Integrand size = 46, antiderivative size = 501 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {512 (2 c d-b e)^5 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2909907 c^7 e^2 (d+e x)^{7/2}}-\frac {256 (2 c d-b e)^4 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{415701 c^6 e^2 (d+e x)^{5/2}}-\frac {64 (2 c d-b e)^3 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{46189 c^5 e^2 (d+e x)^{3/2}}-\frac {32 (2 c d-b e)^2 (19 c e f+5 c d g-12 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12597 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (19 c e f+5 c d g-12 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{969 c^3 e^2}-\frac {2 (19 c e f+5 c d g-12 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{323 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2} \]
-512/2909907*(-b*e+2*c*d)^5*(-12*b*e*g+5*c*d*g+19*c*e*f)*(d*(-b*e+c*d)-b*e ^2*x-c*e^2*x^2)^(7/2)/c^7/e^2/(e*x+d)^(7/2)-256/415701*(-b*e+2*c*d)^4*(-12 *b*e*g+5*c*d*g+19*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^6/e^2/(e *x+d)^(5/2)-64/46189*(-b*e+2*c*d)^3*(-12*b*e*g+5*c*d*g+19*c*e*f)*(d*(-b*e+ c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^5/e^2/(e*x+d)^(3/2)-2/323*(-12*b*e*g+5*c*d *g+19*c*e*f)*(e*x+d)^(3/2)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^2/e^2- 2/19*g*(e*x+d)^(5/2)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c/e^2-32/12597 *(-b*e+2*c*d)^2*(-12*b*e*g+5*c*d*g+19*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(7/2)/c^4/e^2/(e*x+d)^(1/2)-4/969*(-b*e+2*c*d)*(-12*b*e*g+5*c*d*g+19*c *e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)*(e*x+d)^(1/2)/c^3/e^2
Time = 0.45 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.97 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (3072 b^6 e^6 g-256 b^5 c e^5 (19 e f+167 d g+42 e g x)+128 b^4 c^2 e^4 \left (1956 d^2 g+7 e^2 x (19 f+27 g x)+d e (513 f+1085 g x)\right )-32 b^3 c^3 e^3 \left (24701 d^3 g+63 e^3 x^2 (19 f+22 g x)+7 d e^2 x (950 f+1287 g x)+d^2 e (11533 f+23044 g x)\right )+8 b^2 c^4 e^2 \left (177311 d^4 g+231 e^4 x^3 (38 f+39 g x)+42 d e^3 x^2 (1311 f+1441 g x)+42 d^2 e^2 x (3211 f+4080 g x)+2 d^3 e (68609 f+126819 g x)\right )-2 b c^5 e \left (682101 d^5 g+3003 e^5 x^4 (19 f+18 g x)+1617 d e^4 x^3 (228 f+221 g x)+126 d^2 e^3 x^2 (7885 f+8052 g x)+98 d^3 e^2 x (14098 f+16299 g x)+d^4 e (894273 f+1467802 g x)\right )+c^6 \left (525458 d^6 g+9009 e^6 x^5 (19 f+17 g x)+3003 d e^5 x^4 (361 f+321 g x)+462 d^2 e^4 x^3 (6289 f+5590 g x)+42 d^3 e^3 x^2 (100719 f+91135 g x)+7 d^4 e^2 x (499529 f+487215 g x)+d^5 e (1414759 f+1839103 g x)\right )\right )}{2909907 c^7 e^2 \sqrt {d+e x}} \]
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(3072*b ^6*e^6*g - 256*b^5*c*e^5*(19*e*f + 167*d*g + 42*e*g*x) + 128*b^4*c^2*e^4*( 1956*d^2*g + 7*e^2*x*(19*f + 27*g*x) + d*e*(513*f + 1085*g*x)) - 32*b^3*c^ 3*e^3*(24701*d^3*g + 63*e^3*x^2*(19*f + 22*g*x) + 7*d*e^2*x*(950*f + 1287* g*x) + d^2*e*(11533*f + 23044*g*x)) + 8*b^2*c^4*e^2*(177311*d^4*g + 231*e^ 4*x^3*(38*f + 39*g*x) + 42*d*e^3*x^2*(1311*f + 1441*g*x) + 42*d^2*e^2*x*(3 211*f + 4080*g*x) + 2*d^3*e*(68609*f + 126819*g*x)) - 2*b*c^5*e*(682101*d^ 5*g + 3003*e^5*x^4*(19*f + 18*g*x) + 1617*d*e^4*x^3*(228*f + 221*g*x) + 12 6*d^2*e^3*x^2*(7885*f + 8052*g*x) + 98*d^3*e^2*x*(14098*f + 16299*g*x) + d ^4*e*(894273*f + 1467802*g*x)) + c^6*(525458*d^6*g + 9009*e^6*x^5*(19*f + 17*g*x) + 3003*d*e^5*x^4*(361*f + 321*g*x) + 462*d^2*e^4*x^3*(6289*f + 559 0*g*x) + 42*d^3*e^3*x^2*(100719*f + 91135*g*x) + 7*d^4*e^2*x*(499529*f + 4 87215*g*x) + d^5*e*(1414759*f + 1839103*g*x))))/(2909907*c^7*e^2*Sqrt[d + e*x])
Time = 0.74 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1221, 1128, 1128, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int (d+e x)^{5/2} (f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \int (d+e x)^{5/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \left (\frac {10 (2 c d-b e) \int (d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right )}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \left (\frac {10 (2 c d-b e) \left (\frac {8 (2 c d-b e) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right )}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \left (\frac {10 (2 c d-b e) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{\sqrt {d+e x}}dx}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right )}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \left (\frac {10 (2 c d-b e) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right )}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-12 b e g+5 c d g+19 c e f) \left (\frac {10 (2 c d-b e) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right )}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {10 (2 c d-b e) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e}\right ) (-12 b e g+5 c d g+19 c e f)}{19 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{19 c e^2}\) |
(-2*g*(d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(19*c*e ^2) + ((19*c*e*f + 5*c*d*g - 12*b*e*g)*((-2*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(17*c*e) + (10*(2*c*d - b*e)*((-2*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(15*c*e) + (8*(2*c*d - b *e)*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*c*e*Sqrt[d + e*x ]) + (6*(2*c*d - b*e)*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1 1*c*e*(d + e*x)^(3/2)) + (4*(2*c*d - b*e)*((-4*(2*c*d - b*e)*(d*(c*d - b*e ) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*c^2*e*(d + e*x)^(7/2)) - (2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e*(d + e*x)^(5/2))))/(11*c)))/(13* c)))/(15*c)))/(17*c)))/(19*c*e)
3.23.50.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.39 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (x c e +b e -c d \right )^{3} \left (153153 g \,e^{6} x^{6} c^{6}-108108 b \,c^{5} e^{6} g \,x^{5}+963963 c^{6} d \,e^{5} g \,x^{5}+171171 c^{6} e^{6} f \,x^{5}+72072 b^{2} c^{4} e^{6} g \,x^{4}-714714 b \,c^{5} d \,e^{5} g \,x^{4}-114114 b \,c^{5} e^{6} f \,x^{4}+2582580 c^{6} d^{2} e^{4} g \,x^{4}+1084083 c^{6} d \,e^{5} f \,x^{4}-44352 b^{3} c^{3} e^{6} g \,x^{3}+484176 b^{2} c^{4} d \,e^{5} g \,x^{3}+70224 b^{2} c^{4} e^{6} f \,x^{3}-2029104 b \,c^{5} d^{2} e^{4} g \,x^{3}-737352 b \,c^{5} d \,e^{5} f \,x^{3}+3827670 c^{6} d^{3} e^{3} g \,x^{3}+2905518 c^{6} d^{2} e^{4} f \,x^{3}+24192 b^{4} c^{2} e^{6} g \,x^{2}-288288 b^{3} c^{3} d \,e^{5} g \,x^{2}-38304 b^{3} c^{3} e^{6} f \,x^{2}+1370880 b^{2} c^{4} d^{2} e^{4} g \,x^{2}+440496 b^{2} c^{4} d \,e^{5} f \,x^{2}-3194604 b \,c^{5} d^{3} e^{3} g \,x^{2}-1987020 b \,c^{5} d^{2} e^{4} f \,x^{2}+3410505 c^{6} d^{4} e^{2} g \,x^{2}+4230198 c^{6} d^{3} e^{3} f \,x^{2}-10752 b^{5} c \,e^{6} g x +138880 b^{4} c^{2} d \,e^{5} g x +17024 b^{4} c^{2} e^{6} f x -737408 b^{3} c^{3} d^{2} e^{4} g x -212800 b^{3} c^{3} d \,e^{5} f x +2029104 b^{2} c^{4} d^{3} e^{3} g x +1078896 b^{2} c^{4} d^{2} e^{4} f x -2935604 b \,c^{5} d^{4} e^{2} g x -2763208 b \,c^{5} d^{3} e^{3} f x +1839103 c^{6} d^{5} e g x +3496703 c^{6} d^{4} e^{2} f x +3072 b^{6} e^{6} g -42752 b^{5} c d \,e^{5} g -4864 b^{5} c \,e^{6} f +250368 b^{4} c^{2} d^{2} e^{4} g +65664 b^{4} c^{2} d \,e^{5} f -790432 b^{3} c^{3} d^{3} e^{3} g -369056 b^{3} c^{3} d^{2} e^{4} f +1418488 b^{2} c^{4} d^{4} e^{2} g +1097744 b^{2} c^{4} d^{3} e^{3} f -1364202 b \,c^{5} d^{5} e g -1788546 b \,c^{5} d^{4} e^{2} f +525458 c^{6} d^{6} g +1414759 f \,d^{5} c^{6} e \right )}{2909907 \sqrt {e x +d}\, c^{7} e^{2}}\) | \(733\) |
| gosper | \(\frac {2 \left (x c e +b e -c d \right ) \left (153153 g \,e^{6} x^{6} c^{6}-108108 b \,c^{5} e^{6} g \,x^{5}+963963 c^{6} d \,e^{5} g \,x^{5}+171171 c^{6} e^{6} f \,x^{5}+72072 b^{2} c^{4} e^{6} g \,x^{4}-714714 b \,c^{5} d \,e^{5} g \,x^{4}-114114 b \,c^{5} e^{6} f \,x^{4}+2582580 c^{6} d^{2} e^{4} g \,x^{4}+1084083 c^{6} d \,e^{5} f \,x^{4}-44352 b^{3} c^{3} e^{6} g \,x^{3}+484176 b^{2} c^{4} d \,e^{5} g \,x^{3}+70224 b^{2} c^{4} e^{6} f \,x^{3}-2029104 b \,c^{5} d^{2} e^{4} g \,x^{3}-737352 b \,c^{5} d \,e^{5} f \,x^{3}+3827670 c^{6} d^{3} e^{3} g \,x^{3}+2905518 c^{6} d^{2} e^{4} f \,x^{3}+24192 b^{4} c^{2} e^{6} g \,x^{2}-288288 b^{3} c^{3} d \,e^{5} g \,x^{2}-38304 b^{3} c^{3} e^{6} f \,x^{2}+1370880 b^{2} c^{4} d^{2} e^{4} g \,x^{2}+440496 b^{2} c^{4} d \,e^{5} f \,x^{2}-3194604 b \,c^{5} d^{3} e^{3} g \,x^{2}-1987020 b \,c^{5} d^{2} e^{4} f \,x^{2}+3410505 c^{6} d^{4} e^{2} g \,x^{2}+4230198 c^{6} d^{3} e^{3} f \,x^{2}-10752 b^{5} c \,e^{6} g x +138880 b^{4} c^{2} d \,e^{5} g x +17024 b^{4} c^{2} e^{6} f x -737408 b^{3} c^{3} d^{2} e^{4} g x -212800 b^{3} c^{3} d \,e^{5} f x +2029104 b^{2} c^{4} d^{3} e^{3} g x +1078896 b^{2} c^{4} d^{2} e^{4} f x -2935604 b \,c^{5} d^{4} e^{2} g x -2763208 b \,c^{5} d^{3} e^{3} f x +1839103 c^{6} d^{5} e g x +3496703 c^{6} d^{4} e^{2} f x +3072 b^{6} e^{6} g -42752 b^{5} c d \,e^{5} g -4864 b^{5} c \,e^{6} f +250368 b^{4} c^{2} d^{2} e^{4} g +65664 b^{4} c^{2} d \,e^{5} f -790432 b^{3} c^{3} d^{3} e^{3} g -369056 b^{3} c^{3} d^{2} e^{4} f +1418488 b^{2} c^{4} d^{4} e^{2} g +1097744 b^{2} c^{4} d^{3} e^{3} f -1364202 b \,c^{5} d^{5} e g -1788546 b \,c^{5} d^{4} e^{2} f +525458 c^{6} d^{6} g +1414759 f \,d^{5} c^{6} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{2909907 c^{7} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(739\) |
2/2909907/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^3 *(153153*c^6*e^6*g*x^6-108108*b*c^5*e^6*g*x^5+963963*c^6*d*e^5*g*x^5+17117 1*c^6*e^6*f*x^5+72072*b^2*c^4*e^6*g*x^4-714714*b*c^5*d*e^5*g*x^4-114114*b* c^5*e^6*f*x^4+2582580*c^6*d^2*e^4*g*x^4+1084083*c^6*d*e^5*f*x^4-44352*b^3* c^3*e^6*g*x^3+484176*b^2*c^4*d*e^5*g*x^3+70224*b^2*c^4*e^6*f*x^3-2029104*b *c^5*d^2*e^4*g*x^3-737352*b*c^5*d*e^5*f*x^3+3827670*c^6*d^3*e^3*g*x^3+2905 518*c^6*d^2*e^4*f*x^3+24192*b^4*c^2*e^6*g*x^2-288288*b^3*c^3*d*e^5*g*x^2-3 8304*b^3*c^3*e^6*f*x^2+1370880*b^2*c^4*d^2*e^4*g*x^2+440496*b^2*c^4*d*e^5* f*x^2-3194604*b*c^5*d^3*e^3*g*x^2-1987020*b*c^5*d^2*e^4*f*x^2+3410505*c^6* d^4*e^2*g*x^2+4230198*c^6*d^3*e^3*f*x^2-10752*b^5*c*e^6*g*x+138880*b^4*c^2 *d*e^5*g*x+17024*b^4*c^2*e^6*f*x-737408*b^3*c^3*d^2*e^4*g*x-212800*b^3*c^3 *d*e^5*f*x+2029104*b^2*c^4*d^3*e^3*g*x+1078896*b^2*c^4*d^2*e^4*f*x-2935604 *b*c^5*d^4*e^2*g*x-2763208*b*c^5*d^3*e^3*f*x+1839103*c^6*d^5*e*g*x+3496703 *c^6*d^4*e^2*f*x+3072*b^6*e^6*g-42752*b^5*c*d*e^5*g-4864*b^5*c*e^6*f+25036 8*b^4*c^2*d^2*e^4*g+65664*b^4*c^2*d*e^5*f-790432*b^3*c^3*d^3*e^3*g-369056* b^3*c^3*d^2*e^4*f+1418488*b^2*c^4*d^4*e^2*g+1097744*b^2*c^4*d^3*e^3*f-1364 202*b*c^5*d^5*e*g-1788546*b*c^5*d^4*e^2*f+525458*c^6*d^6*g+1414759*c^6*d^5 *e*f)/c^7/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (459) = 918\).
Time = 0.47 (sec) , antiderivative size = 1370, normalized size of antiderivative = 2.73 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/2909907*(153153*c^9*e^9*g*x^9 + 9009*(19*c^9*e^9*f + (56*c^9*d*e^8 + 39* b*c^8*e^9)*g)*x^8 + 3003*(19*(10*c^9*d*e^8 + 7*b*c^8*e^9)*f + (50*c^9*d^2* e^7 + 527*b*c^8*d*e^8 + 69*b^2*c^7*e^9)*g)*x^7 + 231*(19*(38*c^9*d^2*e^7 + 417*b*c^8*d*e^8 + 55*b^2*c^7*e^9)*f - (5114*c^9*d^3*e^6 - 9585*b*c^8*d^2* e^7 - 5216*b^2*c^7*d*e^8 - 3*b^3*c^6*e^9)*g)*x^6 - 63*(19*(1174*c^9*d^3*e^ 6 - 2179*b*c^8*d^2*e^7 - 1204*b^2*c^7*d*e^8 - b^3*c^6*e^9)*f + (20456*c^9* d^4*e^5 + 4189*b*c^8*d^3*e^6 - 45509*b^2*c^7*d^2*e^7 - 143*b^3*c^6*d*e^8 + 12*b^4*c^5*e^9)*g)*x^5 - 7*(95*(2348*c^9*d^4*e^5 + 587*b*c^8*d^3*e^6 - 53 43*b^2*c^7*d^2*e^7 - 25*b^3*c^6*d*e^8 + 2*b^4*c^5*e^9)*f - (72574*c^9*d^5* e^4 - 530165*b*c^8*d^4*e^5 + 496980*b^2*c^7*d^3*e^6 + 8230*b^3*c^6*d^2*e^7 - 1550*b^4*c^5*d*e^8 + 120*b^5*c^4*e^9)*g)*x^4 + (19*(37354*c^9*d^5*e^4 - 257745*b*c^8*d^4*e^5 + 237200*b^2*c^7*d^3*e^6 + 6070*b^3*c^6*d^2*e^7 - 10 80*b^4*c^5*d*e^8 + 80*b^5*c^4*e^9)*f + (1411994*c^9*d^6*e^3 - 3574809*b*c^ 8*d^5*e^4 + 1981645*b^2*c^7*d^4*e^5 + 247010*b^3*c^6*d^3*e^6 - 78240*b^4*c ^5*d^2*e^7 + 13360*b^5*c^4*d*e^8 - 960*b^6*c^3*e^9)*g)*x^3 + 3*(19*(35362* c^9*d^6*e^3 - 87409*b*c^8*d^5*e^4 + 44825*b^2*c^7*d^4*e^5 + 9650*b^3*c^6*d ^3*e^6 - 2860*b^4*c^5*d^2*e^7 + 464*b^5*c^4*d*e^8 - 32*b^6*c^3*e^9)*f + (1 76810*c^9*d^7*e^2 - 248777*b*c^8*d^6*e^3 - 105344*b^2*c^7*d^5*e^4 + 276115 *b^3*c^6*d^4*e^5 - 130100*b^4*c^5*d^3*e^6 + 36640*b^5*c^4*d^2*e^7 - 5728*b ^6*c^3*d*e^8 + 384*b^7*c^2*e^9)*g)*x^2 - 19*(74461*c^9*d^8*e - 317517*b...
Timed out. \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1364 vs. \(2 (459) = 918\).
Time = 0.36 (sec) , antiderivative size = 1364, normalized size of antiderivative = 2.72 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/153153*(9009*c^8*e^8*x^8 - 74461*c^8*d^8 + 317517*b*c^7*d^7*e - 563561*b ^2*c^6*d^6*e^2 + 549615*b^3*c^5*d^5*e^3 - 329190*b^4*c^4*d^4*e^4 + 126672* b^5*c^3*d^3*e^5 - 30560*b^6*c^2*d^2*e^6 + 4224*b^7*c*d*e^7 - 256*b^8*e^8 + 3003*(10*c^8*d*e^7 + 7*b*c^7*e^8)*x^7 + 231*(38*c^8*d^2*e^6 + 417*b*c^7*d *e^7 + 55*b^2*c^6*e^8)*x^6 - 63*(1174*c^8*d^3*e^5 - 2179*b*c^7*d^2*e^6 - 1 204*b^2*c^6*d*e^7 - b^3*c^5*e^8)*x^5 - 35*(2348*c^8*d^4*e^4 + 587*b*c^7*d^ 3*e^5 - 5343*b^2*c^6*d^2*e^6 - 25*b^3*c^5*d*e^7 + 2*b^4*c^4*e^8)*x^4 + (37 354*c^8*d^5*e^3 - 257745*b*c^7*d^4*e^4 + 237200*b^2*c^6*d^3*e^5 + 6070*b^3 *c^5*d^2*e^6 - 1080*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*x^3 + 3*(35362*c^8*d^6 *e^2 - 87409*b*c^7*d^5*e^3 + 44825*b^2*c^6*d^4*e^4 + 9650*b^3*c^5*d^3*e^5 - 2860*b^4*c^4*d^2*e^6 + 464*b^5*c^3*d*e^7 - 32*b^6*c^2*e^8)*x^2 + (39346* c^8*d^7*e - 31625*b*c^7*d^6*e^2 - 83676*b^2*c^6*d^5*e^3 + 114555*b^3*c^5*d ^4*e^4 - 50040*b^4*c^4*d^3*e^5 + 13296*b^5*c^3*d^2*e^6 - 1984*b^6*c^2*d*e^ 7 + 128*b^7*c*e^8)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^6*e^2*x + c^ 6*d*e) + 2/2909907*(153153*c^9*e^9*x^9 - 525458*c^9*d^9 + 2940576*b*c^8*d^ 8*e - 7087468*b^2*c^7*d^7*e^2 + 9663960*b^3*c^6*d^6*e^3 - 8241330*b^4*c^5* d^5*e^4 + 4583640*b^5*c^4*d^4*e^5 - 1672864*b^6*c^3*d^3*e^6 + 387840*b^7*c ^2*d^2*e^7 - 51968*b^8*c*d*e^8 + 3072*b^9*e^9 + 9009*(56*c^9*d*e^8 + 39*b* c^8*e^9)*x^8 + 3003*(50*c^9*d^2*e^7 + 527*b*c^8*d*e^8 + 69*b^2*c^7*e^9)*x^ 7 - 231*(5114*c^9*d^3*e^6 - 9585*b*c^8*d^2*e^7 - 5216*b^2*c^7*d*e^8 - 3...
Leaf count of result is larger than twice the leaf count of optimal. 29450 vs. \(2 (459) = 918\).
Time = 1.17 (sec) , antiderivative size = 29450, normalized size of antiderivative = 58.78 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
-2/14549535*(4849845*c^2*d^7*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2* sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 9699690*b*c*d^6*e*f*(( -(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c* d - b*e)*b*e)/c) + 4849845*b^2*d^5*e^2*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/ 2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 138567*c^2*d ^5*e^2*f*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^ 2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2 *c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15 *((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2) ) + 1385670*b*c*d^4*e^3*f*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e) *b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70* (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b* e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2* c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c* d - b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) - 1385670*b^2*d^3*e^4*f*((22*sqrt(2*c*d - b*e)*c^3*d...
Time = 14.16 (sec) , antiderivative size = 1307, normalized size of antiderivative = 2.61 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^4\,x^7\,\sqrt {d+e\,x}\,\left (69\,g\,b^2\,e^2+527\,g\,b\,c\,d\,e+133\,f\,b\,c\,e^2+50\,g\,c^2\,d^2+190\,f\,c^2\,d\,e\right )}{969}+\frac {x^5\,\sqrt {d+e\,x}\,\left (-1512\,g\,b^4\,c^5\,e^9+18018\,g\,b^3\,c^6\,d\,e^8+2394\,f\,b^3\,c^6\,e^9+5734134\,g\,b^2\,c^7\,d^2\,e^7+2882376\,f\,b^2\,c^7\,d\,e^8-527814\,g\,b\,c^8\,d^3\,e^6+5216526\,f\,b\,c^8\,d^2\,e^7-2577456\,g\,c^9\,d^4\,e^5-2810556\,f\,c^9\,d^3\,e^6\right )}{2909907\,c^7\,e^3}+\frac {2\,c^2\,e^6\,g\,x^9\,\sqrt {d+e\,x}}{19}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-1920\,g\,b^6\,c^3\,e^9+26720\,g\,b^5\,c^4\,d\,e^8+3040\,f\,b^5\,c^4\,e^9-156480\,g\,b^4\,c^5\,d^2\,e^7-41040\,f\,b^4\,c^5\,d\,e^8+494020\,g\,b^3\,c^6\,d^3\,e^6+230660\,f\,b^3\,c^6\,d^2\,e^7+3963290\,g\,b^2\,c^7\,d^4\,e^5+9013600\,f\,b^2\,c^7\,d^3\,e^6-7149618\,g\,b\,c^8\,d^5\,e^4-9794310\,f\,b\,c^8\,d^4\,e^5+2823988\,g\,c^9\,d^6\,e^3+1419452\,f\,c^9\,d^5\,e^4\right )}{2909907\,c^7\,e^3}+\frac {x^6\,\sqrt {d+e\,x}\,\left (1386\,g\,b^3\,c^6\,e^9+2409792\,g\,b^2\,c^7\,d\,e^8+482790\,f\,b^2\,c^7\,e^9+4428270\,g\,b\,c^8\,d^2\,e^7+3660426\,f\,b\,c^8\,d\,e^8-2362668\,g\,c^9\,d^3\,e^6+333564\,f\,c^9\,d^2\,e^7\right )}{2909907\,c^7\,e^3}+\frac {2\,c\,e^5\,x^8\,\sqrt {d+e\,x}\,\left (39\,b\,e\,g+56\,c\,d\,g+19\,c\,e\,f\right )}{323}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (3072\,g\,b^6\,e^6-42752\,g\,b^5\,c\,d\,e^5-4864\,f\,b^5\,c\,e^6+250368\,g\,b^4\,c^2\,d^2\,e^4+65664\,f\,b^4\,c^2\,d\,e^5-790432\,g\,b^3\,c^3\,d^3\,e^3-369056\,f\,b^3\,c^3\,d^2\,e^4+1418488\,g\,b^2\,c^4\,d^4\,e^2+1097744\,f\,b^2\,c^4\,d^3\,e^3-1364202\,g\,b\,c^5\,d^5\,e-1788546\,f\,b\,c^5\,d^4\,e^2+525458\,g\,c^6\,d^6+1414759\,f\,c^6\,d^5\,e\right )}{2909907\,c^7\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (1680\,g\,b^5\,c^4\,e^9-21700\,g\,b^4\,c^5\,d\,e^8-2660\,f\,b^4\,c^5\,e^9+115220\,g\,b^3\,c^6\,d^2\,e^7+33250\,f\,b^3\,c^6\,d\,e^8+6957720\,g\,b^2\,c^7\,d^3\,e^6+7106190\,f\,b^2\,c^7\,d^2\,e^7-7422310\,g\,b\,c^8\,d^4\,e^5-780710\,f\,b\,c^8\,d^3\,e^6+1016036\,g\,c^9\,d^5\,e^4-3122840\,f\,c^9\,d^4\,e^5\right )}{2909907\,c^7\,e^3}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (384\,g\,b^6\,e^6-5344\,g\,b^5\,c\,d\,e^5-608\,f\,b^5\,c\,e^6+31296\,g\,b^4\,c^2\,d^2\,e^4+8208\,f\,b^4\,c^2\,d\,e^5-98804\,g\,b^3\,c^3\,d^3\,e^3-46132\,f\,b^3\,c^3\,d^2\,e^4+177311\,g\,b^2\,c^4\,d^4\,e^2+137218\,f\,b^2\,c^4\,d^3\,e^3+71967\,g\,b\,c^5\,d^5\,e+988893\,f\,b\,c^5\,d^4\,e^2-176810\,g\,c^6\,d^6-671878\,f\,c^6\,d^5\,e\right )}{969969\,c^5\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-1536\,g\,b^6\,e^6+21376\,g\,b^5\,c\,d\,e^5+2432\,f\,b^5\,c\,e^6-125184\,g\,b^4\,c^2\,d^2\,e^4-32832\,f\,b^4\,c^2\,d\,e^5+395216\,g\,b^3\,c^3\,d^3\,e^3+184528\,f\,b^3\,c^3\,d^2\,e^4-709244\,g\,b^2\,c^4\,d^4\,e^2-548872\,f\,b^2\,c^4\,d^3\,e^3+682101\,g\,b\,c^5\,d^5\,e+894273\,f\,b\,c^5\,d^4\,e^2-262729\,g\,c^6\,d^6+747574\,f\,c^6\,d^5\,e\right )}{2909907\,c^6\,e^2}\right )}{x+\frac {d}{e}} \]
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^4*x^7*(d + e*x)^(1/2)*( 69*b^2*e^2*g + 50*c^2*d^2*g + 133*b*c*e^2*f + 190*c^2*d*e*f + 527*b*c*d*e* g))/969 + (x^5*(d + e*x)^(1/2)*(2394*b^3*c^6*e^9*f - 1512*b^4*c^5*e^9*g - 2810556*c^9*d^3*e^6*f - 2577456*c^9*d^4*e^5*g + 5216526*b*c^8*d^2*e^7*f + 2882376*b^2*c^7*d*e^8*f - 527814*b*c^8*d^3*e^6*g + 18018*b^3*c^6*d*e^8*g + 5734134*b^2*c^7*d^2*e^7*g))/(2909907*c^7*e^3) + (2*c^2*e^6*g*x^9*(d + e*x )^(1/2))/19 + (x^3*(d + e*x)^(1/2)*(3040*b^5*c^4*e^9*f - 1920*b^6*c^3*e^9* g + 1419452*c^9*d^5*e^4*f + 2823988*c^9*d^6*e^3*g - 9794310*b*c^8*d^4*e^5* f - 41040*b^4*c^5*d*e^8*f - 7149618*b*c^8*d^5*e^4*g + 26720*b^5*c^4*d*e^8* g + 9013600*b^2*c^7*d^3*e^6*f + 230660*b^3*c^6*d^2*e^7*f + 3963290*b^2*c^7 *d^4*e^5*g + 494020*b^3*c^6*d^3*e^6*g - 156480*b^4*c^5*d^2*e^7*g))/(290990 7*c^7*e^3) + (x^6*(d + e*x)^(1/2)*(482790*b^2*c^7*e^9*f + 1386*b^3*c^6*e^9 *g + 333564*c^9*d^2*e^7*f - 2362668*c^9*d^3*e^6*g + 3660426*b*c^8*d*e^8*f + 4428270*b*c^8*d^2*e^7*g + 2409792*b^2*c^7*d*e^8*g))/(2909907*c^7*e^3) + (2*c*e^5*x^8*(d + e*x)^(1/2)*(39*b*e*g + 56*c*d*g + 19*c*e*f))/323 + (2*(b *e - c*d)^3*(d + e*x)^(1/2)*(3072*b^6*e^6*g + 525458*c^6*d^6*g - 4864*b^5* c*e^6*f + 1414759*c^6*d^5*e*f - 1364202*b*c^5*d^5*e*g - 42752*b^5*c*d*e^5* g - 1788546*b*c^5*d^4*e^2*f + 65664*b^4*c^2*d*e^5*f + 1097744*b^2*c^4*d^3* e^3*f - 369056*b^3*c^3*d^2*e^4*f + 1418488*b^2*c^4*d^4*e^2*g - 790432*b^3* c^3*d^3*e^3*g + 250368*b^4*c^2*d^2*e^4*g))/(2909907*c^7*e^3) + (x^4*(d ...