Integrand size = 37, antiderivative size = 79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt {d+e x}}-\frac {4 c d \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^3}+\frac {2 c^2 d^2 (d+e x)^{3/2}}{3 e^3} \] Output:
-2*(-a*e^2+c*d^2)^2/e^3/(e*x+d)^(1/2)-4*c*d*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/e ^3+2/3*c^2*d^2*(e*x+d)^(3/2)/e^3
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-3 a^2 e^4+6 a c d e^2 (2 d+e x)+c^2 d^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt {d+e x}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(7/2),x]
Output:
(2*(-3*a^2*e^4 + 6*a*c*d*e^2*(2*d + e*x) + c^2*d^2*(-8*d^2 - 4*d*e*x + e^2 *x^2)))/(3*e^3*Sqrt[d + e*x])
Time = 0.35 (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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 \sqrt {d+e x}}+\frac {\left (a e^2-c d^2\right )^2}{e^2 (d+e x)^{3/2}}+\frac {c^2 d^2 \sqrt {d+e x}}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 c d \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^3}-\frac {2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt {d+e x}}+\frac {2 c^2 d^2 (d+e x)^{3/2}}{3 e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(7/2),x]
Output:
(-2*(c*d^2 - a*e^2)^2)/(e^3*Sqrt[d + e*x]) - (4*c*d*(c*d^2 - a*e^2)*Sqrt[d + e*x])/e^3 + (2*c^2*d^2*(d + e*x)^(3/2))/(3*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 = 1.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(-\frac {2 \left (a^{2} e^{4}-2 x a c d \,e^{3}-4 \left (\frac {c \,x^{2}}{12}+a \right ) c \,d^{2} e^{2}+\frac {4 x \,c^{2} d^{3} e}{3}+\frac {8 c^{2} d^{4}}{3}\right )}{\sqrt {e x +d}\, e^{3}}\) | \(65\) |
risch | \(\frac {2 c d \left (c d x e +6 a \,e^{2}-5 c \,d^{2}\right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{3} \sqrt {e x +d}}\) | \(71\) |
gosper | \(-\frac {2 \left (-x^{2} c^{2} d^{2} e^{2}-6 x a c d \,e^{3}+4 x \,c^{2} d^{3} e +3 a^{2} e^{4}-12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(73\) |
trager | \(-\frac {2 \left (-x^{2} c^{2} d^{2} e^{2}-6 x a c d \,e^{3}+4 x \,c^{2} d^{3} e +3 a^{2} e^{4}-12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(73\) |
derivativedivides | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c d \,e^{2} \sqrt {e x +d}-4 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(86\) |
default | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c d \,e^{2} \sqrt {e x +d}-4 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(86\) |
orering | \(-\frac {2 \left (-x^{2} c^{2} d^{2} e^{2}-6 x a c d \,e^{3}+4 x \,c^{2} d^{3} e +3 a^{2} e^{4}-12 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{3 e^{3} \left (c d x +a e \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(110\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2/(e*x+d)^(7/2),x,method=_RETURNVERB OSE)
Output:
-2/(e*x+d)^(1/2)*(a^2*e^4-2*x*a*c*d*e^3-4*(1/12*c*x^2+a)*c*d^2*e^2+4/3*x*c ^2*d^3*e+8/3*c^2*d^4)/e^3
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e^{2} x^{2} - 8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 2 \, {\left (2 \, c^{2} d^{3} e - 3 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(7/2),x, algorithm=" fricas")
Output:
2/3*(c^2*d^2*e^2*x^2 - 8*c^2*d^4 + 12*a*c*d^2*e^2 - 3*a^2*e^4 - 2*(2*c^2*d ^3*e - 3*a*c*d*e^3)*x)*sqrt(e*x + d)/(e^4*x + d*e^3)
Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {2 a^{2} e}{\sqrt {d + e x}} + \frac {8 a c d^{2}}{e \sqrt {d + e x}} + \frac {4 a c d x}{\sqrt {d + e x}} - \frac {16 c^{2} d^{4}}{3 e^{3} \sqrt {d + e x}} - \frac {8 c^{2} d^{3} x}{3 e^{2} \sqrt {d + e x}} + \frac {2 c^{2} d^{2} x^{2}}{3 e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} \sqrt {d} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**(7/2),x)
Output:
Piecewise((-2*a**2*e/sqrt(d + e*x) + 8*a*c*d**2/(e*sqrt(d + e*x)) + 4*a*c* d*x/sqrt(d + e*x) - 16*c**2*d**4/(3*e**3*sqrt(d + e*x)) - 8*c**2*d**3*x/(3 *e**2*sqrt(d + e*x)) + 2*c**2*d**2*x**2/(3*e*sqrt(d + e*x)), Ne(e, 0)), (c **2*sqrt(d)*x**3/3, True))
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} - 6 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{e^{2}} - \frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(7/2),x, algorithm=" maxima")
Output:
2/3*(((e*x + d)^(3/2)*c^2*d^2 - 6*(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/e^2 - 3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^2))/e
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{6} - 6 \, \sqrt {e x + d} c^{2} d^{3} e^{6} + 6 \, \sqrt {e x + d} a c d e^{8}\right )}}{3 \, e^{9}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(7/2),x, algorithm=" giac")
Output:
-2*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^3) + 2/3*((e*x + d )^(3/2)*c^2*d^2*e^6 - 6*sqrt(e*x + d)*c^2*d^3*e^6 + 6*sqrt(e*x + d)*a*c*d* e^8)/e^9
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {6\,a^2\,e^4+6\,c^2\,d^4-2\,c^2\,d^2\,{\left (d+e\,x\right )}^2+12\,c^2\,d^3\,\left (d+e\,x\right )-12\,a\,c\,d^2\,e^2-12\,a\,c\,d\,e^2\,\left (d+e\,x\right )}{3\,e^3\,\sqrt {d+e\,x}} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^(7/2),x)
Output:
-(6*a^2*e^4 + 6*c^2*d^4 - 2*c^2*d^2*(d + e*x)^2 + 12*c^2*d^3*(d + e*x) - 1 2*a*c*d^2*e^2 - 12*a*c*d*e^2*(d + e*x))/(3*e^3*(d + e*x)^(1/2))
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {\frac {2}{3} c^{2} d^{2} e^{2} x^{2}+4 a c d \,e^{3} x -\frac {8}{3} c^{2} d^{3} e x -2 a^{2} e^{4}+8 a c \,d^{2} e^{2}-\frac {16}{3} c^{2} d^{4}}{\sqrt {e x +d}\, e^{3}} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(7/2),x)
Output:
(2*( - 3*a**2*e**4 + 12*a*c*d**2*e**2 + 6*a*c*d*e**3*x - 8*c**2*d**4 - 4*c **2*d**3*e*x + c**2*d**2*e**2*x**2))/(3*sqrt(d + e*x)*e**3)