Integrand size = 26, antiderivative size = 132 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^4}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{5 e^4}-\frac {6 c (2 c d-b e) (d+e x)^{7/2}}{7 e^4}+\frac {4 c^2 (d+e x)^{9/2}}{9 e^4} \]
-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(3/2)/e^4+2/5*(6*c^2*d^2+b^2 *e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^(5/2)/e^4-6/7*c*(-b*e+2*c*d)*(e*x+d)^(7/2 )/e^4+4/9*c^2*(e*x+d)^(9/2)/e^4
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (21 b e^2 (-2 b d+5 a e+3 b e x)+c^2 \left (-32 d^3+48 d^2 e x-60 d e^2 x^2+70 e^3 x^3\right )+3 c e \left (14 a e (-2 d+3 e x)+3 b \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )\right )}{315 e^4} \]
(2*(d + e*x)^(3/2)*(21*b*e^2*(-2*b*d + 5*a*e + 3*b*e*x) + c^2*(-32*d^3 + 4 8*d^2*e*x - 60*d*e^2*x^2 + 70*e^3*x^3) + 3*c*e*(14*a*e*(-2*d + 3*e*x) + 3* b*(8*d^2 - 12*d*e*x + 15*e^2*x^2))))/(315*e^4)
Time = 0.28 (sec) , antiderivative size = 132, 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.
\(\displaystyle \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {(d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^3}+\frac {\sqrt {d+e x} (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {3 c (d+e x)^{5/2} (2 c d-b e)}{e^3}+\frac {2 c^2 (d+e x)^{7/2}}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {6 c (d+e x)^{7/2} (2 c d-b e)}{7 e^4}+\frac {4 c^2 (d+e x)^{9/2}}{9 e^4}\) |
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2))/(3*e^4) + (2*(6 *c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(5/2))/(5*e^4) - (6*c* (2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^4) + (4*c^2*(d + e*x)^(9/2))/(9*e^4)
3.16.97.3.1 Defintions of rubi rules used
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 = 0.58 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {2 c^{2} x^{3}}{3}+\left (\frac {9}{7} b \,x^{2}+\frac {6}{5} a x \right ) c +b \left (\frac {3 b x}{5}+a \right )\right ) e^{3}-\frac {4 d \left (\frac {5 c^{2} x^{2}}{7}+\left (\frac {9 b x}{7}+a \right ) c +\frac {b^{2}}{2}\right ) e^{2}}{5}+\frac {24 d^{2} \left (\frac {2 c x}{3}+b \right ) c e}{35}-\frac {32 c^{2} d^{3}}{105}\right )}{3 e^{4}}\) | \(97\) |
derivativedivides | \(\frac {\frac {4 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(109\) |
default | \(\frac {\frac {4 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(109\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (70 c^{2} x^{3} e^{3}+135 x^{2} b c \,e^{3}-60 x^{2} c^{2} d \,e^{2}+126 x a c \,e^{3}+63 x \,b^{2} e^{3}-108 x b c d \,e^{2}+48 x \,c^{2} d^{2} e +105 a b \,e^{3}-84 a c d \,e^{2}-42 b^{2} d \,e^{2}+72 b c \,d^{2} e -32 c^{2} d^{3}\right )}{315 e^{4}}\) | \(123\) |
trager | \(\frac {2 \left (70 c^{2} e^{4} x^{4}+135 c \,e^{4} b \,x^{3}+10 c^{2} d \,e^{3} x^{3}+126 a c \,e^{4} x^{2}+63 b^{2} e^{4} x^{2}+27 b c d \,e^{3} x^{2}-12 c^{2} d^{2} e^{2} x^{2}+105 a b \,e^{4} x +42 a c d \,e^{3} x +21 b^{2} d \,e^{3} x -36 b c \,d^{2} e^{2} x +16 c^{2} d^{3} e x +105 a b d \,e^{3}-84 a c \,d^{2} e^{2}-42 b^{2} d^{2} e^{2}+72 b c \,d^{3} e -32 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(186\) |
risch | \(\frac {2 \left (70 c^{2} e^{4} x^{4}+135 c \,e^{4} b \,x^{3}+10 c^{2} d \,e^{3} x^{3}+126 a c \,e^{4} x^{2}+63 b^{2} e^{4} x^{2}+27 b c d \,e^{3} x^{2}-12 c^{2} d^{2} e^{2} x^{2}+105 a b \,e^{4} x +42 a c d \,e^{3} x +21 b^{2} d \,e^{3} x -36 b c \,d^{2} e^{2} x +16 c^{2} d^{3} e x +105 a b d \,e^{3}-84 a c \,d^{2} e^{2}-42 b^{2} d^{2} e^{2}+72 b c \,d^{3} e -32 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(186\) |
2/3*(e*x+d)^(3/2)*((2/3*c^2*x^3+(9/7*b*x^2+6/5*a*x)*c+b*(3/5*b*x+a))*e^3-4 /5*d*(5/7*c^2*x^2+(9/7*b*x+a)*c+1/2*b^2)*e^2+24/35*d^2*(2/3*c*x+b)*c*e-32/ 105*c^2*d^3)/e^4
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.27 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\frac {2 \, {\left (70 \, c^{2} e^{4} x^{4} - 32 \, c^{2} d^{4} + 72 \, b c d^{3} e + 105 \, a b d e^{3} - 42 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 5 \, {\left (2 \, c^{2} d e^{3} + 27 \, b c e^{4}\right )} x^{3} - 3 \, {\left (4 \, c^{2} d^{2} e^{2} - 9 \, b c d e^{3} - 21 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + {\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 105 \, a b e^{4} + 21 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \]
2/315*(70*c^2*e^4*x^4 - 32*c^2*d^4 + 72*b*c*d^3*e + 105*a*b*d*e^3 - 42*(b^ 2 + 2*a*c)*d^2*e^2 + 5*(2*c^2*d*e^3 + 27*b*c*e^4)*x^3 - 3*(4*c^2*d^2*e^2 - 9*b*c*d*e^3 - 21*(b^2 + 2*a*c)*e^4)*x^2 + (16*c^2*d^3*e - 36*b*c*d^2*e^2 + 105*a*b*e^4 + 21*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/e^4
Time = 0.98 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.31 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 b c e - 6 c^{2} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}\right )}{3 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} \left (a + b x + c x^{2}\right )^{2}}{2} & \text {otherwise} \end {cases} \]
Piecewise((2*(2*c**2*(d + e*x)**(9/2)/(9*e**3) + (d + e*x)**(7/2)*(3*b*c*e - 6*c**2*d)/(7*e**3) + (d + e*x)**(5/2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d *e + 6*c**2*d**2)/(5*e**3) + (d + e*x)**(3/2)*(a*b*e**3 - 2*a*c*d*e**2 - b **2*d*e**2 + 3*b*c*d**2*e - 2*c**2*d**3)/(3*e**3))/e, Ne(e, 0)), (sqrt(d)* (a + b*x + c*x**2)**2/2, True))
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\frac {2 \, {\left (70 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{2} - 135 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \]
2/315*(70*(e*x + d)^(9/2)*c^2 - 135*(2*c^2*d - b*c*e)*(e*x + d)^(7/2) + 63 *(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d)^(5/2) - 105*(2*c^2* d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^(3/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (116) = 232\).
Time = 0.25 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.88 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a b d + 105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b + \frac {105 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b^{2} d}{e} + \frac {210 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a c d}{e} + \frac {63 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b c d}{e^{2}} + \frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e} + \frac {42 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c}{e} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c^{2} d}{e^{3}} + \frac {27 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{2}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{3}}\right )}}{315 \, e} \]
2/315*(315*sqrt(e*x + d)*a*b*d + 105*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d) *a*b + 105*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*b^2*d/e + 210*((e*x + d)^ (3/2) - 3*sqrt(e*x + d)*d)*a*c*d/e + 63*(3*(e*x + d)^(5/2) - 10*(e*x + d)^ (3/2)*d + 15*sqrt(e*x + d)*d^2)*b*c*d/e^2 + 21*(3*(e*x + d)^(5/2) - 10*(e* x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e + 42*(3*(e*x + d)^(5/2) - 10* (e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c/e + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*c^2* d/e^3 + 27*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)* d^2 - 35*sqrt(e*x + d)*d^3)*b*c/e^2 + 2*(35*(e*x + d)^(9/2) - 180*(e*x + d )^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e *x + d)*d^4)*c^2/e^3)/e
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx=\frac {4\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{5\,e^4}-\frac {\left (12\,c^2\,d-6\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,e^4} \]