^4 - 4*a*c*d^2*e^2 - a^2*e^4 + 6*(2*c^2*d^ 3*e - a*c*d*e^3)*x)*sqrt(e*x + d)/(e^5*x^2 + 2*d*e^4*x + d^2*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (73) = 146\).
Time = 0.71 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.34 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\begin {cases} - \frac {2 a^{2} e^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {8 a c d^{2} e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {12 a c d e^{3} x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 c^{2} d^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 c^{2} d^{3} e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 c^{2} d^{2} e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 \sqrt {d}} & \text {otherwise} \end {cases} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**(9/2),x)
Output:
Piecewise((-2*a**2*e**4/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 8*a*c*d**2*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 12*a *c*d*e**3*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*c**2*d* *4/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*c**2*d**3*e*x/(3 *d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 6*c**2*d**2*e**2*x**2/(3 *d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)), Ne(e, 0)), (c**2*x**3/(3* sqrt(d)), True))
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {e x + d} c^{2} d^{2}}{e^{2}} - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 6 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{3 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(9/2),x, algorithm=" maxima")
Output:
2/3*(3*sqrt(e*x + d)*c^2*d^2/e^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4 - 6* (c^2*d^3 - a*c*d*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^2))/e
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 \, \sqrt {e x + d} c^{2} d^{2}}{e^{3}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )} c^{2} d^{3} - c^{2} d^{4} - 6 \, {\left (e x + d\right )} a c d e^{2} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{3}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(9/2),x, algorithm=" giac")
Output:
2*sqrt(e*x + d)*c^2*d^2/e^3 + 2/3*(6*(e*x + d)*c^2*d^3 - c^2*d^4 - 6*(e*x + d)*a*c*d*e^2 + 2*a*c*d^2*e^2 - a^2*e^4)/((e*x + d)^(3/2)*e^3)
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=-\frac {2\,a^2\,e^4+2\,c^2\,d^4-6\,c^2\,d^2\,{\left (d+e\,x\right )}^2-12\,c^2\,d^3\,\left (d+e\,x\right )-4\,a\,c\,d^2\,e^2+12\,a\,c\,d\,e^2\,\left (d+e\,x\right )}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^(9/2),x)
Output:
-(2*a^2*e^4 + 2*c^2*d^4 - 6*c^2*d^2*(d + e*x)^2 - 12*c^2*d^3*(d + e*x) - 4 *a*c*d^2*e^2 + 12*a*c*d*e^2*(d + e*x))/(3*e^3*(d + e*x)^(3/2))
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x +8 c^{2} d^{3} e x -\frac {2}{3} a^{2} e^{4}-\frac {8}{3} a c \,d^{2} e^{2}+\frac {16}{3} c^{2} d^{4}}{\sqrt {e x +d}\, e^{3} \left (e x +d \right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(9/2),x)
Output:
(2*( - a**2*e**4 - 4*a*c*d**2*e**2 - 6*a*c*d*e**3*x + 8*c**2*d**4 + 12*c** 2*d**3*e*x + 3*c**2*d**2*e**2*x**2))/(3*sqrt(d + e*x)*e**3*(d + e*x))