Integrand size = 21, antiderivative size = 240 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}-\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \] Output:
-2/5*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^(5/2)+2*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^ 7/(e*x+d)^(3/2)-6*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^( 1/2)-2*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(1/2)/e^7+2*c* (b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(3/2)/e^7-6/5*c^2*(-b*e+2*c*d)*(e*x+ d)^(5/2)/e^7+2/7*c^3*(e*x+d)^(7/2)/e^7
Time = 0.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \left (7 b^3 e^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-7 b^2 c e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+7 b c^2 e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )-c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \] Input:
Integrate[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]
Output:
(2*(7*b^3*e^3*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) - 7*b^2*c*e ^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + 7*b*c^2*e*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d *e^4*x^4 + 3*e^5*x^5) - c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))
Time = 0.62 (sec) , antiderivative size = 240, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c \sqrt {d+e x} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6 \sqrt {d+e x}}+\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^{3/2}}-\frac {3 c^2 (d+e x)^{3/2} (2 c d-b e)}{e^6}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^{7/2}}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^{5/2}}+\frac {c^3 (d+e x)^{5/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac {6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}-\frac {2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}\) |
Input:
Int[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]
Output:
(-2*d^3*(c*d - b*e)^3)/(5*e^7*(d + e*x)^(5/2)) + (2*d^2*(c*d - b*e)^2*(2*c *d - b*e))/(e^7*(d + e*x)^(3/2)) - (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*Sqrt[d + e*x]) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d *e + b^2*e^2)*Sqrt[d + e*x])/e^7 + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)* (d + e*x)^(3/2))/e^7 - (6*c^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) + (2* c^3*(d + e*x)^(7/2))/(7*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 = 0.48 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {2 x^{3} \left (\frac {1}{7} c^{3} x^{3}+\frac {3}{5} b \,c^{2} x^{2}+b^{2} c x +b^{3}\right ) e^{6}+12 \left (-\frac {2}{35} c^{3} x^{3}-\frac {1}{3} b \,c^{2} x^{2}-\frac {4}{3} b^{2} c x +b^{3}\right ) x^{2} d \,e^{5}+16 x \,d^{2} \left (\frac {1}{7} c^{3} x^{3}+2 b \,c^{2} x^{2}-6 b^{2} c x +b^{3}\right ) e^{4}+\frac {32 d^{3} \left (-\frac {20}{7} c^{3} x^{3}+30 b \,c^{2} x^{2}-20 b^{2} c x +b^{3}\right ) e^{3}}{5}-\frac {256 c \,d^{4} \left (\frac {15}{7} c^{2} x^{2}-5 c b x +b^{2}\right ) e^{2}}{5}+\frac {512 \left (-\frac {10 c x}{7}+b \right ) c^{2} d^{5} e}{5}-\frac {2048 d^{6} c^{3}}{35}}{\left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(207\) |
| risch | \(\frac {2 \left (5 c^{3} x^{3} e^{3}+21 e^{3} x^{2} b \,c^{2}-27 c^{3} d \,e^{2} x^{2}+35 x \,b^{2} c \,e^{3}-133 x b \,c^{2} d \,e^{2}+106 c^{3} d^{2} e x +35 b^{3} e^{3}-385 d \,e^{2} b^{2} c +896 d^{2} e b \,c^{2}-562 d^{3} c^{3}\right ) \sqrt {e x +d}}{35 e^{7}}+\frac {2 \left (15 x^{2} b^{2} e^{4}-75 x^{2} b c d \,e^{3}+75 x^{2} c^{2} d^{2} e^{2}+25 x \,b^{2} d \,e^{3}-135 x b c \,d^{2} e^{2}+140 x \,c^{2} d^{3} e +11 d^{2} e^{2} b^{2}-62 d^{3} e b c +66 c^{2} d^{4}\right ) d \left (b e -c d \right )}{5 e^{7} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(251\) |
| gosper | \(\frac {\frac {2}{7} x^{6} c^{3} e^{6}+\frac {6}{5} x^{5} b \,c^{2} e^{6}-\frac {24}{35} x^{5} c^{3} d \,e^{5}+2 x^{4} b^{2} c \,e^{6}-4 x^{4} b \,c^{2} d \,e^{5}+\frac {16}{7} x^{4} c^{3} d^{2} e^{4}+2 x^{3} b^{3} e^{6}-16 x^{3} b^{2} c d \,e^{5}+32 x^{3} b \,c^{2} d^{2} e^{4}-\frac {128}{7} x^{3} c^{3} d^{3} e^{3}+12 x^{2} b^{3} d \,e^{5}-96 x^{2} b^{2} c \,d^{2} e^{4}+192 x^{2} b \,c^{2} d^{3} e^{3}-\frac {768}{7} x^{2} c^{3} d^{4} e^{2}+16 x \,b^{3} d^{2} e^{4}-128 x \,b^{2} c \,d^{3} e^{3}+256 x b \,c^{2} d^{4} e^{2}-\frac {1024}{7} x \,c^{3} d^{5} e +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {256}{5} b^{2} c \,d^{4} e^{2}+\frac {512}{5} b \,c^{2} d^{5} e -\frac {2048}{35} d^{6} c^{3}}{\left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(286\) |
| trager | \(\frac {\frac {2}{7} x^{6} c^{3} e^{6}+\frac {6}{5} x^{5} b \,c^{2} e^{6}-\frac {24}{35} x^{5} c^{3} d \,e^{5}+2 x^{4} b^{2} c \,e^{6}-4 x^{4} b \,c^{2} d \,e^{5}+\frac {16}{7} x^{4} c^{3} d^{2} e^{4}+2 x^{3} b^{3} e^{6}-16 x^{3} b^{2} c d \,e^{5}+32 x^{3} b \,c^{2} d^{2} e^{4}-\frac {128}{7} x^{3} c^{3} d^{3} e^{3}+12 x^{2} b^{3} d \,e^{5}-96 x^{2} b^{2} c \,d^{2} e^{4}+192 x^{2} b \,c^{2} d^{3} e^{3}-\frac {768}{7} x^{2} c^{3} d^{4} e^{2}+16 x \,b^{3} d^{2} e^{4}-128 x \,b^{2} c \,d^{3} e^{3}+256 x b \,c^{2} d^{4} e^{2}-\frac {1024}{7} x \,c^{3} d^{5} e +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {256}{5} b^{2} c \,d^{4} e^{2}+\frac {512}{5} b \,c^{2} d^{5} e -\frac {2048}{35} d^{6} c^{3}}{\left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(286\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-10 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}+\frac {2 d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {6 d \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +10 d^{2} e b \,c^{2}-5 d^{3} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(296\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-10 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}+\frac {2 d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {6 d \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +10 d^{2} e b \,c^{2}-5 d^{3} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(296\) |
| orering | \(\frac {2 \left (5 x^{6} c^{3} e^{6}+21 x^{5} b \,c^{2} e^{6}-12 x^{5} c^{3} d \,e^{5}+35 x^{4} b^{2} c \,e^{6}-70 x^{4} b \,c^{2} d \,e^{5}+40 x^{4} c^{3} d^{2} e^{4}+35 x^{3} b^{3} e^{6}-280 x^{3} b^{2} c d \,e^{5}+560 x^{3} b \,c^{2} d^{2} e^{4}-320 x^{3} c^{3} d^{3} e^{3}+210 x^{2} b^{3} d \,e^{5}-1680 x^{2} b^{2} c \,d^{2} e^{4}+3360 x^{2} b \,c^{2} d^{3} e^{3}-1920 x^{2} c^{3} d^{4} e^{2}+280 x \,b^{3} d^{2} e^{4}-2240 x \,b^{2} c \,d^{3} e^{3}+4480 x b \,c^{2} d^{4} e^{2}-2560 x \,c^{3} d^{5} e +112 b^{3} d^{3} e^{3}-896 b^{2} c \,d^{4} e^{2}+1792 b \,c^{2} d^{5} e -1024 d^{6} c^{3}\right ) \left (c \,x^{2}+b x \right )^{3}}{35 e^{7} \left (c x +b \right )^{3} \left (e x +d \right )^{\frac {5}{2}} x^{3}}\) | \(307\) |
Input:
int((c*x^2+b*x)^3/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
Output:
32/5/(e*x+d)^(5/2)*(5/16*x^3*(1/7*c^3*x^3+3/5*b*c^2*x^2+b^2*c*x+b^3)*e^6+1 5/8*(-2/35*c^3*x^3-1/3*b*c^2*x^2-4/3*b^2*c*x+b^3)*x^2*d*e^5+5/2*x*d^2*(1/7 *c^3*x^3+2*b*c^2*x^2-6*b^2*c*x+b^3)*e^4+d^3*(-20/7*c^3*x^3+30*b*c^2*x^2-20 *b^2*c*x+b^3)*e^3-8*c*d^4*(15/7*c^2*x^2-5*c*b*x+b^2)*e^2+16*(-10/7*c*x+b)* c^2*d^5*e-64/7*d^6*c^3)/e^7
Time = 0.09 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.26 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 896 \, b^{2} c d^{4} e^{2} + 112 \, b^{3} d^{3} e^{3} - 3 \, {\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, b^{2} c e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, b^{2} c d e^{5} - 7 \, b^{3} e^{6}\right )} x^{3} - 30 \, {\left (64 \, c^{3} d^{4} e^{2} - 112 \, b c^{2} d^{3} e^{3} + 56 \, b^{2} c d^{2} e^{4} - 7 \, b^{3} d e^{5}\right )} x^{2} - 40 \, {\left (64 \, c^{3} d^{5} e - 112 \, b c^{2} d^{4} e^{2} + 56 \, b^{2} c d^{3} e^{3} - 7 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="fricas")
Output:
2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 896*b^2*c*d^4*e^2 + 112*b^3*d^3*e^3 - 3*(4*c^3*d*e^5 - 7*b*c^2*e^6)*x^5 + 5*(8*c^3*d^2*e^4 - 14*b*c^2*d*e^5 + 7*b^2*c*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e^4 + 56*b^2*c*d*e^5 - 7*b^3*e^6)*x^3 - 30*(64*c^3*d^4*e^2 - 112*b*c^2*d^3*e^ 3 + 56*b^2*c*d^2*e^4 - 7*b^3*d*e^5)*x^2 - 40*(64*c^3*d^5*e - 112*b*c^2*d^4 *e^2 + 56*b^2*c*d^3*e^3 - 7*b^3*d^2*e^4)*x)*sqrt(e*x + d)/(e^10*x^3 + 3*d* e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
Time = 4.93 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.24 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {d^{3} \left (b e - c d\right )^{3}}{5 e^{6} \left (d + e x\right )^{\frac {5}{2}}} - \frac {d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{6} \left (d + e x\right )^{\frac {3}{2}}} + \frac {3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+b*x)**3/(e*x+d)**(7/2),x)
Output:
Piecewise((2*(c**3*(d + e*x)**(7/2)/(7*e**6) + d**3*(b*e - c*d)**3/(5*e**6 *(d + e*x)**(5/2)) - d**2*(b*e - 2*c*d)*(b*e - c*d)**2/(e**6*(d + e*x)**(3 /2)) + 3*d*(b*e - c*d)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**6*sqrt(d + e*x)) + (d + e*x)**(5/2)*(3*b*c**2*e - 6*c**3*d)/(5*e**6) + (d + e*x)**( 3/2)*(3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(3*e**6) + sqrt(d + e* x)*(b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/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)/d**(7/2), True))
Time = 0.04 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.15 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 15 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="maxima")
Output:
2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35 *(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(3/2) - 35*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*sqrt(e*x + d))/e^6 - 7*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 15*(5*c^3*d^4 - 10*b*c^ 2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^2 - 5*(2*c^3*d^5 - 5*b*c^ 2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d))/((e*x + d)^(5/2)*e^6)) /e
Time = 0.20 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.48 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (75 \, {\left (e x + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (e x + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \, {\left (e x + d\right )}^{2} b c^{2} d^{3} e + 25 \, {\left (e x + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \, {\left (e x + d\right )}^{2} b^{2} c d^{2} e^{2} - 20 \, {\left (e x + d\right )} b^{2} c d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} - 15 \, {\left (e x + d\right )}^{2} b^{3} d e^{3} + 5 \, {\left (e x + d\right )} b^{3} d^{2} e^{3} - b^{3} d^{3} e^{3}\right )}}{5 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{7}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {e x + d} c^{3} d^{3} e^{42} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{2} e^{43} - 175 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt {e x + d} b c^{2} d^{2} e^{43} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c e^{44} - 420 \, \sqrt {e x + d} b^{2} c d e^{44} + 35 \, \sqrt {e x + d} b^{3} e^{45}\right )}}{35 \, e^{49}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="giac")
Output:
-2/5*(75*(e*x + d)^2*c^3*d^4 - 10*(e*x + d)*c^3*d^5 + c^3*d^6 - 150*(e*x + d)^2*b*c^2*d^3*e + 25*(e*x + d)*b*c^2*d^4*e - 3*b*c^2*d^5*e + 90*(e*x + d )^2*b^2*c*d^2*e^2 - 20*(e*x + d)*b^2*c*d^3*e^2 + 3*b^2*c*d^4*e^2 - 15*(e*x + d)^2*b^3*d*e^3 + 5*(e*x + d)*b^3*d^2*e^3 - b^3*d^3*e^3)/((e*x + d)^(5/2 )*e^7) + 2/35*(5*(e*x + d)^(7/2)*c^3*e^42 - 42*(e*x + d)^(5/2)*c^3*d*e^42 + 175*(e*x + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt(e*x + d)*c^3*d^3*e^42 + 21*( e*x + d)^(5/2)*b*c^2*e^43 - 175*(e*x + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(e *x + d)*b*c^2*d^2*e^43 + 35*(e*x + d)^(3/2)*b^2*c*e^44 - 420*sqrt(e*x + d) *b^2*c*d*e^44 + 35*sqrt(e*x + d)*b^3*e^45)/e^49
Time = 0.05 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.16 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{e^7}+\frac {\left (d+e\,x\right )\,\left (-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )-{\left (d+e\,x\right )}^2\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )-\frac {2\,c^3\,d^6}{5}+\frac {2\,b^3\,d^3\,e^3}{5}-\frac {6\,b^2\,c\,d^4\,e^2}{5}+\frac {6\,b\,c^2\,d^5\,e}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{3\,e^7} \] Input:
int((b*x + c*x^2)^3/(d + e*x)^(7/2),x)
Output:
((d + e*x)^(1/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2 ))/e^7 + ((d + e*x)*(4*c^3*d^5 - 2*b^3*d^2*e^3 + 8*b^2*c*d^3*e^2 - 10*b*c^ 2*d^4*e) - (d + e*x)^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) - (2*c^3*d^6)/5 + (2*b^3*d^3*e^3)/5 - (6*b^2*c*d^4*e^2)/5 + (6 *b*c^2*d^5*e)/5)/(e^7*(d + e*x)^(5/2)) + (2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(5/2))/(5*e^7) + ((d + e*x)^(3/2)*(30*c ^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(3*e^7)
Time = 0.23 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.27 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {\frac {2}{7} c^{3} e^{6} x^{6}+\frac {6}{5} b \,c^{2} e^{6} x^{5}-\frac {24}{35} c^{3} d \,e^{5} x^{5}+2 b^{2} c \,e^{6} x^{4}-4 b \,c^{2} d \,e^{5} x^{4}+\frac {16}{7} c^{3} d^{2} e^{4} x^{4}+2 b^{3} e^{6} x^{3}-16 b^{2} c d \,e^{5} x^{3}+32 b \,c^{2} d^{2} e^{4} x^{3}-\frac {128}{7} c^{3} d^{3} e^{3} x^{3}+12 b^{3} d \,e^{5} x^{2}-96 b^{2} c \,d^{2} e^{4} x^{2}+192 b \,c^{2} d^{3} e^{3} x^{2}-\frac {768}{7} c^{3} d^{4} e^{2} x^{2}+16 b^{3} d^{2} e^{4} x -128 b^{2} c \,d^{3} e^{3} x +256 b \,c^{2} d^{4} e^{2} x -\frac {1024}{7} c^{3} d^{5} e x +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {256}{5} b^{2} c \,d^{4} e^{2}+\frac {512}{5} b \,c^{2} d^{5} e -\frac {2048}{35} c^{3} d^{6}}{\sqrt {e x +d}\, e^{7} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int((c*x^2+b*x)^3/(e*x+d)^(7/2),x)
Output:
(2*(112*b**3*d**3*e**3 + 280*b**3*d**2*e**4*x + 210*b**3*d*e**5*x**2 + 35* b**3*e**6*x**3 - 896*b**2*c*d**4*e**2 - 2240*b**2*c*d**3*e**3*x - 1680*b** 2*c*d**2*e**4*x**2 - 280*b**2*c*d*e**5*x**3 + 35*b**2*c*e**6*x**4 + 1792*b *c**2*d**5*e + 4480*b*c**2*d**4*e**2*x + 3360*b*c**2*d**3*e**3*x**2 + 560* b*c**2*d**2*e**4*x**3 - 70*b*c**2*d*e**5*x**4 + 21*b*c**2*e**6*x**5 - 1024 *c**3*d**6 - 2560*c**3*d**5*e*x - 1920*c**3*d**4*e**2*x**2 - 320*c**3*d**3 *e**3*x**3 + 40*c**3*d**2*e**4*x**4 - 12*c**3*d*e**5*x**5 + 5*c**3*e**6*x* *6))/(35*sqrt(d + e*x)*e**7*(d**2 + 2*d*e*x + e**2*x**2))