Integrand size = 19, antiderivative size = 204 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7} \]
2/5*(a*e^2+c*d^2)^3*(e*x+d)^(5/2)/e^7-12/7*c*d*(a*e^2+c*d^2)^2*(e*x+d)^(7/ 2)/e^7+2/3*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)*(e*x+d)^(9/2)/e^7-8/11*c^2*d*(3 *a*e^2+5*c*d^2)*(e*x+d)^(11/2)/e^7+6/13*c^2*(a*e^2+5*c*d^2)*(e*x+d)^(13/2) /e^7-4/5*c^3*d*(e*x+d)^(15/2)/e^7+2/17*c^3*(e*x+d)^(17/2)/e^7
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (51051 a^3 e^6+2431 a^2 c e^4 \left (8 d^2-20 d e x+35 e^2 x^2\right )+51 a c^2 e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^3 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]
(2*(d + e*x)^(5/2)*(51051*a^3*e^6 + 2431*a^2*c*e^4*(8*d^2 - 20*d*e*x + 35* e^2*x^2) + 51*a*c^2*e^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e ^3*x^3 + 1155*e^4*x^4) + c^3*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6)))/ (255255*e^7)
Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {476, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{e^6}-\frac {4 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{e^6}+\frac {3 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6}-\frac {6 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{e^6}+\frac {(d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{e^6}+\frac {c^3 (d+e x)^{15/2}}{e^6}-\frac {6 c^3 d (d+e x)^{13/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac {8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac {2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac {4 c^3 d (d+e x)^{15/2}}{5 e^7}\) |
(2*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (12*c*d*(c*d^2 + a*e^2)^2* (d + e*x)^(7/2))/(7*e^7) + (2*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x )^(9/2))/(3*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/(11*e^7) + (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(13/2))/(13*e^7) - (4*c^3*d*(d + e*x )^(15/2))/(5*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7)
3.7.6.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 2.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\frac {\left (5 e^{6} x^{6}-4 d \,e^{5} x^{5}+\frac {40}{13} d^{2} e^{4} x^{4}-\frac {320}{143} x^{3} d^{3} e^{3}+\frac {640}{429} d^{4} e^{2} x^{2}-\frac {2560}{3003} d^{5} e x +\frac {1024}{3003} d^{6}\right ) c^{3}}{17}+\frac {128 e^{2} a \left (\frac {1155}{128} e^{4} x^{4}-\frac {105}{16} d \,e^{3} x^{3}+\frac {35}{8} d^{2} e^{2} x^{2}-\frac {5}{2} d^{3} e x +d^{4}\right ) c^{2}}{1001}+\frac {8 e^{4} \left (\frac {35}{8} x^{2} e^{2}-\frac {5}{2} d e x +d^{2}\right ) a^{2} c}{21}+e^{6} a^{3}\right )}{5 e^{7}}\) | \(163\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (15015 x^{6} c^{3} e^{6}-12012 x^{5} c^{3} d \,e^{5}+58905 x^{4} a \,c^{2} e^{6}+9240 x^{4} c^{3} d^{2} e^{4}-42840 x^{3} a \,c^{2} d \,e^{5}-6720 x^{3} c^{3} d^{3} e^{3}+85085 x^{2} a^{2} c \,e^{6}+28560 x^{2} a \,c^{2} d^{2} e^{4}+4480 x^{2} c^{3} d^{4} e^{2}-48620 x \,a^{2} c d \,e^{5}-16320 x a \,c^{2} d^{3} e^{3}-2560 x \,c^{3} d^{5} e +51051 e^{6} a^{3}+19448 d^{2} e^{4} a^{2} c +6528 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{255255 e^{7}}\) | \(205\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {15}{2}}}{5}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{7}}\) | \(269\) |
| trager | \(\frac {2 \left (15015 e^{8} c^{3} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 a \,c^{2} d \,e^{7} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
| risch | \(\frac {2 \left (15015 e^{8} c^{3} x^{8}+18018 c^{3} d \,e^{7} x^{7}+58905 a \,c^{2} e^{8} x^{6}+231 c^{3} d^{2} e^{6} x^{6}+74970 a \,c^{2} d \,e^{7} x^{5}-252 c^{3} d^{3} e^{5} x^{5}+85085 a^{2} c \,e^{8} x^{4}+1785 a \,c^{2} d^{2} e^{6} x^{4}+280 c^{3} d^{4} e^{4} x^{4}+121550 a^{2} c d \,e^{7} x^{3}-2040 a \,c^{2} d^{3} e^{5} x^{3}-320 c^{3} d^{5} e^{3} x^{3}+51051 a^{3} e^{8} x^{2}+7293 a^{2} c \,d^{2} e^{6} x^{2}+2448 a \,c^{2} d^{4} e^{4} x^{2}+384 c^{3} d^{6} e^{2} x^{2}+102102 a^{3} d \,e^{7} x -9724 a^{2} c \,d^{3} e^{5} x -3264 a \,c^{2} d^{5} e^{3} x -512 c^{3} d^{7} e x +51051 a^{3} d^{2} e^{6}+19448 a^{2} c \,d^{4} e^{4}+6528 a \,c^{2} d^{6} e^{2}+1024 c^{3} d^{8}\right ) \sqrt {e x +d}}{255255 e^{7}}\) | \(317\) |
2/5*(e*x+d)^(5/2)*(1/17*(5*e^6*x^6-4*d*e^5*x^5+40/13*d^2*e^4*x^4-320/143*x ^3*d^3*e^3+640/429*d^4*e^2*x^2-2560/3003*d^5*e*x+1024/3003*d^6)*c^3+128/10 01*e^2*a*(1155/128*e^4*x^4-105/16*d*e^3*x^3+35/8*d^2*e^2*x^2-5/2*d^3*e*x+d ^4)*c^2+8/21*e^4*(35/8*x^2*e^2-5/2*d*e*x+d^2)*a^2*c+e^6*a^3)/e^7
Time = 0.27 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.49 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, c^{3} e^{8} x^{8} + 18018 \, c^{3} d e^{7} x^{7} + 1024 \, c^{3} d^{8} + 6528 \, a c^{2} d^{6} e^{2} + 19448 \, a^{2} c d^{4} e^{4} + 51051 \, a^{3} d^{2} e^{6} + 231 \, {\left (c^{3} d^{2} e^{6} + 255 \, a c^{2} e^{8}\right )} x^{6} - 126 \, {\left (2 \, c^{3} d^{3} e^{5} - 595 \, a c^{2} d e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{4} e^{4} + 51 \, a c^{2} d^{2} e^{6} + 2431 \, a^{2} c e^{8}\right )} x^{4} - 10 \, {\left (32 \, c^{3} d^{5} e^{3} + 204 \, a c^{2} d^{3} e^{5} - 12155 \, a^{2} c d e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{6} e^{2} + 816 \, a c^{2} d^{4} e^{4} + 2431 \, a^{2} c d^{2} e^{6} + 17017 \, a^{3} e^{8}\right )} x^{2} - 2 \, {\left (256 \, c^{3} d^{7} e + 1632 \, a c^{2} d^{5} e^{3} + 4862 \, a^{2} c d^{3} e^{5} - 51051 \, a^{3} d e^{7}\right )} x\right )} \sqrt {e x + d}}{255255 \, e^{7}} \]
2/255255*(15015*c^3*e^8*x^8 + 18018*c^3*d*e^7*x^7 + 1024*c^3*d^8 + 6528*a* c^2*d^6*e^2 + 19448*a^2*c*d^4*e^4 + 51051*a^3*d^2*e^6 + 231*(c^3*d^2*e^6 + 255*a*c^2*e^8)*x^6 - 126*(2*c^3*d^3*e^5 - 595*a*c^2*d*e^7)*x^5 + 35*(8*c^ 3*d^4*e^4 + 51*a*c^2*d^2*e^6 + 2431*a^2*c*e^8)*x^4 - 10*(32*c^3*d^5*e^3 + 204*a*c^2*d^3*e^5 - 12155*a^2*c*d*e^7)*x^3 + 3*(128*c^3*d^6*e^2 + 816*a*c^ 2*d^4*e^4 + 2431*a^2*c*d^2*e^6 + 17017*a^3*e^8)*x^2 - 2*(256*c^3*d^7*e + 1 632*a*c^2*d^5*e^3 + 4862*a^2*c*d^3*e^5 - 51051*a^3*d*e^7)*x)*sqrt(e*x + d) /e^7
Time = 0.79 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.49 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (- \frac {2 c^{3} d \left (d + e x\right )^{\frac {15}{2}}}{5 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{6}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{5 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(-2*c**3*d*(d + e*x)**(15/2)/(5*e**6) + c**3*(d + e*x)**(17/2 )/(17*e**6) + (d + e*x)**(13/2)*(3*a*c**2*e**2 + 15*c**3*d**2)/(13*e**6) + (d + e*x)**(11/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(11*e**6) + (d + e*x )**(9/2)*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(9*e**6) + ( d + e*x)**(7/2)*(-6*a**2*c*d*e**4 - 12*a*c**2*d**3*e**2 - 6*c**3*d**5)/(7* e**6) + (d + e*x)**(5/2)*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e **2 + c**3*d**6)/(5*e**6))/e, Ne(e, 0)), (d**(3/2)*(a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7), True))
Time = 0.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (15015 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{3} - 102102 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} d + 58905 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 92820 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 218790 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 51051 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{255255 \, e^{7}} \]
2/255255*(15015*(e*x + d)^(17/2)*c^3 - 102102*(e*x + d)^(15/2)*c^3*d + 589 05*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(13/2) - 92820*(5*c^3*d^3 + 3*a*c^2*d *e^2)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*( e*x + d)^(9/2) - 218790*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d )^(7/2) + 51051*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e *x + d)^(5/2))/e^7
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (176) = 352\).
Time = 0.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.84 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*a^3*d^2 + 510510*((e*x + d)^(3/2) - 3*sqrt( e*x + d)*d)*a^3*d + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*s qrt(e*x + d)*d^2)*a^3 + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*c*d^2/e^2 + 131274*(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)*a^2*c*d/e^2 + 7293*(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)*a*c^2*d^2/e^4 + 729 3*(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)*a^2*c/e^2 + 6630*(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)*a*c^2* d/e^4 + 255*(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^3*d^2/e^6 + 765*(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)*a*c^2/e^4 + 238*(429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 3 2175*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/ 2)*d^6 - 6435*sqrt(e*x + d)*d^7)*c^3*d/e^6 + 7*(6435*(e*x + d)^(17/2) -...
Time = 9.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{9\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{15/2}}{5\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]
((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + ((d + e*x)^(9/2)* (30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2*d^2*e^2))/(9*e^7) + (2*c^3*(d + e*x)^ (17/2))/(17*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(5/2))/(5*e^7) - ((40*c^ 3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(11/2))/(11*e^7) - (4*c^3*d*(d + e*x)^(1 5/2))/(5*e^7) - (12*c*d*(a*e^2 + c*d^2)^2*(d + e*x)^(7/2))/(7*e^7)