Integrand size = 21, antiderivative size = 147 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 d^2 (c d-b e)^2 (d+e x)^{9/2}}{9 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{15/2}}{15 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5} \]
2/9*d^2*(-b*e+c*d)^2*(e*x+d)^(9/2)/e^5-4/11*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x +d)^(11/2)/e^5+2/13*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(13/2)/e^5-4/15* c*(-b*e+2*c*d)*(e*x+d)^(15/2)/e^5+2/17*c^2*(e*x+d)^(17/2)/e^5
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{9/2} \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5} \]
(2*(d + e*x)^(9/2)*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 34*b*c*e* (-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + c^2*(128*d^4 - 576* d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)))/(109395*e^5)
Time = 0.29 (sec) , antiderivative size = 147, 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.
| \(\displaystyle \int \left (b x+c x^2\right )^2 (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^4}+\frac {d^2 (d+e x)^{7/2} (c d-b e)^2}{e^4}-\frac {2 c (d+e x)^{13/2} (2 c d-b e)}{e^4}+\frac {2 d (d+e x)^{9/2} (c d-b e) (b e-2 c d)}{e^4}+\frac {c^2 (d+e x)^{15/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac {4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac {4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5}\) |
(2*d^2*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(15/2))/(15*e^5) + (2*c^2*(d + e*x)^(17/2))/(17*e^5)
3.4.45.3.1 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 = 2.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (e x +d \right )^{\frac {9}{2}} \left (\frac {99 x^{2} \left (\frac {13}{17} c^{2} x^{2}+\frac {26}{15} b c x +b^{2}\right ) e^{4}}{8}-\frac {9 x \left (\frac {286}{255} c^{2} x^{2}+\frac {11}{5} b c x +b^{2}\right ) d \,e^{3}}{2}+d^{2} \left (\frac {198}{85} c^{2} x^{2}+\frac {18}{5} b c x +b^{2}\right ) e^{2}-\frac {4 c \left (\frac {18 c x}{17}+b \right ) d^{3} e}{5}+\frac {16 c^{2} d^{4}}{85}\right )}{1287 e^{5}}\) | \(108\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 c^{2} x^{4} e^{4}+14586 x^{3} b c \,e^{4}-3432 x^{3} c^{2} d \,e^{3}+8415 x^{2} b^{2} e^{4}-6732 x^{2} b c d \,e^{3}+1584 x^{2} c^{2} d^{2} e^{2}-3060 x \,b^{2} d \,e^{3}+2448 x b c \,d^{2} e^{2}-576 x \,c^{2} d^{3} e +680 b^{2} d^{2} e^{2}-544 d^{3} e b c +128 c^{2} d^{4}\right )}{109395 e^{5}}\) | \(141\) |
| derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (c^{2} d^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(144\) |
| default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (c^{2} d^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(144\) |
| trager | \(\frac {2 \left (6435 c^{2} e^{8} x^{8}+14586 b c \,e^{8} x^{7}+22308 c^{2} d \,e^{7} x^{7}+8415 b^{2} e^{8} x^{6}+51612 b c d \,e^{7} x^{6}+26466 c^{2} d^{2} e^{6} x^{6}+30600 b^{2} d \,e^{7} x^{5}+63036 b c \,d^{2} e^{6} x^{5}+10908 c^{2} d^{3} e^{5} x^{5}+38930 b^{2} d^{2} e^{6} x^{4}+27200 b c \,d^{3} e^{5} x^{4}+35 c^{2} d^{4} e^{4} x^{4}+18020 b^{2} d^{3} e^{5} x^{3}+170 b c \,d^{4} e^{4} x^{3}-40 c^{2} d^{5} e^{3} x^{3}+255 b^{2} d^{4} e^{4} x^{2}-204 b c \,d^{5} e^{3} x^{2}+48 c^{2} d^{6} e^{2} x^{2}-340 b^{2} d^{5} e^{3} x +272 b c \,d^{6} e^{2} x -64 c^{2} d^{7} e x +680 b^{2} d^{6} e^{2}-544 b c \,d^{7} e +128 c^{2} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(305\) |
| risch | \(\frac {2 \left (6435 c^{2} e^{8} x^{8}+14586 b c \,e^{8} x^{7}+22308 c^{2} d \,e^{7} x^{7}+8415 b^{2} e^{8} x^{6}+51612 b c d \,e^{7} x^{6}+26466 c^{2} d^{2} e^{6} x^{6}+30600 b^{2} d \,e^{7} x^{5}+63036 b c \,d^{2} e^{6} x^{5}+10908 c^{2} d^{3} e^{5} x^{5}+38930 b^{2} d^{2} e^{6} x^{4}+27200 b c \,d^{3} e^{5} x^{4}+35 c^{2} d^{4} e^{4} x^{4}+18020 b^{2} d^{3} e^{5} x^{3}+170 b c \,d^{4} e^{4} x^{3}-40 c^{2} d^{5} e^{3} x^{3}+255 b^{2} d^{4} e^{4} x^{2}-204 b c \,d^{5} e^{3} x^{2}+48 c^{2} d^{6} e^{2} x^{2}-340 b^{2} d^{5} e^{3} x +272 b c \,d^{6} e^{2} x -64 c^{2} d^{7} e x +680 b^{2} d^{6} e^{2}-544 b c \,d^{7} e +128 c^{2} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(305\) |
16/1287*(e*x+d)^(9/2)*(99/8*x^2*(13/17*c^2*x^2+26/15*b*c*x+b^2)*e^4-9/2*x* (286/255*c^2*x^2+11/5*b*c*x+b^2)*d*e^3+d^2*(198/85*c^2*x^2+18/5*b*c*x+b^2) *e^2-4/5*c*(18/17*c*x+b)*d^3*e+16/85*c^2*d^4)/e^5
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (127) = 254\).
Time = 0.48 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.97 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6435 \, c^{2} e^{8} x^{8} + 128 \, c^{2} d^{8} - 544 \, b c d^{7} e + 680 \, b^{2} d^{6} e^{2} + 858 \, {\left (26 \, c^{2} d e^{7} + 17 \, b c e^{8}\right )} x^{7} + 33 \, {\left (802 \, c^{2} d^{2} e^{6} + 1564 \, b c d e^{7} + 255 \, b^{2} e^{8}\right )} x^{6} + 36 \, {\left (303 \, c^{2} d^{3} e^{5} + 1751 \, b c d^{2} e^{6} + 850 \, b^{2} d e^{7}\right )} x^{5} + 5 \, {\left (7 \, c^{2} d^{4} e^{4} + 5440 \, b c d^{3} e^{5} + 7786 \, b^{2} d^{2} e^{6}\right )} x^{4} - 10 \, {\left (4 \, c^{2} d^{5} e^{3} - 17 \, b c d^{4} e^{4} - 1802 \, b^{2} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{6} e^{2} - 68 \, b c d^{5} e^{3} + 85 \, b^{2} d^{4} e^{4}\right )} x^{2} - 4 \, {\left (16 \, c^{2} d^{7} e - 68 \, b c d^{6} e^{2} + 85 \, b^{2} d^{5} e^{3}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \]
2/109395*(6435*c^2*e^8*x^8 + 128*c^2*d^8 - 544*b*c*d^7*e + 680*b^2*d^6*e^2 + 858*(26*c^2*d*e^7 + 17*b*c*e^8)*x^7 + 33*(802*c^2*d^2*e^6 + 1564*b*c*d* e^7 + 255*b^2*e^8)*x^6 + 36*(303*c^2*d^3*e^5 + 1751*b*c*d^2*e^6 + 850*b^2* d*e^7)*x^5 + 5*(7*c^2*d^4*e^4 + 5440*b*c*d^3*e^5 + 7786*b^2*d^2*e^6)*x^4 - 10*(4*c^2*d^5*e^3 - 17*b*c*d^4*e^4 - 1802*b^2*d^3*e^5)*x^3 + 3*(16*c^2*d^ 6*e^2 - 68*b*c*d^5*e^3 + 85*b^2*d^4*e^4)*x^2 - 4*(16*c^2*d^7*e - 68*b*c*d^ 6*e^2 + 85*b^2*d^5*e^3)*x)*sqrt(e*x + d)/e^5
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (144) = 288\).
Time = 0.73 (sec) , antiderivative size = 590, normalized size of antiderivative = 4.01 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\begin {cases} \frac {16 b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {64 b c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {32 b c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {8 b c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {4 b c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {640 b c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {824 b c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {184 b c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {4 b c e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 c^{2} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 c^{2} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 c^{2} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 c^{2} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 c^{2} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 c^{2} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 c^{2} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 c^{2} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 c^{2} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Piecewise((16*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*b**2*d**3 *x**3*sqrt(d + e*x)/1287 + 916*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*b* *2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64* b*c*d**7*sqrt(d + e*x)/(6435*e**4) + 32*b*c*d**6*x*sqrt(d + e*x)/(6435*e** 3) - 8*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4*b*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 640*b*c*d**3*x**4*sqrt(d + e*x)/1287 + 824*b*c*d**2*e*x**5 *sqrt(d + e*x)/715 + 184*b*c*d*e**2*x**6*sqrt(d + e*x)/195 + 4*b*c*e**3*x* *7*sqrt(d + e*x)/15 + 256*c**2*d**8*sqrt(d + e*x)/(109395*e**5) - 128*c**2 *d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*c**2*d**6*x**2*sqrt(d + e*x)/(364 65*e**3) - 16*c**2*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*c**2*d**4*x** 4*sqrt(d + e*x)/(21879*e) + 2424*c**2*d**3*x**5*sqrt(d + e*x)/12155 + 1604 *c**2*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*c**2*d*e**2*x**7*sqrt(d + e*x)/ 255 + 2*c**2*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**(7/2)*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), True))
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.95 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{2} - 14586 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 8415 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 19890 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 12155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{109395 \, e^{5}} \]
2/109395*(6435*(e*x + d)^(17/2)*c^2 - 14586*(2*c^2*d - b*c*e)*(e*x + d)^(1 5/2) + 8415*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d)^(13/2) - 19890*(2* c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(11/2) + 12155*(c^2*d^4 - 2*b *c*d^3*e + b^2*d^2*e^2)*(e*x + d)^(9/2))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 1171 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 1171, normalized size of antiderivative = 7.97 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
2/765765*(51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2*d^4/e^2 + 43758*(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*d^4/e^3 + 87516*(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^2*d^3/e^2 + 2431*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 3 78*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)* c^2*d^4/e^4 + 19448*(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)*b*c*d^3 /e^3 + 14586*(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)*b^2*d^2/e^2 + 4420*(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)*c^2*d^3/e^4 + 13260*(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*d^2/e^3 + 4420*(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^2*d/e^2 + 1530*(231*(e *x + d)^(13/2) - 1638*(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^2*d^2/e^4 + 2040*(231*(e*x + d)^(13/2) - 1...
Time = 9.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}-\frac {{\left (d+e\,x\right )}^{11/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{13\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \]