Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]
2/3*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^(3/2)/e^5-4/5*c*(-b*e+2 *c*d)*(e*x+d)^(5/2)/e^5+2/7*c^2*(e*x+d)^(7/2)/e^5-2*(a*e^2-b*d*e+c*d^2)^2/ e^5/(e*x+d)^(1/2)-4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(1/2)/e^5
Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {-6 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-70 e^2 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )+28 c e \left (5 a e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 b \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{105 e^5 \sqrt {d+e x}} \]
(-6*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) - 70*e^2*(3*a^2*e^2 - 6*a*b*e*(2*d + e*x) + b^2*(8*d^2 + 4*d*e*x - e^2*x^2 )) + 28*c*e*(5*a*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*b*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(105*e^5*Sqrt[d + e*x])
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {\sqrt {d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4 \sqrt {d+e x}}+\frac {\left (a e^2-b d e+c d^2\right )^2}{e^4 (d+e x)^{3/2}}-\frac {2 c (d+e x)^{3/2} (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^{5/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}\) |
(-2*(c*d^2 - b*d*e + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(c*d ^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e^5 + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*( 3*b*d - a*e))*(d + e*x)^(3/2))/(3*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(5/2 ))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)
3.23.79.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {\left (30 c^{2} x^{4}+\left (84 b \,x^{3}+140 x^{2} a \right ) c +70 b^{2} x^{2}+420 a b x -210 a^{2}\right ) e^{4}+840 \left (-\frac {2 c^{2} x^{3}}{35}+\left (-\frac {1}{5} b \,x^{2}-\frac {2}{3} a x \right ) c +b \left (-\frac {b x}{3}+a \right )\right ) d \,e^{3}-1120 \left (-\frac {3 c^{2} x^{2}}{35}+\left (-\frac {3 b x}{5}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}+1344 \left (-\frac {2 c x}{7}+b \right ) c \,d^{3} e -768 c^{2} d^{4}}{105 \sqrt {e x +d}\, e^{5}}\) | \(147\) |
| risch | \(\frac {2 \left (15 c^{2} x^{3} e^{3}+42 b c \,e^{3} x^{2}-39 c^{2} d \,e^{2} x^{2}+70 a c \,e^{3} x +35 b^{2} e^{3} x -126 b c d \,e^{2} x +87 c^{2} d^{2} e x +210 a b \,e^{3}-350 a c d \,e^{2}-175 b^{2} d \,e^{2}+462 b c \,d^{2} e -279 c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{5}}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{e^{5} \sqrt {e x +d}}\) | \(187\) |
| gosper | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 a b d \,e^{3}+560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(194\) |
| trager | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 a b d \,e^{3}+560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(194\) |
| derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(236\) |
| default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(236\) |
1/105*((30*c^2*x^4+(84*b*x^3+140*a*x^2)*c+70*b^2*x^2+420*a*b*x-210*a^2)*e^ 4+840*(-2/35*c^2*x^3+(-1/5*b*x^2-2/3*a*x)*c+b*(-1/3*b*x+a))*d*e^3-1120*(-3 /35*c^2*x^2+(-3/5*b*x+a)*c+1/2*b^2)*d^2*e^2+1344*(-2/7*c*x+b)*c*d^3*e-768* c^2*d^4)/(e*x+d)^(1/2)/e^5
Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e + 420 \, a b d e^{3} - 105 \, a^{2} e^{4} - 280 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \, {\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} + {\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{3} e - 168 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 70 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + d e^{5}\right )}} \]
2/105*(15*c^2*e^4*x^4 - 384*c^2*d^4 + 672*b*c*d^3*e + 420*a*b*d*e^3 - 105* a^2*e^4 - 280*(b^2 + 2*a*c)*d^2*e^2 - 6*(4*c^2*d*e^3 - 7*b*c*e^4)*x^3 + (4 8*c^2*d^2*e^2 - 84*b*c*d*e^3 + 35*(b^2 + 2*a*c)*e^4)*x^2 - 2*(96*c^2*d^3*e - 168*b*c*d^2*e^2 - 105*a*b*e^4 + 70*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d )/(e^6*x + d*e^5)
Time = 4.62 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{e^{4}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{4} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(c**2*(d + e*x)**(7/2)/(7*e**4) + (d + e*x)**(5/2)*(2*b*c*e - 4*c**2*d)/(5*e**4) + (d + e*x)**(3/2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(3*e**4) + sqrt(d + e*x)*(2*a*b*e**3 - 4*a*c*d*e**2 - 2*b* *2*d*e**2 + 6*b*c*d**2*e - 4*c**2*d**3)/e**4 - (a*e**2 - b*d*e + c*d**2)** 2/(e**4*sqrt(d + e*x)))/e, Ne(e, 0)), ((a**2*x + a*b*x**2 + b*c*x**4/2 + c **2*x**5/5 + x**3*(2*a*c + b**2)/3)/d**(3/2), True))
Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 42 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \]
2/105*((15*(e*x + d)^(7/2)*c^2 - 42*(2*c^2*d - b*c*e)*(e*x + d)^(5/2) + 35 *(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d)^(3/2) - 210*(2*c^2* d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*sqrt(e*x + d))/e^4 - 10 5*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)/ (sqrt(e*x + d)*e^4))/e
Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{5}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {e x + d} c^{2} d^{3} e^{30} + 42 \, {\left (e x + d\right )}^{\frac {5}{2}} b c e^{31} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} b c d e^{31} + 630 \, \sqrt {e x + d} b c d^{2} e^{31} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{32} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c e^{32} - 210 \, \sqrt {e x + d} b^{2} d e^{32} - 420 \, \sqrt {e x + d} a c d e^{32} + 210 \, \sqrt {e x + d} a b e^{33}\right )}}{105 \, e^{35}} \]
-2*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^ 2*e^4)/(sqrt(e*x + d)*e^5) + 2/105*(15*(e*x + d)^(7/2)*c^2*e^30 - 84*(e*x + d)^(5/2)*c^2*d*e^30 + 210*(e*x + d)^(3/2)*c^2*d^2*e^30 - 420*sqrt(e*x + d)*c^2*d^3*e^30 + 42*(e*x + d)^(5/2)*b*c*e^31 - 210*(e*x + d)^(3/2)*b*c*d* e^31 + 630*sqrt(e*x + d)*b*c*d^2*e^31 + 35*(e*x + d)^(3/2)*b^2*e^32 + 70*( e*x + d)^(3/2)*a*c*e^32 - 210*sqrt(e*x + d)*b^2*d*e^32 - 420*sqrt(e*x + d) *a*c*d*e^32 + 210*sqrt(e*x + d)*a*b*e^33)/e^35
Time = 0.05 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{3\,e^5}-\frac {2\,a^2\,e^4-4\,a\,b\,d\,e^3+4\,a\,c\,d^2\,e^2+2\,b^2\,d^2\,e^2-4\,b\,c\,d^3\,e+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^5} \]
(2*c^2*(d + e*x)^(7/2))/(7*e^5) + ((d + e*x)^(3/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(3*e^5) - (2*a^2*e^4 + 2*c^2*d^4 + 2*b^2*d^2*e ^2 - 4*a*b*d*e^3 - 4*b*c*d^3*e + 4*a*c*d^2*e^2)/(e^5*(d + e*x)^(1/2)) - (( 8*c^2*d - 4*b*c*e)*(d + e*x)^(5/2))/(5*e^5) + (4*(b*e - 2*c*d)*(d + e*x)^( 1/2)*(a*e^2 + c*d^2 - b*d*e))/e^5