Integrand size = 21, antiderivative size = 244 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 d^3 (c d-b e)^3 \sqrt {d+e x}}{e^7}-\frac {2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \] Output:
2*d^3*(-b*e+c*d)^3*(e*x+d)^(1/2)/e^7-2*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+ d)^(3/2)/e^7+6/5*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(5/2)/ e^7-2/7*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(7/2)/e^7+2/3 *c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(9/2)/e^7-6/11*c^2*(-b*e+2*c*d)*( e*x+d)^(11/2)/e^7+2/13*c^3*(e*x+d)^(13/2)/e^7
Time = 0.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.95 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (429 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 b^2 c e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+65 b c^2 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \] Input:
Integrate[(b*x + c*x^2)^3/Sqrt[d + e*x],x]
Output:
(2*Sqrt[d + e*x]*(429*b^3*e^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x ^3) + 143*b^2*c*e^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 65*b*c^2*e*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d ^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231* e^6*x^6)))/(15015*e^7)
Time = 0.61 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{5/2} (2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6}+\frac {3 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}-\frac {3 c^2 (d+e x)^{9/2} (2 c d-b e)}{e^6}+\frac {d^3 (c d-b e)^3}{e^6 \sqrt {d+e x}}-\frac {3 d^2 \sqrt {d+e x} (c d-b e)^2 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{11/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac {6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 d^3 \sqrt {d+e x} (c d-b e)^3}{e^7}-\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}\) |
Input:
Int[(b*x + c*x^2)^3/Sqrt[d + e*x],x]
Output:
(2*d^3*(c*d - b*e)^3*Sqrt[d + e*x])/e^7 - (2*d^2*(c*d - b*e)^2*(2*c*d - b* e)*(d + e*x)^(3/2))/e^7 + (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^ 2)*(d + e*x)^(5/2))/(5*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) *(d + e*x)^(9/2))/(3*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7 ) + (2*c^3*(d + e*x)^(13/2))/(13*e^7)
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 = 0.74 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (-\frac {5 \left (\frac {7}{13} c^{3} x^{3}+\frac {21}{11} b \,c^{2} x^{2}+\frac {7}{3} b^{2} c x +b^{3}\right ) x^{3} e^{6}}{16}+\frac {3 x^{2} d \left (\frac {70}{143} c^{3} x^{3}+\frac {175}{99} b \,c^{2} x^{2}+\frac {20}{9} b^{2} c x +b^{3}\right ) e^{5}}{8}-\frac {x \left (\frac {175}{429} c^{3} x^{3}+\frac {50}{33} b \,c^{2} x^{2}+2 b^{2} c x +b^{3}\right ) d^{2} e^{4}}{2}+d^{3} \left (\frac {100}{429} c^{3} x^{3}+\frac {10}{11} b \,c^{2} x^{2}+\frac {4}{3} b^{2} c x +b^{3}\right ) e^{3}-\frac {8 c \,d^{4} \left (\frac {15}{143} c^{2} x^{2}+\frac {5}{11} c b x +b^{2}\right ) e^{2}}{3}+\frac {80 c^{2} \left (\frac {2 c x}{13}+b \right ) d^{5} e}{33}-\frac {320 d^{6} c^{3}}{429}\right ) \sqrt {e x +d}}{35 e^{7}}\) | \(208\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-3 d \,c^{3}+3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 c^{3} d^{2}-9 d \left (b e -c d \right ) c^{2}+3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-d^{3} c^{3}+9 d^{2} \left (b e -c d \right ) c^{2}-9 d \left (b e -c d \right )^{2} c +\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-3 d^{3} \left (b e -c d \right ) c^{2}+9 d^{2} \left (b e -c d \right )^{2} c -3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-3 d^{3} \left (b e -c d \right )^{2} c +3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}-2 d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(269\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}-\frac {2 \left (3 d \,c^{3}-3 \left (b e -c d \right ) c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-3 c^{3} d^{2}+9 d \left (b e -c d \right ) c^{2}-3 \left (b e -c d \right )^{2} c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 \left (d^{3} c^{3}-9 d^{2} \left (b e -c d \right ) c^{2}+9 d \left (b e -c d \right )^{2} c -\left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {2 \left (3 d^{3} \left (b e -c d \right ) c^{2}-9 d^{2} \left (b e -c d \right )^{2} c +3 d \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (3 d^{3} \left (b e -c d \right )^{2} c -3 d^{2} \left (b e -c d \right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}-2 d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(270\) |
| gosper | \(-\frac {2 \left (-1155 x^{6} c^{3} e^{6}-4095 x^{5} b \,c^{2} e^{6}+1260 x^{5} c^{3} d \,e^{5}-5005 x^{4} b^{2} c \,e^{6}+4550 x^{4} b \,c^{2} d \,e^{5}-1400 x^{4} c^{3} d^{2} e^{4}-2145 x^{3} b^{3} e^{6}+5720 x^{3} b^{2} c d \,e^{5}-5200 x^{3} b \,c^{2} d^{2} e^{4}+1600 x^{3} c^{3} d^{3} e^{3}+2574 x^{2} b^{3} d \,e^{5}-6864 x^{2} b^{2} c \,d^{2} e^{4}+6240 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}-3432 x \,b^{3} d^{2} e^{4}+9152 x \,b^{2} c \,d^{3} e^{3}-8320 x b \,c^{2} d^{4} e^{2}+2560 x \,c^{3} d^{5} e +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right ) \sqrt {e x +d}}{15015 e^{7}}\) | \(286\) |
| trager | \(-\frac {2 \left (-1155 x^{6} c^{3} e^{6}-4095 x^{5} b \,c^{2} e^{6}+1260 x^{5} c^{3} d \,e^{5}-5005 x^{4} b^{2} c \,e^{6}+4550 x^{4} b \,c^{2} d \,e^{5}-1400 x^{4} c^{3} d^{2} e^{4}-2145 x^{3} b^{3} e^{6}+5720 x^{3} b^{2} c d \,e^{5}-5200 x^{3} b \,c^{2} d^{2} e^{4}+1600 x^{3} c^{3} d^{3} e^{3}+2574 x^{2} b^{3} d \,e^{5}-6864 x^{2} b^{2} c \,d^{2} e^{4}+6240 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}-3432 x \,b^{3} d^{2} e^{4}+9152 x \,b^{2} c \,d^{3} e^{3}-8320 x b \,c^{2} d^{4} e^{2}+2560 x \,c^{3} d^{5} e +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right ) \sqrt {e x +d}}{15015 e^{7}}\) | \(286\) |
| risch | \(-\frac {2 \left (-1155 x^{6} c^{3} e^{6}-4095 x^{5} b \,c^{2} e^{6}+1260 x^{5} c^{3} d \,e^{5}-5005 x^{4} b^{2} c \,e^{6}+4550 x^{4} b \,c^{2} d \,e^{5}-1400 x^{4} c^{3} d^{2} e^{4}-2145 x^{3} b^{3} e^{6}+5720 x^{3} b^{2} c d \,e^{5}-5200 x^{3} b \,c^{2} d^{2} e^{4}+1600 x^{3} c^{3} d^{3} e^{3}+2574 x^{2} b^{3} d \,e^{5}-6864 x^{2} b^{2} c \,d^{2} e^{4}+6240 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}-3432 x \,b^{3} d^{2} e^{4}+9152 x \,b^{2} c \,d^{3} e^{3}-8320 x b \,c^{2} d^{4} e^{2}+2560 x \,c^{3} d^{5} e +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right ) \sqrt {e x +d}}{15015 e^{7}}\) | \(286\) |
| orering | \(-\frac {2 \left (-1155 x^{6} c^{3} e^{6}-4095 x^{5} b \,c^{2} e^{6}+1260 x^{5} c^{3} d \,e^{5}-5005 x^{4} b^{2} c \,e^{6}+4550 x^{4} b \,c^{2} d \,e^{5}-1400 x^{4} c^{3} d^{2} e^{4}-2145 x^{3} b^{3} e^{6}+5720 x^{3} b^{2} c d \,e^{5}-5200 x^{3} b \,c^{2} d^{2} e^{4}+1600 x^{3} c^{3} d^{3} e^{3}+2574 x^{2} b^{3} d \,e^{5}-6864 x^{2} b^{2} c \,d^{2} e^{4}+6240 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}-3432 x \,b^{3} d^{2} e^{4}+9152 x \,b^{2} c \,d^{3} e^{3}-8320 x b \,c^{2} d^{4} e^{2}+2560 x \,c^{3} d^{5} e +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right ) \sqrt {e x +d}\, \left (c \,x^{2}+b x \right )^{3}}{15015 e^{7} \left (c x +b \right )^{3} x^{3}}\) | \(307\) |
Input:
int((c*x^2+b*x)^3/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
-32/35*(-5/16*(7/13*c^3*x^3+21/11*b*c^2*x^2+7/3*b^2*c*x+b^3)*x^3*e^6+3/8*x ^2*d*(70/143*c^3*x^3+175/99*b*c^2*x^2+20/9*b^2*c*x+b^3)*e^5-1/2*x*(175/429 *c^3*x^3+50/33*b*c^2*x^2+2*b^2*c*x+b^3)*d^2*e^4+d^3*(100/429*c^3*x^3+10/11 *b*c^2*x^2+4/3*b^2*c*x+b^3)*e^3-8/3*c*d^4*(15/143*c^2*x^2+5/11*c*b*x+b^2)* e^2+80/33*c^2*(2/13*c*x+b)*d^5*e-320/429*d^6*c^3)*(e*x+d)^(1/2)/e^7
Time = 0.08 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e + 18304 \, b^{2} c d^{4} e^{2} - 6864 \, b^{3} d^{3} e^{3} - 315 \, {\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, b^{2} c e^{6}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, b^{2} c d e^{5} - 429 \, b^{3} e^{6}\right )} x^{3} + 6 \, {\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, b^{2} c d^{2} e^{4} - 429 \, b^{3} d e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{5} e - 1040 \, b c^{2} d^{4} e^{2} + 1144 \, b^{2} c d^{3} e^{3} - 429 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/15015*(1155*c^3*e^6*x^6 + 5120*c^3*d^6 - 16640*b*c^2*d^5*e + 18304*b^2*c *d^4*e^2 - 6864*b^3*d^3*e^3 - 315*(4*c^3*d*e^5 - 13*b*c^2*e^6)*x^5 + 35*(4 0*c^3*d^2*e^4 - 130*b*c^2*d*e^5 + 143*b^2*c*e^6)*x^4 - 5*(320*c^3*d^3*e^3 - 1040*b*c^2*d^2*e^4 + 1144*b^2*c*d*e^5 - 429*b^3*e^6)*x^3 + 6*(320*c^3*d^ 4*e^2 - 1040*b*c^2*d^3*e^3 + 1144*b^2*c*d^2*e^4 - 429*b^3*d*e^5)*x^2 - 8*( 320*c^3*d^5*e - 1040*b*c^2*d^4*e^2 + 1144*b^2*c*d^3*e^3 - 429*b^3*d^2*e^4) *x)*sqrt(e*x + d)/e^7
Time = 0.76 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.51 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+b*x)**3/(e*x+d)**(1/2),x)
Output:
Piecewise((2*(c**3*(d + e*x)**(13/2)/(13*e**6) + (d + e*x)**(11/2)*(3*b*c* *2*e - 6*c**3*d)/(11*e**6) + (d + e*x)**(9/2)*(3*b**2*c*e**2 - 15*b*c**2*d *e + 15*c**3*d**2)/(9*e**6) + (d + e*x)**(7/2)*(b**3*e**3 - 12*b**2*c*d*e* *2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(7*e**6) + (d + e*x)**(5/2)*(-3*b**3 *d*e**3 + 18*b**2*c*d**2*e**2 - 30*b*c**2*d**3*e + 15*c**3*d**4)/(5*e**6) + (d + e*x)**(3/2)*(3*b**3*d**2*e**3 - 12*b**2*c*d**3*e**2 + 15*b*c**2*d** 4*e - 6*c**3*d**5)/(3*e**6) + sqrt(d + e*x)*(-b**3*d**3*e**3 + 3*b**2*c*d* *4*e**2 - 3*b*c**2*d**5*e + c**3*d**6)/e**6)/e, Ne(e, 0)), ((b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7)/sqrt(d), True))
Time = 0.03 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/15015*(429*(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^3/e^3 + 143*(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*sq rt(e*x + d)*d^4)*b^2*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2) *d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^( 3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2) - 16 38*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^ 3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e
Time = 0.12 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="giac")
Output:
2/15015*(429*(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^3/e^3 + 143*(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*sq rt(e*x + d)*d^4)*b^2*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2) *d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^( 3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2) - 16 38*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^ 3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e
Time = 5.40 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.98 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{5\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}}{e^7}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{e^7} \] Input:
int((b*x + c*x^2)^3/(d + e*x)^(1/2),x)
Output:
((d + e*x)^(7/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2 ))/(7*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7) - ((12*c^3*d - 6*b*c^2*e)*( d + e*x)^(11/2))/(11*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 3 0*b*c^2*d*e))/(9*e^7) + ((d + e*x)^(5/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^ 2*c*d^2*e^2 - 60*b*c^2*d^3*e))/(5*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(1 /2))/e^7 + (2*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(3/2))/e^7
Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.16 \[ \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} e^{6} x^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x -6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}} \] Input:
int((c*x^2+b*x)^3/(e*x+d)^(1/2),x)
Output:
(2*sqrt(d + e*x)*( - 6864*b**3*d**3*e**3 + 3432*b**3*d**2*e**4*x - 2574*b* *3*d*e**5*x**2 + 2145*b**3*e**6*x**3 + 18304*b**2*c*d**4*e**2 - 9152*b**2* c*d**3*e**3*x + 6864*b**2*c*d**2*e**4*x**2 - 5720*b**2*c*d*e**5*x**3 + 500 5*b**2*c*e**6*x**4 - 16640*b*c**2*d**5*e + 8320*b*c**2*d**4*e**2*x - 6240* b*c**2*d**3*e**3*x**2 + 5200*b*c**2*d**2*e**4*x**3 - 4550*b*c**2*d*e**5*x* *4 + 4095*b*c**2*e**6*x**5 + 5120*c**3*d**6 - 2560*c**3*d**5*e*x + 1920*c* *3*d**4*e**2*x**2 - 1600*c**3*d**3*e**3*x**3 + 1400*c**3*d**2*e**4*x**4 - 1260*c**3*d*e**5*x**5 + 1155*c**3*e**6*x**6))/(15015*e**7)