Integrand size = 46, antiderivative size = 424 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {256 (2 c d-b e)^4 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{765765 c^6 e^2 (d+e x)^{7/2}}-\frac {128 (2 c d-b e)^3 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{109395 c^5 e^2 (d+e x)^{5/2}}-\frac {32 (2 c d-b e)^2 (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{12155 c^4 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e) (17 c e f+3 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3315 c^3 e^2 \sqrt {d+e x}}-\frac {2 (17 c e f+3 c d g-10 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{255 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2} \]
-256/765765*(-b*e+2*c*d)^4*(-10*b*e*g+3*c*d*g+17*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)^(7/2)-128/109395*(-b*e+2*c*d)^3*(-10* b*e*g+3*c*d*g+17*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)^(5/2)-32/12155*(-b*e+2*c*d)^2*(-10*b*e*g+3*c*d*g+17*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)^(3/2)-2/17*g*(e*x+d)^(3/2)*(d *(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c/e^2-16/3315*(-b*e+2*c*d)*(-10*b*e*g +3*c*d*g+17*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^3/e^2/(e*x+d)^ (1/2)-2/255*(-10*b*e*g+3*c*d*g+17*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^2/e^2
Time = 0.37 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (17 e f+118 d g+35 e g x)-32 b^3 c^2 e^3 \left (2253 d^2 g+7 e^2 x (34 f+45 g x)+2 d e (391 f+756 g x)\right )+16 b^2 c^3 e^2 \left (10864 d^3 g+294 d e^2 x (17 f+21 g x)+21 e^3 x^2 (51 f+55 g x)+3 d^2 e (2397 f+4249 g x)\right )-2 b c^4 e \left (104843 d^4 g+231 e^4 x^3 (68 f+65 g x)+84 d e^3 x^2 (969 f+968 g x)+42 d^2 e^2 x (3842 f+4287 g x)+4 d^3 e (32623 f+50554 g x)\right )+c^5 \left (94134 d^5 g+3003 e^5 x^4 (17 f+15 g x)+462 d e^4 x^3 (578 f+507 g x)+126 d^2 e^3 x^2 (4471 f+3949 g x)+28 d^3 e^2 x (21097 f+19638 g x)+d^4 e (278171 f+329469 g x)\right )\right )}{765765 c^6 e^2 \sqrt {d+e x}} \]
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-1280* b^5*e^5*g + 128*b^4*c*e^4*(17*e*f + 118*d*g + 35*e*g*x) - 32*b^3*c^2*e^3*( 2253*d^2*g + 7*e^2*x*(34*f + 45*g*x) + 2*d*e*(391*f + 756*g*x)) + 16*b^2*c ^3*e^2*(10864*d^3*g + 294*d*e^2*x*(17*f + 21*g*x) + 21*e^3*x^2*(51*f + 55* g*x) + 3*d^2*e*(2397*f + 4249*g*x)) - 2*b*c^4*e*(104843*d^4*g + 231*e^4*x^ 3*(68*f + 65*g*x) + 84*d*e^3*x^2*(969*f + 968*g*x) + 42*d^2*e^2*x*(3842*f + 4287*g*x) + 4*d^3*e*(32623*f + 50554*g*x)) + c^5*(94134*d^5*g + 3003*e^5 *x^4*(17*f + 15*g*x) + 462*d*e^4*x^3*(578*f + 507*g*x) + 126*d^2*e^3*x^2*( 4471*f + 3949*g*x) + 28*d^3*e^2*x*(21097*f + 19638*g*x) + d^4*e*(278171*f + 329469*g*x))))/(765765*c^6*e^2*Sqrt[d + e*x])
Time = 0.67 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1221, 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.
| \(\displaystyle \int (d+e x)^{3/2} (f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \int (d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{\sqrt {d+e x}}dx}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right ) (-10 b e g+3 c d g+17 c e f)}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\) |
(-2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(17*c*e ^2) + ((17*c*e*f + 3*c*d*g - 10*b*e*g)*((-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))/(11*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*e )
3.23.51.3.1 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 = 0.36 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (x c e +b e -c d \right )^{3} \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 d^{4} f \,c^{5} e \right )}{765765 \sqrt {e x +d}\, c^{6} e^{2}}\) | \(529\) |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 d^{4} f \,c^{5} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{765765 c^{6} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(535\) |
-2/765765/(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 *(-45045*c^5*e^5*g*x^5+30030*b*c^4*e^5*g*x^4-234234*c^5*d*e^4*g*x^4-51051* c^5*e^5*f*x^4-18480*b^2*c^3*e^5*g*x^3+162624*b*c^4*d*e^4*g*x^3+31416*b*c^4 *e^5*f*x^3-497574*c^5*d^2*e^3*g*x^3-267036*c^5*d*e^4*f*x^3+10080*b^3*c^2*e ^5*g*x^2-98784*b^2*c^3*d*e^4*g*x^2-17136*b^2*c^3*e^5*f*x^2+360108*b*c^4*d^ 2*e^3*g*x^2+162792*b*c^4*d*e^4*f*x^2-549864*c^5*d^3*e^2*g*x^2-563346*c^5*d ^2*e^3*f*x^2-4480*b^4*c*e^5*g*x+48384*b^3*c^2*d*e^4*g*x+7616*b^3*c^2*e^5*f *x-203952*b^2*c^3*d^2*e^3*g*x-79968*b^2*c^3*d*e^4*f*x+404432*b*c^4*d^3*e^2 *g*x+322728*b*c^4*d^2*e^3*f*x-329469*c^5*d^4*e*g*x-590716*c^5*d^3*e^2*f*x+ 1280*b^5*e^5*g-15104*b^4*c*d*e^4*g-2176*b^4*c*e^5*f+72096*b^3*c^2*d^2*e^3* g+25024*b^3*c^2*d*e^4*f-173824*b^2*c^3*d^3*e^2*g-115056*b^2*c^3*d^2*e^3*f+ 209686*b*c^4*d^4*e*g+260984*b*c^4*d^3*e^2*f-94134*c^5*d^5*g-278171*c^5*d^4 *e*f)/c^6/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (388) = 776\).
Time = 0.39 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.62 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/765765*(45045*c^8*e^8*g*x^8 + 3003*(17*c^8*e^8*f + (33*c^8*d*e^7 + 35*b* c^7*e^8)*g)*x^7 + 231*(17*(29*c^8*d*e^7 + 31*b*c^7*e^8)*f - (303*c^8*d^2*e ^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*g)*x^6 - 63*(17*(79*c^8*d^2*e^6 - 398*b*c^7*d*e^7 - 71*b^2*c^6*e^8)*f + (4527*c^8*d^3*e^5 - 4129*b*c^7*d^2* e^6 - 4813*b^2*c^6*d*e^7 - 5*b^3*c^5*e^8)*g)*x^5 - 35*(17*(587*c^8*d^3*e^5 - 525*b*c^7*d^2*e^6 - 633*b^2*c^6*d*e^7 - b^3*c^5*e^8)*f + (1761*c^8*d^4* e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3*c^5*d*e^7 + 10 *b^4*c^4*e^8)*g)*x^4 - 5*(17*(835*c^8*d^4*e^4 + 6548*b*c^7*d^3*e^5 - 8586* b^2*c^6*d^2*e^6 - 92*b^3*c^5*d*e^7 + 8*b^4*c^4*e^8)*f - (51549*c^8*d^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238*b^2*c^6*d^3*e^5 + 4506*b^3*c^5*d^2*e^6 - 9 44*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*g)*x^3 + 3*(17*(7339*c^8*d^5*e^3 - 2043 5*b*c^7*d^4*e^4 + 12250*b^2*c^6*d^3*e^5 + 1030*b^3*c^5*d^2*e^6 - 200*b^4*c ^4*d*e^7 + 16*b^5*c^3*e^8)*f + (52047*c^8*d^6*e^2 - 89650*b*c^7*d^5*e^3 + 15875*b^2*c^6*d^4*e^4 + 30740*b^3*c^5*d^3*e^5 - 10900*b^4*c^4*d^2*e^6 + 20 48*b^5*c^3*d*e^7 - 160*b^6*c^2*e^8)*g)*x^2 - 17*(16363*c^8*d^7*e - 64441*b *c^7*d^6*e^2 + 101913*b^2*c^6*d^5*e^3 - 84195*b^3*c^5*d^4*e^4 + 40200*b^4* c^4*d^3*e^5 - 11568*b^5*c^3*d^2*e^6 + 1856*b^6*c^2*d*e^7 - 128*b^7*c*e^8)* f - 2*(47067*c^8*d^8 - 246044*b*c^7*d^7*e + 542642*b^2*c^6*d^6*e^2 - 65838 0*b^3*c^5*d^5*e^3 + 481275*b^4*c^4*d^4*e^4 - 218352*b^5*c^3*d^3*e^5 + 6062 4*b^6*c^2*d^2*e^6 - 9472*b^7*c*d*e^7 + 640*b^8*e^8)*g + (17*(14341*c^8*...
\[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (388) = 776\).
Time = 0.32 (sec) , antiderivative size = 1108, normalized size of antiderivative = 2.61 \[ \int (d+e x)^{3/2} (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 \, {\left (3003 \, c^{7} e^{7} x^{7} - 16363 \, c^{7} d^{7} + 64441 \, b c^{6} d^{6} e - 101913 \, b^{2} c^{5} d^{5} e^{2} + 84195 \, b^{3} c^{4} d^{4} e^{3} - 40200 \, b^{4} c^{3} d^{3} e^{4} + 11568 \, b^{5} c^{2} d^{2} e^{5} - 1856 \, b^{6} c d e^{6} + 128 \, b^{7} e^{7} + 231 \, {\left (29 \, c^{7} d e^{6} + 31 \, b c^{6} e^{7}\right )} x^{6} - 63 \, {\left (79 \, c^{7} d^{2} e^{5} - 398 \, b c^{6} d e^{6} - 71 \, b^{2} c^{5} e^{7}\right )} x^{5} - 35 \, {\left (587 \, c^{7} d^{3} e^{4} - 525 \, b c^{6} d^{2} e^{5} - 633 \, b^{2} c^{5} d e^{6} - b^{3} c^{4} e^{7}\right )} x^{4} - 5 \, {\left (835 \, c^{7} d^{4} e^{3} + 6548 \, b c^{6} d^{3} e^{4} - 8586 \, b^{2} c^{5} d^{2} e^{5} - 92 \, b^{3} c^{4} d e^{6} + 8 \, b^{4} c^{3} e^{7}\right )} x^{3} + 3 \, {\left (7339 \, c^{7} d^{5} e^{2} - 20435 \, b c^{6} d^{4} e^{3} + 12250 \, b^{2} c^{5} d^{3} e^{4} + 1030 \, b^{3} c^{4} d^{2} e^{5} - 200 \, b^{4} c^{3} d e^{6} + 16 \, b^{5} c^{2} e^{7}\right )} x^{2} + {\left (14341 \, c^{7} d^{6} e - 21006 \, b c^{6} d^{5} e^{2} - 4395 \, b^{2} c^{5} d^{4} e^{3} + 15180 \, b^{3} c^{4} d^{3} e^{4} - 4920 \, b^{4} c^{3} d^{2} e^{5} + 864 \, b^{5} c^{2} d e^{6} - 64 \, b^{6} c e^{7}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{45045 \, {\left (c^{5} e^{2} x + c^{5} d e\right )}} + \frac {2 \, {\left (45045 \, c^{8} e^{8} x^{8} - 94134 \, c^{8} d^{8} + 492088 \, b c^{7} d^{7} e - 1085284 \, b^{2} c^{6} d^{6} e^{2} + 1316760 \, b^{3} c^{5} d^{5} e^{3} - 962550 \, b^{4} c^{4} d^{4} e^{4} + 436704 \, b^{5} c^{3} d^{3} e^{5} - 121248 \, b^{6} c^{2} d^{2} e^{6} + 18944 \, b^{7} c d e^{7} - 1280 \, b^{8} e^{8} + 3003 \, {\left (33 \, c^{8} d e^{7} + 35 \, b c^{7} e^{8}\right )} x^{7} - 231 \, {\left (303 \, c^{8} d^{2} e^{6} - 1558 \, b c^{7} d e^{7} - 275 \, b^{2} c^{6} e^{8}\right )} x^{6} - 63 \, {\left (4527 \, c^{8} d^{3} e^{5} - 4129 \, b c^{7} d^{2} e^{6} - 4813 \, b^{2} c^{6} d e^{7} - 5 \, b^{3} c^{5} e^{8}\right )} x^{5} - 35 \, {\left (1761 \, c^{8} d^{4} e^{4} + 11860 \, b c^{7} d^{3} e^{5} - 15954 \, b^{2} c^{6} d^{2} e^{6} - 108 \, b^{3} c^{5} d e^{7} + 10 \, b^{4} c^{4} e^{8}\right )} x^{4} + 5 \, {\left (51549 \, c^{8} d^{5} e^{3} - 146429 \, b c^{7} d^{4} e^{4} + 91238 \, b^{2} c^{6} d^{3} e^{5} + 4506 \, b^{3} c^{5} d^{2} e^{6} - 944 \, b^{4} c^{4} d e^{7} + 80 \, b^{5} c^{3} e^{8}\right )} x^{3} + 3 \, {\left (52047 \, c^{8} d^{6} e^{2} - 89650 \, b c^{7} d^{5} e^{3} + 15875 \, b^{2} c^{6} d^{4} e^{4} + 30740 \, b^{3} c^{5} d^{3} e^{5} - 10900 \, b^{4} c^{4} d^{2} e^{6} + 2048 \, b^{5} c^{3} d e^{7} - 160 \, b^{6} c^{2} e^{8}\right )} x^{2} - {\left (47067 \, c^{8} d^{7} e - 198977 \, b c^{7} d^{6} e^{2} + 343665 \, b^{2} c^{6} d^{5} e^{3} - 314715 \, b^{3} c^{5} d^{4} e^{4} + 166560 \, b^{4} c^{4} d^{3} e^{5} - 51792 \, b^{5} c^{3} d^{2} e^{6} + 8832 \, b^{6} c^{2} d e^{7} - 640 \, b^{7} c e^{8}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{765765 \, {\left (c^{6} e^{3} x + c^{6} d e^{2}\right )}} \]
2/45045*(3003*c^7*e^7*x^7 - 16363*c^7*d^7 + 64441*b*c^6*d^6*e - 101913*b^2 *c^5*d^5*e^2 + 84195*b^3*c^4*d^4*e^3 - 40200*b^4*c^3*d^3*e^4 + 11568*b^5*c ^2*d^2*e^5 - 1856*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(29*c^7*d*e^6 + 31*b*c^6 *e^7)*x^6 - 63*(79*c^7*d^2*e^5 - 398*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 3 5*(587*c^7*d^3*e^4 - 525*b*c^6*d^2*e^5 - 633*b^2*c^5*d*e^6 - b^3*c^4*e^7)* x^4 - 5*(835*c^7*d^4*e^3 + 6548*b*c^6*d^3*e^4 - 8586*b^2*c^5*d^2*e^5 - 92* b^3*c^4*d*e^6 + 8*b^4*c^3*e^7)*x^3 + 3*(7339*c^7*d^5*e^2 - 20435*b*c^6*d^4 *e^3 + 12250*b^2*c^5*d^3*e^4 + 1030*b^3*c^4*d^2*e^5 - 200*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*x^2 + (14341*c^7*d^6*e - 21006*b*c^6*d^5*e^2 - 4395*b^2*c^ 5*d^4*e^3 + 15180*b^3*c^4*d^3*e^4 - 4920*b^4*c^3*d^2*e^5 + 864*b^5*c^2*d*e ^6 - 64*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^ 5*d*e) + 2/765765*(45045*c^8*e^8*x^8 - 94134*c^8*d^8 + 492088*b*c^7*d^7*e - 1085284*b^2*c^6*d^6*e^2 + 1316760*b^3*c^5*d^5*e^3 - 962550*b^4*c^4*d^4*e ^4 + 436704*b^5*c^3*d^3*e^5 - 121248*b^6*c^2*d^2*e^6 + 18944*b^7*c*d*e^7 - 1280*b^8*e^8 + 3003*(33*c^8*d*e^7 + 35*b*c^7*e^8)*x^7 - 231*(303*c^8*d^2* e^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*x^6 - 63*(4527*c^8*d^3*e^5 - 412 9*b*c^7*d^2*e^6 - 4813*b^2*c^6*d*e^7 - 5*b^3*c^5*e^8)*x^5 - 35*(1761*c^8*d ^4*e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3*c^5*d*e^7 + 10*b^4*c^4*e^8)*x^4 + 5*(51549*c^8*d^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238 *b^2*c^6*d^3*e^5 + 4506*b^3*c^5*d^2*e^6 - 944*b^4*c^4*d*e^7 + 80*b^5*c^...
Leaf count of result is larger than twice the leaf count of optimal. 21262 vs. \(2 (388) = 776\).
Time = 0.95 (sec) , antiderivative size = 21262, normalized size of antiderivative = 50.15 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
-2/765765*(255255*c^2*d^6*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqr t(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 510510*b*c*d^5*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) + 255255*b^2*d^4*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) + 7293*c^2*d^4*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*s qrt(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) - (3 5*(-(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)) + 291 72*b*c*d^3*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)) - 43758*b^2*d^2*e^4*f*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqr...
Time = 13.39 (sec) , antiderivative size = 1023, normalized size of antiderivative = 2.41 \[ \int (d+e x)^{3/2} (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^3\,x^6\,\sqrt {d+e\,x}\,\left (275\,g\,b^2\,e^2+1558\,g\,b\,c\,d\,e+527\,f\,b\,c\,e^2-303\,g\,c^2\,d^2+493\,f\,c^2\,d\,e\right )}{3315}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (-1280\,g\,b^5\,e^5+15104\,g\,b^4\,c\,d\,e^4+2176\,f\,b^4\,c\,e^5-72096\,g\,b^3\,c^2\,d^2\,e^3-25024\,f\,b^3\,c^2\,d\,e^4+173824\,g\,b^2\,c^3\,d^3\,e^2+115056\,f\,b^2\,c^3\,d^2\,e^3-209686\,g\,b\,c^4\,d^4\,e-260984\,f\,b\,c^4\,d^3\,e^2+94134\,g\,c^5\,d^5+278171\,f\,c^5\,d^4\,e\right )}{765765\,c^6\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (-700\,g\,b^4\,c^4\,e^8+7560\,g\,b^3\,c^5\,d\,e^7+1190\,f\,b^3\,c^5\,e^8+1116780\,g\,b^2\,c^6\,d^2\,e^6+753270\,f\,b^2\,c^6\,d\,e^7-830200\,g\,b\,c^7\,d^3\,e^5+624750\,f\,b\,c^7\,d^2\,e^6-123270\,g\,c^8\,d^4\,e^4-698530\,f\,c^8\,d^3\,e^5\right )}{765765\,c^6\,e^3}+\frac {2\,c^2\,e^5\,g\,x^8\,\sqrt {d+e\,x}}{17}+\frac {x^5\,\sqrt {d+e\,x}\,\left (630\,g\,b^3\,c^5\,e^8+606438\,g\,b^2\,c^6\,d\,e^7+152082\,f\,b^2\,c^6\,e^8+520254\,g\,b\,c^7\,d^2\,e^6+852516\,f\,b\,c^7\,d\,e^7-570402\,g\,c^8\,d^3\,e^5-169218\,f\,c^8\,d^2\,e^6\right )}{765765\,c^6\,e^3}+\frac {2\,c\,e^4\,x^7\,\sqrt {d+e\,x}\,\left (35\,b\,e\,g+33\,c\,d\,g+17\,c\,e\,f\right )}{255}+\frac {x^3\,\sqrt {d+e\,x}\,\left (800\,g\,b^5\,c^3\,e^8-9440\,g\,b^4\,c^4\,d\,e^7-1360\,f\,b^4\,c^4\,e^8+45060\,g\,b^3\,c^5\,d^2\,e^6+15640\,f\,b^3\,c^5\,d\,e^7+912380\,g\,b^2\,c^6\,d^3\,e^5+1459620\,f\,b^2\,c^6\,d^2\,e^6-1464290\,g\,b\,c^7\,d^4\,e^4-1113160\,f\,b\,c^7\,d^3\,e^5+515490\,g\,c^8\,d^5\,e^3-141950\,f\,c^8\,d^4\,e^4\right )}{765765\,c^6\,e^3}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-160\,g\,b^5\,e^5+1888\,g\,b^4\,c\,d\,e^4+272\,f\,b^4\,c\,e^5-9012\,g\,b^3\,c^2\,d^2\,e^3-3128\,f\,b^3\,c^2\,d\,e^4+21728\,g\,b^2\,c^3\,d^3\,e^2+14382\,f\,b^2\,c^3\,d^2\,e^3+37603\,g\,b\,c^4\,d^4\,e+222632\,f\,b\,c^4\,d^3\,e^2-52047\,g\,c^5\,d^5-124763\,f\,c^5\,d^4\,e\right )}{255255\,c^4\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (640\,g\,b^5\,e^5-7552\,g\,b^4\,c\,d\,e^4-1088\,f\,b^4\,c\,e^5+36048\,g\,b^3\,c^2\,d^2\,e^3+12512\,f\,b^3\,c^2\,d\,e^4-86912\,g\,b^2\,c^3\,d^3\,e^2-57528\,f\,b^2\,c^3\,d^2\,e^3+104843\,g\,b\,c^4\,d^4\,e+130492\,f\,b\,c^4\,d^3\,e^2-47067\,g\,c^5\,d^5+243797\,f\,c^5\,d^4\,e\right )}{765765\,c^5\,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^3*x^6*(d + e*x)^(1/2)*( 275*b^2*e^2*g - 303*c^2*d^2*g + 527*b*c*e^2*f + 493*c^2*d*e*f + 1558*b*c*d *e*g))/3315 + (2*(b*e - c*d)^3*(d + e*x)^(1/2)*(94134*c^5*d^5*g - 1280*b^5 *e^5*g + 2176*b^4*c*e^5*f + 278171*c^5*d^4*e*f - 209686*b*c^4*d^4*e*g + 15 104*b^4*c*d*e^4*g - 260984*b*c^4*d^3*e^2*f - 25024*b^3*c^2*d*e^4*f + 11505 6*b^2*c^3*d^2*e^3*f + 173824*b^2*c^3*d^3*e^2*g - 72096*b^3*c^2*d^2*e^3*g)) /(765765*c^6*e^3) + (x^4*(d + e*x)^(1/2)*(1190*b^3*c^5*e^8*f - 700*b^4*c^4 *e^8*g - 698530*c^8*d^3*e^5*f - 123270*c^8*d^4*e^4*g + 624750*b*c^7*d^2*e^ 6*f + 753270*b^2*c^6*d*e^7*f - 830200*b*c^7*d^3*e^5*g + 7560*b^3*c^5*d*e^7 *g + 1116780*b^2*c^6*d^2*e^6*g))/(765765*c^6*e^3) + (2*c^2*e^5*g*x^8*(d + e*x)^(1/2))/17 + (x^5*(d + e*x)^(1/2)*(152082*b^2*c^6*e^8*f + 630*b^3*c^5* e^8*g - 169218*c^8*d^2*e^6*f - 570402*c^8*d^3*e^5*g + 852516*b*c^7*d*e^7*f + 520254*b*c^7*d^2*e^6*g + 606438*b^2*c^6*d*e^7*g))/(765765*c^6*e^3) + (2 *c*e^4*x^7*(d + e*x)^(1/2)*(35*b*e*g + 33*c*d*g + 17*c*e*f))/255 + (x^3*(d + e*x)^(1/2)*(800*b^5*c^3*e^8*g - 1360*b^4*c^4*e^8*f - 141950*c^8*d^4*e^4 *f + 515490*c^8*d^5*e^3*g - 1113160*b*c^7*d^3*e^5*f + 15640*b^3*c^5*d*e^7* f - 1464290*b*c^7*d^4*e^4*g - 9440*b^4*c^4*d*e^7*g + 1459620*b^2*c^6*d^2*e ^6*f + 912380*b^2*c^6*d^3*e^5*g + 45060*b^3*c^5*d^2*e^6*g))/(765765*c^6*e^ 3) + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(272*b^4*c*e^5*f - 52047*c^5*d^5*g - 160*b^5*e^5*g - 124763*c^5*d^4*e*f + 37603*b*c^4*d^4*e*g + 1888*b^4*...