Integrand size = 27, antiderivative size = 320 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )^2 \sqrt {a+b x}}{b^7}+\frac {4 \left (b^2 d-2 a b e+3 a^2 f\right ) \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) (a+b x)^{3/2}}{3 b^7}+\frac {2 \left (b^4 \left (d^2+2 c e\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (d e+c f)+6 a^2 b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{5/2}}{5 b^7}+\frac {4 \left (10 a^2 b e f-10 a^3 f^2+b^3 (d e+c f)-2 a b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{7/2}}{7 b^7}-\frac {2 \left (10 a b e f-15 a^2 f^2-b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{9/2}}{9 b^7}+\frac {4 f (b e-3 a f) (a+b x)^{11/2}}{11 b^7}+\frac {2 f^2 (a+b x)^{13/2}}{13 b^7} \] Output:
2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)^2*(b*x+a)^(1/2)/b^7+4/3*(3*a^2*f-2*a*b*e+ b^2*d)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(b*x+a)^(3/2)/b^7+2/5*(b^4*(2*c*e+d^ 2)-20*a^3*b*e*f+15*a^4*f^2-6*a*b^3*(c*f+d*e)+6*a^2*b^2*(2*d*f+e^2))*(b*x+a )^(5/2)/b^7+4/7*(10*a^2*b*e*f-10*a^3*f^2+b^3*(c*f+d*e)-2*a*b^2*(2*d*f+e^2) )*(b*x+a)^(7/2)/b^7-2/9*(10*a*b*e*f-15*a^2*f^2-b^2*(2*d*f+e^2))*(b*x+a)^(9 /2)/b^7+4/11*f*(-3*a*f+b*e)*(b*x+a)^(11/2)/b^7+2/13*f^2*(b*x+a)^(13/2)/b^7
Time = 0.21 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (15360 a^6 f^2-2560 a^5 b f (13 e+3 f x)+128 a^4 b^2 \left (143 e^2+130 e f x+f \left (286 d+45 f x^2\right )\right )-32 a^3 b^3 \left (1287 c f+143 d (9 e+4 f x)+2 x \left (143 e^2+195 e f x+75 f^2 x^2\right )\right )+8 a^2 b^4 \left (3003 d^2+858 d x (3 e+2 f x)+858 c (7 e+3 f x)+x^2 \left (858 e^2+1300 e f x+525 f^2 x^2\right )\right )+b^6 \left (45045 c^2+858 c x (35 d+3 x (7 e+5 f x))+x^2 \left (9009 d^2+1430 d x (9 e+7 f x)+35 x^2 \left (143 e^2+234 e f x+99 f^2 x^2\right )\right )\right )-4 a b^5 \left (429 c (35 d+x (14 e+9 f x))+x \left (3003 d^2+143 d x (27 e+20 f x)+5 x^2 \left (286 e^2+455 e f x+189 f^2 x^2\right )\right )\right )\right )}{45045 b^7} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]
Output:
(2*Sqrt[a + b*x]*(15360*a^6*f^2 - 2560*a^5*b*f*(13*e + 3*f*x) + 128*a^4*b^ 2*(143*e^2 + 130*e*f*x + f*(286*d + 45*f*x^2)) - 32*a^3*b^3*(1287*c*f + 14 3*d*(9*e + 4*f*x) + 2*x*(143*e^2 + 195*e*f*x + 75*f^2*x^2)) + 8*a^2*b^4*(3 003*d^2 + 858*d*x*(3*e + 2*f*x) + 858*c*(7*e + 3*f*x) + x^2*(858*e^2 + 130 0*e*f*x + 525*f^2*x^2)) + b^6*(45045*c^2 + 858*c*x*(35*d + 3*x*(7*e + 5*f* x)) + x^2*(9009*d^2 + 1430*d*x*(9*e + 7*f*x) + 35*x^2*(143*e^2 + 234*e*f*x + 99*f^2*x^2))) - 4*a*b^5*(429*c*(35*d + x*(14*e + 9*f*x)) + x*(3003*d^2 + 143*d*x*(27*e + 20*f*x) + 5*x^2*(286*e^2 + 455*e*f*x + 189*f^2*x^2)))))/ (45045*b^7)
Time = 0.55 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2389, 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.
| \(\displaystyle \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (\frac {(a+b x)^{7/2} \left (15 a^2 f^2-10 a b e f+b^2 \left (2 d f+e^2\right )\right )}{b^6}+\frac {2 (a+b x)^{5/2} \left (-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{b^6}+\frac {2 \sqrt {a+b x} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}+\frac {\left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )^2}{b^6 \sqrt {a+b x}}+\frac {(a+b x)^{3/2} \left (15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left (2 d f+e^2\right )-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{b^6}+\frac {2 f (a+b x)^{9/2} (b e-3 a f)}{b^6}+\frac {f^2 (a+b x)^{11/2}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f-\left (b^2 \left (2 d f+e^2\right )\right )\right )}{9 b^7}+\frac {4 (a+b x)^{7/2} \left (-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{7 b^7}+\frac {4 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^7}+\frac {2 \sqrt {a+b x} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )^2}{b^7}+\frac {2 (a+b x)^{5/2} \left (15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left (2 d f+e^2\right )-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{5 b^7}+\frac {4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac {2 f^2 (a+b x)^{13/2}}{13 b^7}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]
Output:
(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x])/b^7 + (4*(b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a + b*x)^(3/2))/(3 *b^7) + (2*(b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e + c*f) + 6*a^2*b^2*(e^2 + 2*d*f))*(a + b*x)^(5/2))/(5*b^7) + (4*(10*a^2*b*e *f - 10*a^3*f^2 + b^3*(d*e + c*f) - 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2) )/(7*b^7) - (2*(10*a*b*e*f - 15*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)^(9/ 2))/(9*b^7) + (4*f*(b*e - 3*a*f)*(a + b*x)^(11/2))/(11*b^7) + (2*f^2*(a + b*x)^(13/2))/(13*b^7)
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.44 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {\frac {2 f^{2} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {4 \left (-3 a f +b e \right ) f \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (3 a^{2} f -2 a e b +d \,b^{2}\right ) f +\left (-3 a f +b e \right )^{2}\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) f +2 \left (3 a^{2} f -2 a e b +d \,b^{2}\right ) \left (-3 a f +b e \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) \left (-3 a f +b e \right )+\left (3 a^{2} f -2 a e b +d \,b^{2}\right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right ) \left (3 a^{2} f -2 a e b +d \,b^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (-a^{3} f +a^{2} b e -a \,b^{2} d +b^{3} c \right )^{2} \sqrt {b x +a}}{b^{7}}\) | \(292\) |
| pseudoelliptic | \(\frac {2048 \left (\frac {\left (\frac {231 f^{2} x^{6}}{2}+273 e f \,x^{5}+\frac {1001 \left (d f +\frac {e^{2}}{2}\right ) x^{4}}{3}+429 \left (c f +d e \right ) x^{3}+\frac {3003 \left (c e +\frac {d^{2}}{2}\right ) x^{2}}{5}+1001 c d x +\frac {3003 c^{2}}{2}\right ) b^{6}}{512}-\frac {1001 a \left (\frac {9 f^{2} x^{5}}{143}+\frac {5 e f \,x^{4}}{33}+\frac {2 \left (2 d f +e^{2}\right ) x^{3}}{21}+\frac {9 \left (c f +d e \right ) x^{2}}{35}+\frac {\left (2 c e +d^{2}\right ) x}{5}+c d \right ) b^{5}}{256}+\frac {1001 \left (\frac {25 f^{2} x^{4}}{286}+\frac {50 e f \,x^{3}}{231}+\frac {\left (2 d f +e^{2}\right ) x^{2}}{7}+\frac {3 \left (c f +d e \right ) x}{7}+c e +\frac {d^{2}}{2}\right ) a^{2} b^{4}}{320}-\frac {429 a^{3} \left (\frac {50 f^{2} x^{3}}{429}+\frac {10 e f \,x^{2}}{33}+\frac {2 \left (2 d f +e^{2}\right ) x}{9}+c f +d e \right ) b^{3}}{160}+\frac {143 a^{4} \left (\frac {45}{286} f^{2} x^{2}+\frac {5}{11} e f x +d f +\frac {1}{2} e^{2}\right ) b^{2}}{60}-\frac {13 a^{5} f \left (\frac {3 f x}{13}+e \right ) b}{6}+a^{6} f^{2}\right ) \sqrt {b x +a}}{3003 b^{7}}\) | \(292\) |
| default | \(\frac {\frac {2 f^{2} \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {4 \left (3 a f -b e \right ) f \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 \left (-3 a^{2} f +2 a e b -d \,b^{2}\right ) f +\left (3 a f -b e \right )^{2}\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) f +2 \left (-3 a^{2} f +2 a e b -d \,b^{2}\right ) \left (3 a f -b e \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \left (3 a f -b e \right )+\left (-3 a^{2} f +2 a e b -d \,b^{2}\right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \left (-3 a^{2} f +2 a e b -d \,b^{2}\right ) \left (b x +a \right )^{\frac {3}{2}}}{3}+2 \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right )^{2} \sqrt {b x +a}}{b^{7}}\) | \(300\) |
| gosper | \(\frac {2 \sqrt {b x +a}\, \left (3465 f^{2} x^{6} b^{6}-3780 a \,b^{5} f^{2} x^{5}+8190 b^{6} e f \,x^{5}+4200 a^{2} b^{4} f^{2} x^{4}-9100 a \,b^{5} e f \,x^{4}+10010 b^{6} d f \,x^{4}+5005 b^{6} e^{2} x^{4}-4800 a^{3} b^{3} f^{2} x^{3}+10400 a^{2} b^{4} e f \,x^{3}-11440 a \,b^{5} d f \,x^{3}-5720 a \,b^{5} e^{2} x^{3}+12870 b^{6} c f \,x^{3}+12870 b^{6} d e \,x^{3}+5760 a^{4} b^{2} f^{2} x^{2}-12480 a^{3} b^{3} e f \,x^{2}+13728 a^{2} b^{4} d f \,x^{2}+6864 a^{2} b^{4} e^{2} x^{2}-15444 a \,b^{5} c f \,x^{2}-15444 a \,b^{5} d e \,x^{2}+18018 b^{6} c e \,x^{2}+9009 b^{6} d^{2} x^{2}-7680 a^{5} b \,f^{2} x +16640 a^{4} b^{2} e f x -18304 a^{3} b^{3} d f x -9152 a^{3} b^{3} e^{2} x +20592 a^{2} b^{4} c f x +20592 a^{2} b^{4} d e x -24024 a \,b^{5} c e x -12012 a \,b^{5} d^{2} x +30030 b^{6} c d x +15360 a^{6} f^{2}-33280 a^{5} b e f +36608 a^{4} b^{2} d f +18304 a^{4} b^{2} e^{2}-41184 a^{3} b^{3} c f -41184 a^{3} b^{3} d e +48048 a^{2} b^{4} c e +24024 a^{2} b^{4} d^{2}-60060 a \,b^{5} c d +45045 c^{2} b^{6}\right )}{45045 b^{7}}\) | \(447\) |
| trager | \(\frac {2 \sqrt {b x +a}\, \left (3465 f^{2} x^{6} b^{6}-3780 a \,b^{5} f^{2} x^{5}+8190 b^{6} e f \,x^{5}+4200 a^{2} b^{4} f^{2} x^{4}-9100 a \,b^{5} e f \,x^{4}+10010 b^{6} d f \,x^{4}+5005 b^{6} e^{2} x^{4}-4800 a^{3} b^{3} f^{2} x^{3}+10400 a^{2} b^{4} e f \,x^{3}-11440 a \,b^{5} d f \,x^{3}-5720 a \,b^{5} e^{2} x^{3}+12870 b^{6} c f \,x^{3}+12870 b^{6} d e \,x^{3}+5760 a^{4} b^{2} f^{2} x^{2}-12480 a^{3} b^{3} e f \,x^{2}+13728 a^{2} b^{4} d f \,x^{2}+6864 a^{2} b^{4} e^{2} x^{2}-15444 a \,b^{5} c f \,x^{2}-15444 a \,b^{5} d e \,x^{2}+18018 b^{6} c e \,x^{2}+9009 b^{6} d^{2} x^{2}-7680 a^{5} b \,f^{2} x +16640 a^{4} b^{2} e f x -18304 a^{3} b^{3} d f x -9152 a^{3} b^{3} e^{2} x +20592 a^{2} b^{4} c f x +20592 a^{2} b^{4} d e x -24024 a \,b^{5} c e x -12012 a \,b^{5} d^{2} x +30030 b^{6} c d x +15360 a^{6} f^{2}-33280 a^{5} b e f +36608 a^{4} b^{2} d f +18304 a^{4} b^{2} e^{2}-41184 a^{3} b^{3} c f -41184 a^{3} b^{3} d e +48048 a^{2} b^{4} c e +24024 a^{2} b^{4} d^{2}-60060 a \,b^{5} c d +45045 c^{2} b^{6}\right )}{45045 b^{7}}\) | \(447\) |
| risch | \(\frac {2 \sqrt {b x +a}\, \left (3465 f^{2} x^{6} b^{6}-3780 a \,b^{5} f^{2} x^{5}+8190 b^{6} e f \,x^{5}+4200 a^{2} b^{4} f^{2} x^{4}-9100 a \,b^{5} e f \,x^{4}+10010 b^{6} d f \,x^{4}+5005 b^{6} e^{2} x^{4}-4800 a^{3} b^{3} f^{2} x^{3}+10400 a^{2} b^{4} e f \,x^{3}-11440 a \,b^{5} d f \,x^{3}-5720 a \,b^{5} e^{2} x^{3}+12870 b^{6} c f \,x^{3}+12870 b^{6} d e \,x^{3}+5760 a^{4} b^{2} f^{2} x^{2}-12480 a^{3} b^{3} e f \,x^{2}+13728 a^{2} b^{4} d f \,x^{2}+6864 a^{2} b^{4} e^{2} x^{2}-15444 a \,b^{5} c f \,x^{2}-15444 a \,b^{5} d e \,x^{2}+18018 b^{6} c e \,x^{2}+9009 b^{6} d^{2} x^{2}-7680 a^{5} b \,f^{2} x +16640 a^{4} b^{2} e f x -18304 a^{3} b^{3} d f x -9152 a^{3} b^{3} e^{2} x +20592 a^{2} b^{4} c f x +20592 a^{2} b^{4} d e x -24024 a \,b^{5} c e x -12012 a \,b^{5} d^{2} x +30030 b^{6} c d x +15360 a^{6} f^{2}-33280 a^{5} b e f +36608 a^{4} b^{2} d f +18304 a^{4} b^{2} e^{2}-41184 a^{3} b^{3} c f -41184 a^{3} b^{3} d e +48048 a^{2} b^{4} c e +24024 a^{2} b^{4} d^{2}-60060 a \,b^{5} c d +45045 c^{2} b^{6}\right )}{45045 b^{7}}\) | \(447\) |
| orering | \(\frac {2 \sqrt {b x +a}\, \left (3465 f^{2} x^{6} b^{6}-3780 a \,b^{5} f^{2} x^{5}+8190 b^{6} e f \,x^{5}+4200 a^{2} b^{4} f^{2} x^{4}-9100 a \,b^{5} e f \,x^{4}+10010 b^{6} d f \,x^{4}+5005 b^{6} e^{2} x^{4}-4800 a^{3} b^{3} f^{2} x^{3}+10400 a^{2} b^{4} e f \,x^{3}-11440 a \,b^{5} d f \,x^{3}-5720 a \,b^{5} e^{2} x^{3}+12870 b^{6} c f \,x^{3}+12870 b^{6} d e \,x^{3}+5760 a^{4} b^{2} f^{2} x^{2}-12480 a^{3} b^{3} e f \,x^{2}+13728 a^{2} b^{4} d f \,x^{2}+6864 a^{2} b^{4} e^{2} x^{2}-15444 a \,b^{5} c f \,x^{2}-15444 a \,b^{5} d e \,x^{2}+18018 b^{6} c e \,x^{2}+9009 b^{6} d^{2} x^{2}-7680 a^{5} b \,f^{2} x +16640 a^{4} b^{2} e f x -18304 a^{3} b^{3} d f x -9152 a^{3} b^{3} e^{2} x +20592 a^{2} b^{4} c f x +20592 a^{2} b^{4} d e x -24024 a \,b^{5} c e x -12012 a \,b^{5} d^{2} x +30030 b^{6} c d x +15360 a^{6} f^{2}-33280 a^{5} b e f +36608 a^{4} b^{2} d f +18304 a^{4} b^{2} e^{2}-41184 a^{3} b^{3} c f -41184 a^{3} b^{3} d e +48048 a^{2} b^{4} c e +24024 a^{2} b^{4} d^{2}-60060 a \,b^{5} c d +45045 c^{2} b^{6}\right )}{45045 b^{7}}\) | \(447\) |
Input:
int((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/b^7*(1/13*f^2*(b*x+a)^(13/2)+2/11*(-3*a*f+b*e)*f*(b*x+a)^(11/2)+1/9*(2*( 3*a^2*f-2*a*b*e+b^2*d)*f+(-3*a*f+b*e)^2)*(b*x+a)^(9/2)+1/7*(2*(-a^3*f+a^2* b*e-a*b^2*d+b^3*c)*f+2*(3*a^2*f-2*a*b*e+b^2*d)*(-3*a*f+b*e))*(b*x+a)^(7/2) +1/5*(2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(-3*a*f+b*e)+(3*a^2*f-2*a*b*e+b^2*d )^2)*(b*x+a)^(5/2)+2/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(3*a^2*f-2*a*b*e+b^2 *d)*(b*x+a)^(3/2)+(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)^2*(b*x+a)^(1/2))
Time = 0.08 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3465 \, b^{6} f^{2} x^{6} + 45045 \, b^{6} c^{2} - 60060 \, a b^{5} c d + 24024 \, a^{2} b^{4} d^{2} + 18304 \, a^{4} b^{2} e^{2} + 15360 \, a^{6} f^{2} + 630 \, {\left (13 \, b^{6} e f - 6 \, a b^{5} f^{2}\right )} x^{5} + 35 \, {\left (143 \, b^{6} e^{2} + 120 \, a^{2} b^{4} f^{2} + 26 \, {\left (11 \, b^{6} d - 10 \, a b^{5} e\right )} f\right )} x^{4} + 10 \, {\left (1287 \, b^{6} d e - 572 \, a b^{5} e^{2} - 480 \, a^{3} b^{3} f^{2} + 13 \, {\left (99 \, b^{6} c - 88 \, a b^{5} d + 80 \, a^{2} b^{4} e\right )} f\right )} x^{3} + 3 \, {\left (3003 \, b^{6} d^{2} + 2288 \, a^{2} b^{4} e^{2} + 1920 \, a^{4} b^{2} f^{2} + 858 \, {\left (7 \, b^{6} c - 6 \, a b^{5} d\right )} e - 52 \, {\left (99 \, a b^{5} c - 88 \, a^{2} b^{4} d + 80 \, a^{3} b^{3} e\right )} f\right )} x^{2} + 6864 \, {\left (7 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d\right )} e - 416 \, {\left (99 \, a^{3} b^{3} c - 88 \, a^{4} b^{2} d + 80 \, a^{5} b e\right )} f + 2 \, {\left (15015 \, b^{6} c d - 6006 \, a b^{5} d^{2} - 4576 \, a^{3} b^{3} e^{2} - 3840 \, a^{5} b f^{2} - 1716 \, {\left (7 \, a b^{5} c - 6 \, a^{2} b^{4} d\right )} e + 104 \, {\left (99 \, a^{2} b^{4} c - 88 \, a^{3} b^{3} d + 80 \, a^{4} b^{2} e\right )} f\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{7}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/45045*(3465*b^6*f^2*x^6 + 45045*b^6*c^2 - 60060*a*b^5*c*d + 24024*a^2*b^ 4*d^2 + 18304*a^4*b^2*e^2 + 15360*a^6*f^2 + 630*(13*b^6*e*f - 6*a*b^5*f^2) *x^5 + 35*(143*b^6*e^2 + 120*a^2*b^4*f^2 + 26*(11*b^6*d - 10*a*b^5*e)*f)*x ^4 + 10*(1287*b^6*d*e - 572*a*b^5*e^2 - 480*a^3*b^3*f^2 + 13*(99*b^6*c - 8 8*a*b^5*d + 80*a^2*b^4*e)*f)*x^3 + 3*(3003*b^6*d^2 + 2288*a^2*b^4*e^2 + 19 20*a^4*b^2*f^2 + 858*(7*b^6*c - 6*a*b^5*d)*e - 52*(99*a*b^5*c - 88*a^2*b^4 *d + 80*a^3*b^3*e)*f)*x^2 + 6864*(7*a^2*b^4*c - 6*a^3*b^3*d)*e - 416*(99*a ^3*b^3*c - 88*a^4*b^2*d + 80*a^5*b*e)*f + 2*(15015*b^6*c*d - 6006*a*b^5*d^ 2 - 4576*a^3*b^3*e^2 - 3840*a^5*b*f^2 - 1716*(7*a*b^5*c - 6*a^2*b^4*d)*e + 104*(99*a^2*b^4*c - 88*a^3*b^3*d + 80*a^4*b^2*e)*f)*x)*sqrt(b*x + a)/b^7
Time = 1.25 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.76 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {f^{2} \left (a + b x\right )^{\frac {13}{2}}}{13 b^{6}} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (- 6 a f^{2} + 2 b e f\right )}{11 b^{6}} + \frac {\left (a + b x\right )^{\frac {9}{2}} \cdot \left (15 a^{2} f^{2} - 10 a b e f + 2 b^{2} d f + b^{2} e^{2}\right )}{9 b^{6}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 20 a^{3} f^{2} + 20 a^{2} b e f - 8 a b^{2} d f - 4 a b^{2} e^{2} + 2 b^{3} c f + 2 b^{3} d e\right )}{7 b^{6}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (15 a^{4} f^{2} - 20 a^{3} b e f + 12 a^{2} b^{2} d f + 6 a^{2} b^{2} e^{2} - 6 a b^{3} c f - 6 a b^{3} d e + 2 b^{4} c e + b^{4} d^{2}\right )}{5 b^{6}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 6 a^{5} f^{2} + 10 a^{4} b e f - 8 a^{3} b^{2} d f - 4 a^{3} b^{2} e^{2} + 6 a^{2} b^{3} c f + 6 a^{2} b^{3} d e - 4 a b^{4} c e - 2 a b^{4} d^{2} + 2 b^{5} c d\right )}{3 b^{6}} + \frac {\sqrt {a + b x} \left (a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}\right )}{b^{6}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {c^{2} x + c d x^{2} + \frac {e f x^{6}}{3} + \frac {f^{2} x^{7}}{7} + \frac {x^{5} \cdot \left (2 d f + e^{2}\right )}{5} + \frac {x^{4} \cdot \left (2 c f + 2 d e\right )}{4} + \frac {x^{3} \cdot \left (2 c e + d^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((f*x**3+e*x**2+d*x+c)**2/(b*x+a)**(1/2),x)
Output:
Piecewise((2*(f**2*(a + b*x)**(13/2)/(13*b**6) + (a + b*x)**(11/2)*(-6*a*f **2 + 2*b*e*f)/(11*b**6) + (a + b*x)**(9/2)*(15*a**2*f**2 - 10*a*b*e*f + 2 *b**2*d*f + b**2*e**2)/(9*b**6) + (a + b*x)**(7/2)*(-20*a**3*f**2 + 20*a** 2*b*e*f - 8*a*b**2*d*f - 4*a*b**2*e**2 + 2*b**3*c*f + 2*b**3*d*e)/(7*b**6) + (a + b*x)**(5/2)*(15*a**4*f**2 - 20*a**3*b*e*f + 12*a**2*b**2*d*f + 6*a **2*b**2*e**2 - 6*a*b**3*c*f - 6*a*b**3*d*e + 2*b**4*c*e + b**4*d**2)/(5*b **6) + (a + b*x)**(3/2)*(-6*a**5*f**2 + 10*a**4*b*e*f - 8*a**3*b**2*d*f - 4*a**3*b**2*e**2 + 6*a**2*b**3*c*f + 6*a**2*b**3*d*e - 4*a*b**4*c*e - 2*a* b**4*d**2 + 2*b**5*c*d)/(3*b**6) + sqrt(a + b*x)*(a**6*f**2 - 2*a**5*b*e*f + 2*a**4*b**2*d*f + a**4*b**2*e**2 - 2*a**3*b**3*c*f - 2*a**3*b**3*d*e + 2*a**2*b**4*c*e + a**2*b**4*d**2 - 2*a*b**5*c*d + b**6*c**2)/b**6)/b, Ne(b , 0)), ((c**2*x + c*d*x**2 + e*f*x**6/3 + f**2*x**7/7 + x**5*(2*d*f + e**2 )/5 + x**4*(2*c*f + 2*d*e)/4 + x**3*(2*c*e + d**2)/3)/sqrt(a), True))
Time = 0.05 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.56 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:
integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
2/45045*(45045*sqrt(b*x + a)*c^2 + 858*c*(35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2 + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a )^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*f/b^3) + 3003*(3*(b*x + a)^(5/2) - 10* (b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*d^2/b^2 + 143*(35*(b*x + a)^(9/2 ) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)* a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4 + 286*(35*(b*x + a)^(9/2)*f + 45*(b*e - 4*a*f)*(b*x + a)^(7/2) - 189*(a*b*e - 2*a^2*f)*(b*x + a)^(5/2) + 105*(3 *a^2*b*e - 4*a^3*f)*(b*x + a)^(3/2) - 315*(a^3*b*e - a^4*f)*sqrt(b*x + a)) *d/b^4 + 130*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^ (7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt (b*x + a)*a^5)*e*f/b^5 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)* a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^( 5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*f^2/b^6)/b
Time = 0.12 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.61 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:
integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(b*x + a)*c^2 + 30030*((b*x + a)^(3/2) - 3*sqrt(b*x + a )*a)*c*d/b + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*d^2/b^2 + 6006*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sq rt(b*x + a)*a^2)*c*e/b^2 + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d*e/b^3 + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a )*a^3)*c*f/b^3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b* x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^ 4 + 286*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)* a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*d*f/b^4 + 130*(63*( b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*( b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*e*f /b^5 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a) ^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b *x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*f^2/b^6)/b
Time = 3.11 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\frac {2\,\sqrt {a+b\,x}\,{\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}^2}{b^7}+\frac {2\,f^2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}-\frac {{\left (a+b\,x\right )}^{7/2}\,\left (40\,a^3\,f^2-40\,a^2\,b\,e\,f+8\,a\,b^2\,e^2+16\,d\,a\,b^2\,f-4\,d\,b^3\,e-4\,c\,b^3\,f\right )}{7\,b^7}+\frac {{\left (a+b\,x\right )}^{9/2}\,\left (30\,a^2\,f^2-20\,a\,b\,e\,f+2\,b^2\,e^2+4\,d\,b^2\,f\right )}{9\,b^7}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (30\,a^4\,f^2-40\,a^3\,b\,e\,f+24\,a^2\,b^2\,d\,f+12\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e-12\,c\,a\,b^3\,f+2\,b^4\,d^2+4\,c\,b^4\,e\right )}{5\,b^7}-\frac {\left (12\,a\,f^2-4\,b\,e\,f\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}+\frac {4\,{\left (a+b\,x\right )}^{3/2}\,\left (3\,f\,a^2-2\,e\,a\,b+d\,b^2\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^7} \] Input:
int((c + d*x + e*x^2 + f*x^3)^2/(a + b*x)^(1/2),x)
Output:
(2*(a + b*x)^(1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e)^2)/b^7 + (2*f^2*(a + b*x)^(13/2))/(13*b^7) - ((a + b*x)^(7/2)*(40*a^3*f^2 + 8*a*b^2*e^2 - 4*b ^3*c*f - 4*b^3*d*e + 16*a*b^2*d*f - 40*a^2*b*e*f))/(7*b^7) + ((a + b*x)^(9 /2)*(30*a^2*f^2 + 2*b^2*e^2 + 4*b^2*d*f - 20*a*b*e*f))/(9*b^7) + ((a + b*x )^(5/2)*(2*b^4*d^2 + 30*a^4*f^2 + 12*a^2*b^2*e^2 + 4*b^4*c*e - 12*a*b^3*c* f - 12*a*b^3*d*e - 40*a^3*b*e*f + 24*a^2*b^2*d*f))/(5*b^7) - ((12*a*f^2 - 4*b*e*f)*(a + b*x)^(11/2))/(11*b^7) + (4*(a + b*x)^(3/2)*(b^2*d + 3*a^2*f - 2*a*b*e)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^7)
Time = 0.14 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.39 \[ \int \frac {\left (c+d x+e x^2+f x^3\right )^2}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {b x +a}\, \left (3465 b^{6} f^{2} x^{6}-3780 a \,b^{5} f^{2} x^{5}+8190 b^{6} e f \,x^{5}+4200 a^{2} b^{4} f^{2} x^{4}-9100 a \,b^{5} e f \,x^{4}+10010 b^{6} d f \,x^{4}+5005 b^{6} e^{2} x^{4}-4800 a^{3} b^{3} f^{2} x^{3}+10400 a^{2} b^{4} e f \,x^{3}-11440 a \,b^{5} d f \,x^{3}-5720 a \,b^{5} e^{2} x^{3}+12870 b^{6} c f \,x^{3}+12870 b^{6} d e \,x^{3}+5760 a^{4} b^{2} f^{2} x^{2}-12480 a^{3} b^{3} e f \,x^{2}+13728 a^{2} b^{4} d f \,x^{2}+6864 a^{2} b^{4} e^{2} x^{2}-15444 a \,b^{5} c f \,x^{2}-15444 a \,b^{5} d e \,x^{2}+18018 b^{6} c e \,x^{2}+9009 b^{6} d^{2} x^{2}-7680 a^{5} b \,f^{2} x +16640 a^{4} b^{2} e f x -18304 a^{3} b^{3} d f x -9152 a^{3} b^{3} e^{2} x +20592 a^{2} b^{4} c f x +20592 a^{2} b^{4} d e x -24024 a \,b^{5} c e x -12012 a \,b^{5} d^{2} x +30030 b^{6} c d x +15360 a^{6} f^{2}-33280 a^{5} b e f +36608 a^{4} b^{2} d f +18304 a^{4} b^{2} e^{2}-41184 a^{3} b^{3} c f -41184 a^{3} b^{3} d e +48048 a^{2} b^{4} c e +24024 a^{2} b^{4} d^{2}-60060 a \,b^{5} c d +45045 b^{6} c^{2}\right )}{45045 b^{7}} \] Input:
int((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x)
Output:
(2*sqrt(a + b*x)*(15360*a**6*f**2 - 33280*a**5*b*e*f - 7680*a**5*b*f**2*x + 36608*a**4*b**2*d*f + 18304*a**4*b**2*e**2 + 16640*a**4*b**2*e*f*x + 576 0*a**4*b**2*f**2*x**2 - 41184*a**3*b**3*c*f - 41184*a**3*b**3*d*e - 18304* a**3*b**3*d*f*x - 9152*a**3*b**3*e**2*x - 12480*a**3*b**3*e*f*x**2 - 4800* a**3*b**3*f**2*x**3 + 48048*a**2*b**4*c*e + 20592*a**2*b**4*c*f*x + 24024* a**2*b**4*d**2 + 20592*a**2*b**4*d*e*x + 13728*a**2*b**4*d*f*x**2 + 6864*a **2*b**4*e**2*x**2 + 10400*a**2*b**4*e*f*x**3 + 4200*a**2*b**4*f**2*x**4 - 60060*a*b**5*c*d - 24024*a*b**5*c*e*x - 15444*a*b**5*c*f*x**2 - 12012*a*b **5*d**2*x - 15444*a*b**5*d*e*x**2 - 11440*a*b**5*d*f*x**3 - 5720*a*b**5*e **2*x**3 - 9100*a*b**5*e*f*x**4 - 3780*a*b**5*f**2*x**5 + 45045*b**6*c**2 + 30030*b**6*c*d*x + 18018*b**6*c*e*x**2 + 12870*b**6*c*f*x**3 + 9009*b**6 *d**2*x**2 + 12870*b**6*d*e*x**3 + 10010*b**6*d*f*x**4 + 5005*b**6*e**2*x* *4 + 8190*b**6*e*f*x**5 + 3465*b**6*f**2*x**6))/(45045*b**7)