Integrand size = 37, antiderivative size = 115 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \] Output:
2/3*(-a*e^2+c*d^2)^3/e^4/(e*x+d)^(3/2)-6*c*d*(-a*e^2+c*d^2)^2/e^4/(e*x+d)^ (1/2)-6*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/e^4+2/3*c^3*d^3*(e*x+d)^(3/2) /e^4
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=-\frac {2 \left (a^3 e^6+3 a^2 c d e^4 (2 d+3 e x)-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(11/2),x]
Output:
(-2*(a^3*e^6 + 3*a^2*c*d*e^4*(2*d + 3*e*x) - 3*a*c^2*d^2*e^2*(8*d^2 + 12*d *e*x + 3*e^2*x^2) + c^3*d^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) )/(3*e^4*(d + e*x)^(3/2))
Time = 0.40 (sec) , antiderivative size = 115, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{3/2}}+\frac {\left (a e^2-c d^2\right )^3}{e^3 (d+e x)^{5/2}}+\frac {c^3 d^3 \sqrt {d+e x}}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6 c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}+\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(11/2),x]
Output:
(2*(c*d^2 - a*e^2)^3)/(3*e^4*(d + e*x)^(3/2)) - (6*c*d*(c*d^2 - a*e^2)^2)/ (e^4*Sqrt[d + e*x]) - (6*c^2*d^2*(c*d^2 - a*e^2)*Sqrt[d + e*x])/e^4 + (2*c ^3*d^3*(d + e*x)^(3/2))/(3*e^4)
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 = 1.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-c d x e +a \,e^{2}-2 c \,d^{2}\right ) \left (a^{2} e^{4}+10 x a c d \,e^{3}+8 \left (\frac {c \,x^{2}}{8}+a \right ) c \,d^{2} e^{2}-8 x \,c^{2} d^{3} e -8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(83\) |
risch | \(\frac {2 c^{2} d^{2} \left (c d x e +9 a \,e^{2}-8 c \,d^{2}\right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (9 c d x e +a \,e^{2}+8 c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(93\) |
gosper | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 x^{2} a \,c^{2} d^{2} e^{4}+6 c^{3} d^{4} e^{2} x^{2}+9 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+24 c^{3} d^{5} e x +e^{6} a^{3}+6 d^{2} e^{4} a^{2} c -24 d^{4} e^{2} a \,c^{2}+16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(130\) |
trager | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 x^{2} a \,c^{2} d^{2} e^{4}+6 c^{3} d^{4} e^{2} x^{2}+9 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+24 c^{3} d^{5} e x +e^{6} a^{3}+6 d^{2} e^{4} a^{2} c -24 d^{4} e^{2} a \,c^{2}+16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(130\) |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(141\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(141\) |
orering | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 x^{2} a \,c^{2} d^{2} e^{4}+6 c^{3} d^{4} e^{2} x^{2}+9 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+24 c^{3} d^{5} e x +e^{6} a^{3}+6 d^{2} e^{4} a^{2} c -24 d^{4} e^{2} a \,c^{2}+16 d^{6} c^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{3 e^{4} \left (c d x +a e \right )^{3} \left (e x +d \right )^{\frac {9}{2}}}\) | \(167\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3/(e*x+d)^(11/2),x,method=_RETURNVER BOSE)
Output:
-2/3/(e*x+d)^(3/2)*(-c*d*e*x+a*e^2-2*c*d^2)*(a^2*e^4+10*x*a*c*d*e^3+8*(1/8 *c*x^2+a)*c*d^2*e^2-8*x*c^2*d^3*e-8*c^2*d^4)/e^4
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 24 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (8 \, c^{3} d^{5} e - 12 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(11/2),x, algorithm= "fricas")
Output:
2/3*(c^3*d^3*e^3*x^3 - 16*c^3*d^6 + 24*a*c^2*d^4*e^2 - 6*a^2*c*d^2*e^4 - a ^3*e^6 - 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 - 3*(8*c^3*d^5*e - 12*a*c ^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^ 4)
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (107) = 214\).
Time = 1.08 (sec) , antiderivative size = 450, normalized size of antiderivative = 3.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=\begin {cases} - \frac {2 a^{3} e^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 c^{3} d^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 c^{3} d^{5} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} \sqrt {d} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**(11/2),x)
Output:
Piecewise((-2*a**3*e**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*a**2*c*d**2*e**4/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 18*a**2*c*d*e**5*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48* a*c**2*d**4*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*a* c**2*d**3*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 18*a* c**2*d**2*e**4*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 32 *c**3*d**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*c**3*d** 5*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*c**3*d**4*e** 2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*c**3*d**3*e** 3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), (c**3 *sqrt(d)*x**4/4, True))
Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} - 9 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt {e x + d}}{e^{3}} + \frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(11/2),x, algorithm= "maxima")
Output:
2/3*(((e*x + d)^(3/2)*c^3*d^3 - 9*(c^3*d^4 - a*c^2*d^2*e^2)*sqrt(e*x + d)) /e^3 + (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6 - 9*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e
Time = 0.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=-\frac {2 \, {\left (9 \, {\left (e x + d\right )} c^{3} d^{5} - c^{3} d^{6} - 18 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 9 \, {\left (e x + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt {e x + d} c^{3} d^{4} e^{8} + 9 \, \sqrt {e x + d} a c^{2} d^{2} e^{10}\right )}}{3 \, e^{12}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(11/2),x, algorithm= "giac")
Output:
-2/3*(9*(e*x + d)*c^3*d^5 - c^3*d^6 - 18*(e*x + d)*a*c^2*d^3*e^2 + 3*a*c^2 *d^4*e^2 + 9*(e*x + d)*a^2*c*d*e^4 - 3*a^2*c*d^2*e^4 + a^3*e^6)/((e*x + d) ^(3/2)*e^4) + 2/3*((e*x + d)^(3/2)*c^3*d^3*e^8 - 9*sqrt(e*x + d)*c^3*d^4*e ^8 + 9*sqrt(e*x + d)*a*c^2*d^2*e^10)/e^12
Time = 5.41 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=-\frac {2\,a^3\,e^6-2\,c^3\,d^6-2\,c^3\,d^3\,{\left (d+e\,x\right )}^3+18\,c^3\,d^4\,{\left (d+e\,x\right )}^2+18\,c^3\,d^5\,\left (d+e\,x\right )+6\,a\,c^2\,d^4\,e^2-6\,a^2\,c\,d^2\,e^4-36\,a\,c^2\,d^3\,e^2\,\left (d+e\,x\right )-18\,a\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2+18\,a^2\,c\,d\,e^4\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^(11/2),x)
Output:
-(2*a^3*e^6 - 2*c^3*d^6 - 2*c^3*d^3*(d + e*x)^3 + 18*c^3*d^4*(d + e*x)^2 + 18*c^3*d^5*(d + e*x) + 6*a*c^2*d^4*e^2 - 6*a^2*c*d^2*e^4 - 36*a*c^2*d^3*e ^2*(d + e*x) - 18*a*c^2*d^2*e^2*(d + e*x)^2 + 18*a^2*c*d*e^4*(d + e*x))/(3 *e^4*(d + e*x)^(3/2))
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {\frac {2}{3} c^{3} d^{3} e^{3} x^{3}+6 a \,c^{2} d^{2} e^{4} x^{2}-4 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x +24 a \,c^{2} d^{3} e^{3} x -16 c^{3} d^{5} e x -\frac {2}{3} a^{3} e^{6}-4 a^{2} c \,d^{2} e^{4}+16 a \,c^{2} d^{4} e^{2}-\frac {32}{3} c^{3} d^{6}}{\sqrt {e x +d}\, e^{4} \left (e x +d \right )} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^(11/2),x)
Output:
(2*( - a**3*e**6 - 6*a**2*c*d**2*e**4 - 9*a**2*c*d*e**5*x + 24*a*c**2*d**4 *e**2 + 36*a*c**2*d**3*e**3*x + 9*a*c**2*d**2*e**4*x**2 - 16*c**3*d**6 - 2 4*c**3*d**5*e*x - 6*c**3*d**4*e**2*x**2 + c**3*d**3*e**3*x**3))/(3*sqrt(d + e*x)*e**4*(d + e*x))