Integrand size = 26, antiderivative size = 128 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^4 \sqrt {d+e x}}-\frac {6 c (2 c d-b e) \sqrt {d+e x}}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4} \] Output:
2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^(3/2)-2*(6*c^2*d^2+b^2*e^ 2-2*c*e*(-a*e+3*b*d))/e^4/(e*x+d)^(1/2)-6*c*(-b*e+2*c*d)*(e*x+d)^(1/2)/e^4 +4/3*c^2*(e*x+d)^(3/2)/e^4
Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (b e^2 (2 b d+a e+3 b e x)+2 c^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )+c e \left (2 a e (2 d+3 e x)-3 b \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \] Input:
Integrate[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^(5/2),x]
Output:
(-2*(b*e^2*(2*b*d + a*e + 3*b*e*x) + 2*c^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2* x^2 - e^3*x^3) + c*e*(2*a*e*(2*d + 3*e*x) - 3*b*(8*d^2 + 12*d*e*x + 3*e^2* x^2))))/(3*e^4*(d + e*x)^(3/2))
Time = 0.48 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^3 (d+e x)^{3/2}}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3 (d+e x)^{5/2}}-\frac {3 c (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c^2 \sqrt {d+e x}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4}\) |
Input:
Int[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^(5/2),x]
Output:
(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^4*(d + e*x)^(3/2)) - (2*(6* c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(e^4*Sqrt[d + e*x]) - (6*c*(2*c* d - b*e)*Sqrt[d + e*x])/e^4 + (4*c^2*(d + e*x)^(3/2))/(3*e^4)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.79 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {\left (4 x^{3} c^{2}+\left (18 b \,x^{2}-12 a x \right ) c -6 b^{2} x -2 a b \right ) e^{3}-8 d \left (3 c^{2} x^{2}+\left (-9 b x +a \right ) c +\frac {b^{2}}{2}\right ) e^{2}+48 c \,d^{2} \left (-2 c x +b \right ) e -64 c^{2} d^{3}}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(99\) |
| risch | \(\frac {2 c \left (2 c e x +9 b e -16 c d \right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (6 a c \,e^{3} x +3 e^{3} x \,b^{2}-18 d \,e^{2} c b x +18 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +2 d \,e^{2} b^{2}-15 d^{2} e b c +16 c^{2} d^{3}\right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(117\) |
| gosper | \(-\frac {2 \left (-2 e^{3} x^{3} c^{2}-9 e^{3} x^{2} c b +12 d \,e^{2} c^{2} x^{2}+6 a c \,e^{3} x +3 e^{3} x \,b^{2}-36 d \,e^{2} c b x +48 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +2 d \,e^{2} b^{2}-24 d^{2} e b c +32 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(122\) |
| trager | \(-\frac {2 \left (-2 e^{3} x^{3} c^{2}-9 e^{3} x^{2} c b +12 d \,e^{2} c^{2} x^{2}+6 a c \,e^{3} x +3 e^{3} x \,b^{2}-36 d \,e^{2} c b x +48 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +2 d \,e^{2} b^{2}-24 d^{2} e b c +32 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(122\) |
| orering | \(-\frac {2 \left (-2 e^{3} x^{3} c^{2}-9 e^{3} x^{2} c b +12 d \,e^{2} c^{2} x^{2}+6 a c \,e^{3} x +3 e^{3} x \,b^{2}-36 d \,e^{2} c b x +48 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +2 d \,e^{2} b^{2}-24 d^{2} e b c +32 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(122\) |
| derivativedivides | \(\frac {\frac {4 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 b c e \sqrt {e x +d}-12 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(131\) |
| default | \(\frac {\frac {4 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 b c e \sqrt {e x +d}-12 c^{2} d \sqrt {e x +d}-\frac {2 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(131\) |
Input:
int((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*((4*x^3*c^2+(18*b*x^2-12*a*x)*c-6*b^2*x-2*a*b)*e^3-8*d*(3*c^2*x^2+(-9* b*x+a)*c+1/2*b^2)*e^2+48*c*d^2*(-2*c*x+b)*e-64*c^2*d^3)/(e*x+d)^(3/2)/e^4
Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (2 \, c^{2} e^{3} x^{3} - 32 \, c^{2} d^{3} + 24 \, b c d^{2} e - a b e^{3} - 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (4 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} - 3 \, {\left (16 \, c^{2} d^{2} e - 12 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/3*(2*c^2*e^3*x^3 - 32*c^2*d^3 + 24*b*c*d^2*e - a*b*e^3 - 2*(b^2 + 2*a*c) *d*e^2 - 3*(4*c^2*d*e^2 - 3*b*c*e^3)*x^2 - 3*(16*c^2*d^2*e - 12*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*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. 536 vs. \(2 (124) = 248\).
Time = 0.36 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.19 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 a b e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 a c d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a c e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 b^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 b^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 b c d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 b c d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 b c e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {64 c^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {96 c^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {24 c^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {4 c^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a b x + a c x^{2} + \frac {b^{2} x^{2}}{2} + b c x^{3} + \frac {c^{2} x^{4}}{2}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*x+b)*(c*x**2+b*x+a)/(e*x+d)**(5/2),x)
Output:
Piecewise((-2*a*b*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 8*a*c*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*a*c*e **3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 4*b**2*d*e**2/(3 *d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*b**2*e**3*x/(3*d*e**4* sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48*b*c*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*b*c*d*e**2*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 18*b*c*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e **5*x*sqrt(d + e*x)) - 64*c**2*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqr t(d + e*x)) - 96*c**2*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 24*c**2*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e *x)) + 4*c**2*e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2)/d** (5/2), True))
Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 9 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {e x + d}}{e^{3}} + \frac {2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/3*((2*(e*x + d)^(3/2)*c^2 - 9*(2*c^2*d - b*c*e)*sqrt(e*x + d))/e^3 + (2* c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2 - 3*(6*c^2*d^2 - 6*b *c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.20 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (18 \, {\left (e x + d\right )} c^{2} d^{2} - 2 \, c^{2} d^{3} - 18 \, {\left (e x + d\right )} b c d e + 3 \, b c d^{2} e + 3 \, {\left (e x + d\right )} b^{2} e^{2} + 6 \, {\left (e x + d\right )} a c e^{2} - b^{2} d e^{2} - 2 \, a c d e^{2} + a b e^{3}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{4}} + \frac {2 \, {\left (2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} e^{8} - 18 \, \sqrt {e x + d} c^{2} d e^{8} + 9 \, \sqrt {e x + d} b c e^{9}\right )}}{3 \, e^{12}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^(5/2),x, algorithm="giac")
Output:
-2/3*(18*(e*x + d)*c^2*d^2 - 2*c^2*d^3 - 18*(e*x + d)*b*c*d*e + 3*b*c*d^2* e + 3*(e*x + d)*b^2*e^2 + 6*(e*x + d)*a*c*e^2 - b^2*d*e^2 - 2*a*c*d*e^2 + a*b*e^3)/((e*x + d)^(3/2)*e^4) + 2/3*(2*(e*x + d)^(3/2)*c^2*e^8 - 18*sqrt( e*x + d)*c^2*d*e^8 + 9*sqrt(e*x + d)*b*c*e^9)/e^12
Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {4\,c^2\,{\left (d+e\,x\right )}^3+4\,c^2\,d^3+2\,b^2\,d\,e^2-6\,b^2\,e^2\,\left (d+e\,x\right )-36\,c^2\,d\,{\left (d+e\,x\right )}^2-36\,c^2\,d^2\,\left (d+e\,x\right )-2\,a\,b\,e^3+4\,a\,c\,d\,e^2-6\,b\,c\,d^2\,e-12\,a\,c\,e^2\,\left (d+e\,x\right )+18\,b\,c\,e\,{\left (d+e\,x\right )}^2+36\,b\,c\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \] Input:
int(((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^(5/2),x)
Output:
(4*c^2*(d + e*x)^3 + 4*c^2*d^3 + 2*b^2*d*e^2 - 6*b^2*e^2*(d + e*x) - 36*c^ 2*d*(d + e*x)^2 - 36*c^2*d^2*(d + e*x) - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d ^2*e - 12*a*c*e^2*(d + e*x) + 18*b*c*e*(d + e*x)^2 + 36*b*c*d*e*(d + e*x)) /(3*e^4*(d + e*x)^(3/2))
Time = 0.21 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.02 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {\frac {4}{3} c^{2} e^{3} x^{3}+6 b c \,e^{3} x^{2}-8 c^{2} d \,e^{2} x^{2}-4 a c \,e^{3} x -2 b^{2} e^{3} x +24 b c d \,e^{2} x -32 c^{2} d^{2} e x -\frac {2}{3} a b \,e^{3}-\frac {8}{3} a c d \,e^{2}-\frac {4}{3} b^{2} d \,e^{2}+16 b c \,d^{2} e -\frac {64}{3} c^{2} d^{3}}{\sqrt {e x +d}\, e^{4} \left (e x +d \right )} \] Input:
int((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^(5/2),x)
Output:
(2*( - a*b*e**3 - 4*a*c*d*e**2 - 6*a*c*e**3*x - 2*b**2*d*e**2 - 3*b**2*e** 3*x + 24*b*c*d**2*e + 36*b*c*d*e**2*x + 9*b*c*e**3*x**2 - 32*c**2*d**3 - 4 8*c**2*d**2*e*x - 12*c**2*d*e**2*x**2 + 2*c**2*e**3*x**3))/(3*sqrt(d + e*x )*e**4*(d + e*x))