Integrand size = 46, antiderivative size = 343 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {2 (2 c d-b e)^3 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^5 e^2 (d+e x)^{7/2}}+\frac {2 (2 c d-b e)^2 (3 c e f+5 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^5 e^2 (d+e x)^{9/2}}-\frac {6 (2 c d-b e) (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^5 e^2 (d+e x)^{11/2}}+\frac {2 (c e f+7 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}{13 c^5 e^2 (d+e x)^{13/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{15/2}}{15 c^5 e^2 (d+e x)^{15/2}} \] Output:
-2/7*(-b*e+2*c*d)^3*(-b*e*g+c*d*g+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)^(7/2)+2/9*(-b*e+2*c*d)^2*(-4*b*e*g+5*c*d*g+3*c*e*f)* (d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(9/2)/c^5/e^2/(e*x+d)^(9/2)-6/11*(-b*e+2* c*d)*(-2*b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(11/2)/c^5/ e^2/(e*x+d)^(11/2)+2/13*(-4*b*e*g+7*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e ^2*x^2)^(13/2)/c^5/e^2/(e*x+d)^(13/2)-2/15*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(15/2)/c^5/e^2/(e*x+d)^(15/2)
Time = 0.38 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.77 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (128 b^4 e^4 g-16 b^3 c e^3 (15 e f+77 d g+28 e g x)+24 b^2 c^2 e^2 \left (187 d^2 g+7 e^2 x (5 f+6 g x)+d e (95 f+161 g x)\right )-2 b c^3 e \left (3611 d^3 g+21 e^3 x^2 (45 f+44 g x)+21 d e^2 x (170 f+183 g x)+d^2 e (4065 f+5922 g x)\right )+c^4 \left (3838 d^4 g+231 e^4 x^3 (15 f+13 g x)+147 d^2 e^2 x (145 f+129 g x)+21 d e^3 x^2 (675 f+583 g x)+d^3 e (12525 f+13433 g x)\right )\right )}{45045 c^5 e^2 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5 /2),x]
Output:
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^ 4*e^4*g - 16*b^3*c*e^3*(15*e*f + 77*d*g + 28*e*g*x) + 24*b^2*c^2*e^2*(187* d^2*g + 7*e^2*x*(5*f + 6*g*x) + d*e*(95*f + 161*g*x)) - 2*b*c^3*e*(3611*d^ 3*g + 21*e^3*x^2*(45*f + 44*g*x) + 21*d*e^2*x*(170*f + 183*g*x) + d^2*e*(4 065*f + 5922*g*x)) + c^4*(3838*d^4*g + 231*e^4*x^3*(15*f + 13*g*x) + 147*d ^2*e^2*x*(145*f + 129*g*x) + 21*d*e^3*x^2*(675*f + 583*g*x) + d^3*e*(12525 *f + 13433*g*x))))/(45045*c^5*e^2*Sqrt[d + e*x])
Time = 1.00 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1221, 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.
| \(\displaystyle \int \sqrt {d+e x} (f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-8 b e g+c d g+15 c e f) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{15 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+c d g+15 c e f) \left (\frac {6 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{\sqrt {d+e x}}dx}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+c d g+15 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+c d g+15 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right ) (-8 b e g+c d g+15 c e f)}{15 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2}\) |
Input:
Int[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
Output:
(-2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(15*c*e^2 ) + ((15*c*e*f + c*d*g - 8*b*e*g)*((-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*e)
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 = 1.99 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{3} \left (3003 g \,e^{4} x^{4} c^{4}-1848 b \,c^{3} e^{4} g \,x^{3}+12243 c^{4} d \,e^{3} g \,x^{3}+3465 c^{4} e^{4} f \,x^{3}+1008 b^{2} c^{2} e^{4} g \,x^{2}-7686 b \,c^{3} d \,e^{3} g \,x^{2}-1890 b \,c^{3} e^{4} f \,x^{2}+18963 c^{4} d^{2} e^{2} g \,x^{2}+14175 c^{4} d \,e^{3} f \,x^{2}-448 b^{3} c \,e^{4} g x +3864 b^{2} c^{2} d \,e^{3} g x +840 b^{2} c^{2} e^{4} f x -11844 b \,c^{3} d^{2} e^{2} g x -7140 b \,c^{3} d \,e^{3} f x +13433 c^{4} d^{3} e g x +21315 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1232 b^{3} c d \,e^{3} g -240 b^{3} c \,e^{4} f +4488 b^{2} c^{2} d^{2} e^{2} g +2280 b^{2} c^{2} d \,e^{3} f -7222 b \,c^{3} d^{3} e g -8130 b \,c^{3} d^{2} e^{2} f +3838 c^{4} d^{4} g +12525 d^{3} f \,c^{4} e \right )}{45045 \sqrt {e x +d}\, c^{5} e^{2}}\) | \(361\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (3003 g \,e^{4} x^{4} c^{4}-1848 b \,c^{3} e^{4} g \,x^{3}+12243 c^{4} d \,e^{3} g \,x^{3}+3465 c^{4} e^{4} f \,x^{3}+1008 b^{2} c^{2} e^{4} g \,x^{2}-7686 b \,c^{3} d \,e^{3} g \,x^{2}-1890 b \,c^{3} e^{4} f \,x^{2}+18963 c^{4} d^{2} e^{2} g \,x^{2}+14175 c^{4} d \,e^{3} f \,x^{2}-448 b^{3} c \,e^{4} g x +3864 b^{2} c^{2} d \,e^{3} g x +840 b^{2} c^{2} e^{4} f x -11844 b \,c^{3} d^{2} e^{2} g x -7140 b \,c^{3} d \,e^{3} f x +13433 c^{4} d^{3} e g x +21315 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1232 b^{3} c d \,e^{3} g -240 b^{3} c \,e^{4} f +4488 b^{2} c^{2} d^{2} e^{2} g +2280 b^{2} c^{2} d \,e^{3} f -7222 b \,c^{3} d^{3} e g -8130 b \,c^{3} d^{2} e^{2} f +3838 c^{4} d^{4} g +12525 d^{3} f \,c^{4} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{45045 c^{5} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(367\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (3003 g \,e^{4} x^{4} c^{4}-1848 b \,c^{3} e^{4} g \,x^{3}+12243 c^{4} d \,e^{3} g \,x^{3}+3465 c^{4} e^{4} f \,x^{3}+1008 b^{2} c^{2} e^{4} g \,x^{2}-7686 b \,c^{3} d \,e^{3} g \,x^{2}-1890 b \,c^{3} e^{4} f \,x^{2}+18963 c^{4} d^{2} e^{2} g \,x^{2}+14175 c^{4} d \,e^{3} f \,x^{2}-448 b^{3} c \,e^{4} g x +3864 b^{2} c^{2} d \,e^{3} g x +840 b^{2} c^{2} e^{4} f x -11844 b \,c^{3} d^{2} e^{2} g x -7140 b \,c^{3} d \,e^{3} f x +13433 c^{4} d^{3} e g x +21315 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1232 b^{3} c d \,e^{3} g -240 b^{3} c \,e^{4} f +4488 b^{2} c^{2} d^{2} e^{2} g +2280 b^{2} c^{2} d \,e^{3} f -7222 b \,c^{3} d^{3} e g -8130 b \,c^{3} d^{2} e^{2} f +3838 c^{4} d^{4} g +12525 d^{3} f \,c^{4} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{45045 c^{5} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(367\) |
Input:
int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method= _RETURNVERBOSE)
Output:
2/45045/(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*( 3003*c^4*e^4*g*x^4-1848*b*c^3*e^4*g*x^3+12243*c^4*d*e^3*g*x^3+3465*c^4*e^4 *f*x^3+1008*b^2*c^2*e^4*g*x^2-7686*b*c^3*d*e^3*g*x^2-1890*b*c^3*e^4*f*x^2+ 18963*c^4*d^2*e^2*g*x^2+14175*c^4*d*e^3*f*x^2-448*b^3*c*e^4*g*x+3864*b^2*c ^2*d*e^3*g*x+840*b^2*c^2*e^4*f*x-11844*b*c^3*d^2*e^2*g*x-7140*b*c^3*d*e^3* f*x+13433*c^4*d^3*e*g*x+21315*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1232*b^3*c*d*e ^3*g-240*b^3*c*e^4*f+4488*b^2*c^2*d^2*e^2*g+2280*b^2*c^2*d*e^3*f-7222*b*c^ 3*d^3*e*g-8130*b*c^3*d^2*e^2*f+3838*c^4*d^4*g+12525*c^4*d^3*e*f)/c^5/e^2
Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (313) = 626\).
Time = 0.12 (sec) , antiderivative size = 881, normalized size of antiderivative = 2.57 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
Output:
2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + (14*c^7*d*e^6 + 31*b*c^6 *e^7)*g)*x^6 + 63*(15*(4*c^7*d*e^6 + 9*b*c^6*e^7)*f - (139*c^7*d^2*e^5 - 2 63*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*g)*x^5 - 35*(3*(103*c^7*d^2*e^5 - 193*b*c ^6*d*e^6 - 53*b^2*c^5*e^7)*f + (278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b ^2*c^5*d*e^6 - b^3*c^4*e^7)*g)*x^4 - 5*(3*(824*c^7*d^3*e^4 + 206*b*c^6*d^2 *e^5 - 1454*b^2*c^5*d*e^6 - 5*b^3*c^4*e^7)*f - (1637*c^7*d^4*e^3 - 5930*b* c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*g)* x^3 + 3*(15*(271*c^7*d^4*e^3 - 954*b*c^6*d^3*e^4 + 664*b^2*c^5*d^2*e^5 + 2 1*b^3*c^4*d*e^6 - 2*b^4*c^3*e^7)*f + (3274*c^7*d^5*e^2 - 6125*b*c^6*d^4*e^ 3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*b^ 5*c^2*e^7)*g)*x^2 - 15*(835*c^7*d^6*e - 3047*b*c^6*d^5*e^2 + 4283*b^2*c^5* d^4*e^3 - 2933*b^3*c^4*d^3*e^4 + 1046*b^4*c^3*d^2*e^5 - 200*b^5*c^2*d*e^6 + 16*b^6*c*e^7)*f - 2*(1919*c^7*d^7 - 9368*b*c^6*d^6*e + 18834*b^2*c^5*d^5 *e^2 - 20100*b^3*c^4*d^4*e^3 + 12255*b^4*c^3*d^3*e^4 - 4284*b^5*c^2*d^2*e^ 5 + 808*b^6*c*d*e^6 - 64*b^7*e^7)*g + (15*(1084*c^7*d^5*e^2 - 1897*b*c^6*d ^4*e^3 + 466*b^2*c^5*d^3*e^4 + 431*b^3*c^4*d^2*e^5 - 92*b^4*c^3*d*e^6 + 8* b^5*c^2*e^7)*f - (1919*c^7*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^5*d^4* e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*b^4*c^3*d^2*e^5 - 744*b^5*c^2*d*e^6 + 64 *b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d) /(c^5*e^3*x + c^5*d*e^2)
\[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \sqrt {d + e x} \left (f + g x\right )\, dx \] Input:
integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/ 2),x)
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*sqrt(d + e*x)*(f + g*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (313) = 626\).
Time = 0.12 (sec) , antiderivative size = 878, normalized size of antiderivative = 2.56 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")
Output:
2/3003*(231*c^6*e^6*x^6 - 835*c^6*d^6 + 3047*b*c^5*d^5*e - 4283*b^2*c^4*d^ 4*e^2 + 2933*b^3*c^3*d^3*e^3 - 1046*b^4*c^2*d^2*e^4 + 200*b^5*c*d*e^5 - 16 *b^6*e^6 + 63*(4*c^6*d*e^5 + 9*b*c^5*e^6)*x^5 - 7*(103*c^6*d^2*e^4 - 193*b *c^5*d*e^5 - 53*b^2*c^4*e^6)*x^4 - (824*c^6*d^3*e^3 + 206*b*c^5*d^2*e^4 - 1454*b^2*c^4*d*e^5 - 5*b^3*c^3*e^6)*x^3 + 3*(271*c^6*d^4*e^2 - 954*b*c^5*d ^3*e^3 + 664*b^2*c^4*d^2*e^4 + 21*b^3*c^3*d*e^5 - 2*b^4*c^2*e^6)*x^2 + (10 84*c^6*d^5*e - 1897*b*c^5*d^4*e^2 + 466*b^2*c^4*d^3*e^3 + 431*b^3*c^3*d^2* e^4 - 92*b^4*c^2*d*e^5 + 8*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d )*f/(c^4*e^2*x + c^4*d*e) + 2/45045*(3003*c^7*e^7*x^7 - 3838*c^7*d^7 + 187 36*b*c^6*d^6*e - 37668*b^2*c^5*d^5*e^2 + 40200*b^3*c^4*d^4*e^3 - 24510*b^4 *c^3*d^3*e^4 + 8568*b^5*c^2*d^2*e^5 - 1616*b^6*c*d*e^6 + 128*b^7*e^7 + 231 *(14*c^7*d*e^6 + 31*b*c^6*e^7)*x^6 - 63*(139*c^7*d^2*e^5 - 263*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^2* c^5*d*e^6 - b^3*c^4*e^7)*x^4 + 5*(1637*c^7*d^4*e^3 - 5930*b*c^6*d^3*e^4 + 4224*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*x^3 + 3*(3274*c^7 *d^5*e^2 - 6125*b*c^6*d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d*e^6 + 16*b^5*c^2*e^7)*x^2 - (1919*c^7*d^6*e - 7449*b*c^6* d^5*e^2 + 11385*b^2*c^5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*b^4*c^3*d^2* e^5 - 744*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)*g/(c^5*e^3*x + c^5*d*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 11336 vs. \(2 (313) = 626\).
Time = 0.43 (sec) , antiderivative size = 11336, normalized size of antiderivative = 33.05 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Output:
-2/45045*(45045*sqrt(-c*e*x + c*d - b*e)*c^3*d^6*e*f - 135135*sqrt(-c*e*x + c*d - b*e)*b*c^2*d^5*e^2*f + 135135*sqrt(-c*e*x + c*d - b*e)*b^2*c*d^4*e ^3*f - 45045*sqrt(-c*e*x + c*d - b*e)*b^3*d^3*e^4*f - 45045*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^ (3/2))*b*c*d^4*e^2*f + 90090*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e *x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b^2*d^3*e^3*f - 45045*(3 *sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b^3*d^2*e^4*f/c + 15015*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*c^2*d^6*g - 45045*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b*c*d^5*e*g + 45045*(3*sqrt(-c*e*x + c*d - b* e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b^2* d^4*e^2*g - 15015*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b^3*d^3*e^3*g/c - 9009*(15*sqrt(-c* e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(- c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e* x + c*d - b*e)^(3/2)*b*e + 3*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e ))*c*d^4*e*f + 18018*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e*x + c*d - b*e)^(3/2)*b*e + 3*(c*e*x - ...
Time = 12.23 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.24 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (71\,g\,b^2\,e^2+263\,g\,b\,c\,d\,e+135\,f\,b\,c\,e^2-139\,g\,c^2\,d^2+60\,f\,c^2\,d\,e\right )}{715}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,g\,b^4\,c^3\,e^7+770\,g\,b^3\,c^4\,d\,e^6+150\,f\,b^3\,c^4\,e^7+42240\,g\,b^2\,c^5\,d^2\,e^5+43620\,f\,b^2\,c^5\,d\,e^6-59300\,g\,b\,c^6\,d^3\,e^4-6180\,f\,b\,c^6\,d^2\,e^5+16370\,g\,c^7\,d^4\,e^3-24720\,f\,c^7\,d^3\,e^4\right )}{45045\,c^5\,e^3}+\frac {2\,c^2\,e^4\,g\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1232\,g\,b^3\,c\,d\,e^3-240\,f\,b^3\,c\,e^4+4488\,g\,b^2\,c^2\,d^2\,e^2+2280\,f\,b^2\,c^2\,d\,e^3-7222\,g\,b\,c^3\,d^3\,e-8130\,f\,b\,c^3\,d^2\,e^2+3838\,g\,c^4\,d^4+12525\,f\,c^4\,d^3\,e\right )}{45045\,c^5\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (70\,g\,b^3\,c^4\,e^7+33180\,g\,b^2\,c^5\,d\,e^6+11130\,f\,b^2\,c^5\,e^7-3780\,g\,b\,c^6\,d^2\,e^5+40530\,f\,b\,c^6\,d\,e^6-19460\,g\,c^7\,d^3\,e^4-21630\,f\,c^7\,d^2\,e^5\right )}{45045\,c^5\,e^3}+\frac {2\,c\,e^3\,x^6\,\sqrt {d+e\,x}\,\left (31\,b\,e\,g+14\,c\,d\,g+15\,c\,e\,f\right )}{195}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (16\,g\,b^4\,e^4-154\,g\,b^3\,c\,d\,e^3-30\,f\,b^3\,c\,e^4+561\,g\,b^2\,c^2\,d^2\,e^2+285\,f\,b^2\,c^2\,d\,e^3+2851\,g\,b\,c^3\,d^3\,e+10245\,f\,b\,c^3\,d^2\,e^2-3274\,g\,c^4\,d^4-4065\,f\,c^4\,d^3\,e\right )}{15015\,c^3\,e}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-64\,g\,b^4\,e^4+616\,g\,b^3\,c\,d\,e^3+120\,f\,b^3\,c\,e^4-2244\,g\,b^2\,c^2\,d^2\,e^2-1140\,f\,b^2\,c^2\,d\,e^3+3611\,g\,b\,c^3\,d^3\,e+4065\,f\,b\,c^3\,d^2\,e^2-1919\,g\,c^4\,d^4+16260\,f\,c^4\,d^3\,e\right )}{45045\,c^4\,e^2}\right )}{x+\frac {d}{e}} \] Input:
int((f + g*x)*(d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2), x)
Output:
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^2*x^5*(d + e*x)^(1/2)*( 71*b^2*e^2*g - 139*c^2*d^2*g + 135*b*c*e^2*f + 60*c^2*d*e*f + 263*b*c*d*e* g))/715 + (x^3*(d + e*x)^(1/2)*(150*b^3*c^4*e^7*f - 80*b^4*c^3*e^7*g - 247 20*c^7*d^3*e^4*f + 16370*c^7*d^4*e^3*g - 6180*b*c^6*d^2*e^5*f + 43620*b^2* c^5*d*e^6*f - 59300*b*c^6*d^3*e^4*g + 770*b^3*c^4*d*e^6*g + 42240*b^2*c^5* d^2*e^5*g))/(45045*c^5*e^3) + (2*c^2*e^4*g*x^7*(d + e*x)^(1/2))/15 + (2*(b *e - c*d)^3*(d + e*x)^(1/2)*(128*b^4*e^4*g + 3838*c^4*d^4*g - 240*b^3*c*e^ 4*f + 12525*c^4*d^3*e*f - 7222*b*c^3*d^3*e*g - 1232*b^3*c*d*e^3*g - 8130*b *c^3*d^2*e^2*f + 2280*b^2*c^2*d*e^3*f + 4488*b^2*c^2*d^2*e^2*g))/(45045*c^ 5*e^3) + (x^4*(d + e*x)^(1/2)*(11130*b^2*c^5*e^7*f + 70*b^3*c^4*e^7*g - 21 630*c^7*d^2*e^5*f - 19460*c^7*d^3*e^4*g + 40530*b*c^6*d*e^6*f - 3780*b*c^6 *d^2*e^5*g + 33180*b^2*c^5*d*e^6*g))/(45045*c^5*e^3) + (2*c*e^3*x^6*(d + e *x)^(1/2)*(31*b*e*g + 14*c*d*g + 15*c*e*f))/195 + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(16*b^4*e^4*g - 3274*c^4*d^4*g - 30*b^3*c*e^4*f - 4065*c^4*d^3* e*f + 2851*b*c^3*d^3*e*g - 154*b^3*c*d*e^3*g + 10245*b*c^3*d^2*e^2*f + 285 *b^2*c^2*d*e^3*f + 561*b^2*c^2*d^2*e^2*g))/(15015*c^3*e) + (2*x*(b*e - c*d )^2*(d + e*x)^(1/2)*(120*b^3*c*e^4*f - 1919*c^4*d^4*g - 64*b^4*e^4*g + 162 60*c^4*d^3*e*f + 3611*b*c^3*d^3*e*g + 616*b^3*c*d*e^3*g + 4065*b*c^3*d^2*e ^2*f - 1140*b^2*c^2*d*e^3*f - 2244*b^2*c^2*d^2*e^2*g))/(45045*c^4*e^2)))/( x + d/e)
Time = 0.26 (sec) , antiderivative size = 944, normalized size of antiderivative = 2.75 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(128*b**7*e**7*g - 1616*b**6*c*d*e**6*g - 24 0*b**6*c*e**7*f - 64*b**6*c*e**7*g*x + 8568*b**5*c**2*d**2*e**5*g + 3000*b **5*c**2*d*e**6*f + 744*b**5*c**2*d*e**6*g*x + 120*b**5*c**2*e**7*f*x + 48 *b**5*c**2*e**7*g*x**2 - 24510*b**4*c**3*d**3*e**4*g - 15690*b**4*c**3*d** 2*e**5*f - 3540*b**4*c**3*d**2*e**5*g*x - 1380*b**4*c**3*d*e**6*f*x - 510* b**4*c**3*d*e**6*g*x**2 - 90*b**4*c**3*e**7*f*x**2 - 40*b**4*c**3*e**7*g*x **3 + 40200*b**3*c**4*d**4*e**3*g + 43995*b**3*c**4*d**3*e**4*f + 8715*b** 3*c**4*d**3*e**4*g*x + 6465*b**3*c**4*d**2*e**5*f*x + 2145*b**3*c**4*d**2* e**5*g*x**2 + 945*b**3*c**4*d*e**6*f*x**2 + 385*b**3*c**4*d*e**6*g*x**3 + 75*b**3*c**4*e**7*f*x**3 + 35*b**3*c**4*e**7*g*x**4 - 37668*b**2*c**5*d**5 *e**2*g - 64245*b**2*c**5*d**4*e**3*f - 11385*b**2*c**5*d**4*e**3*g*x + 69 90*b**2*c**5*d**3*e**4*f*x + 6870*b**2*c**5*d**3*e**4*g*x**2 + 29880*b**2* c**5*d**2*e**5*f*x**2 + 21120*b**2*c**5*d**2*e**5*g*x**3 + 21810*b**2*c**5 *d*e**6*f*x**3 + 16590*b**2*c**5*d*e**6*g*x**4 + 5565*b**2*c**5*e**7*f*x** 4 + 4473*b**2*c**5*e**7*g*x**5 + 18736*b*c**6*d**6*e*g + 45705*b*c**6*d**5 *e**2*f + 7449*b*c**6*d**5*e**2*g*x - 28455*b*c**6*d**4*e**3*f*x - 18375*b *c**6*d**4*e**3*g*x**2 - 42930*b*c**6*d**3*e**4*f*x**2 - 29650*b*c**6*d**3 *e**4*g*x**3 - 3090*b*c**6*d**2*e**5*f*x**3 - 1890*b*c**6*d**2*e**5*g*x**4 + 20265*b*c**6*d*e**6*f*x**4 + 16569*b*c**6*d*e**6*g*x**5 + 8505*b*c**6*e **7*f*x**5 + 7161*b*c**6*e**7*g*x**6 - 3838*c**7*d**7*g - 12525*c**7*d*...