Integrand size = 37, antiderivative size = 81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \] Output:
2*(-a*e^2+c*d^2)^2*(e*x+d)^(1/2)/e^3-4/3*c*d*(-a*e^2+c*d^2)*(e*x+d)^(3/2)/ e^3+2/5*c^2*d^2*(e*x+d)^(5/2)/e^3
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (15 a^2 e^4+10 a c d e^2 (-2 d+e x)+c^2 d^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(5/2),x]
Output:
(2*Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)
Time = 0.36 (sec) , antiderivative size = 81, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^2}+\frac {\left (a e^2-c d^2\right )^2}{e^2 \sqrt {d+e x}}+\frac {c^2 d^2 (d+e x)^{3/2}}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(5/2),x]
Output:
(2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/e^3 - (4*c*d*(c*d^2 - a*e^2)*(d + e*x) ^(3/2))/(3*e^3) + (2*c^2*d^2*(d + e*x)^(5/2))/(5*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 = 1.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {e x +d}\, \left (a^{2} e^{4}+\frac {2 x a c d \,e^{3}}{3}-\frac {4 c \left (-\frac {3 c \,x^{2}}{20}+a \right ) d^{2} e^{2}}{3}-\frac {4 x \,c^{2} d^{3} e}{15}+\frac {8 c^{2} d^{4}}{15}\right )}{e^{3}}\) | \(65\) |
derivativedivides | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a \,e^{2}-c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{3}}\) | \(67\) |
default | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a \,e^{2}-c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{3}}\) | \(67\) |
gosper | \(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c^{2} d^{2} e^{2}+10 x a c d \,e^{3}-4 x \,c^{2} d^{3} e +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\) | \(73\) |
trager | \(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c^{2} d^{2} e^{2}+10 x a c d \,e^{3}-4 x \,c^{2} d^{3} e +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\) | \(73\) |
risch | \(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c^{2} d^{2} e^{2}+10 x a c d \,e^{3}-4 x \,c^{2} d^{3} e +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\) | \(73\) |
orering | \(\frac {2 \left (3 x^{2} c^{2} d^{2} e^{2}+10 x a c d \,e^{3}-4 x \,c^{2} d^{3} e +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{15 e^{3} \left (c d x +a e \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(110\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2/(e*x+d)^(5/2),x,method=_RETURNVERB OSE)
Output:
2*(e*x+d)^(1/2)*(a^2*e^4+2/3*x*a*c*d*e^3-4/3*c*(-3/20*c*x^2+a)*d^2*e^2-4/1 5*x*c^2*d^3*e+8/15*c^2*d^4)/e^3
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \, {\left (2 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{3}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(5/2),x, algorithm=" fricas")
Output:
2/15*(3*c^2*d^2*e^2*x^2 + 8*c^2*d^4 - 20*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(2*c ^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d)/e^3
Time = 0.88 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{3 e^{2}} + \frac {\sqrt {d + e x} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{e^{2}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {3}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**(5/2),x)
Output:
Piecewise((2*(c**2*d**2*(d + e*x)**(5/2)/(5*e**2) + (d + e*x)**(3/2)*(2*a* c*d*e**2 - 2*c**2*d**3)/(3*e**2) + sqrt(d + e*x)*(a**2*e**4 - 2*a*c*d**2*e **2 + c**2*d**4)/e**2)/e, Ne(e, 0)), (c**2*d**(3/2)*x**3/3, True))
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{2} - 10 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {e x + d}\right )}}{15 \, e^{3}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(5/2),x, algorithm=" maxima")
Output:
2/15*(3*(e*x + d)^(5/2)*c^2*d^2 - 10*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^(3/2) + 15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(e*x + d))/e^3
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a^{2} e^{2} + 10 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a c d + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c^{2} d^{2}}{e^{2}}\right )}}{15 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(5/2),x, algorithm=" giac")
Output:
2/15*(15*sqrt(e*x + d)*a^2*e^2 + 10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)* a*c*d + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)* c^2*d^2/e^2)/e
Time = 5.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2\,\sqrt {d+e\,x}\,\left (15\,a^2\,e^4+15\,c^2\,d^4+3\,c^2\,d^2\,{\left (d+e\,x\right )}^2-10\,c^2\,d^3\,\left (d+e\,x\right )-30\,a\,c\,d^2\,e^2+10\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{15\,e^3} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^(5/2),x)
Output:
(2*(d + e*x)^(1/2)*(15*a^2*e^4 + 15*c^2*d^4 + 3*c^2*d^2*(d + e*x)^2 - 10*c ^2*d^3*(d + e*x) - 30*a*c*d^2*e^2 + 10*a*c*d*e^2*(d + e*x)))/(15*e^3)
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {e x +d}\, \left (3 c^{2} d^{2} e^{2} x^{2}+10 a c d \,e^{3} x -4 c^{2} d^{3} e x +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(5/2),x)
Output:
(2*sqrt(d + e*x)*(15*a**2*e**4 - 20*a*c*d**2*e**2 + 10*a*c*d*e**3*x + 8*c* *2*d**4 - 4*c**2*d**3*e*x + 3*c**2*d**2*e**2*x**2))/(15*e**3)