Integrand size = 46, antiderivative size = 267 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=-\frac {2 (2 c d-b e)^2 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^4 e^2 (d+e x)^{7/2}}+\frac {2 (2 c d-b e) (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^4 e^2 (d+e x)^{9/2}}-\frac {2 (c e f+5 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^4 e^2 (d+e x)^{11/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}{13 c^4 e^2 (d+e x)^{13/2}} \] Output:
-2/7*(-b*e+2*c*d)^2*(-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^4/e^2/(e*x+d)^(7/2)+2/9*(-b*e+2*c*d)*(-3*b*e*g+4*c*d*g+2*c*e*f)*(d *(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(9/2)/c^4/e^2/(e*x+d)^(9/2)-2/11*(-3*b*e*g+ 5*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(11/2)/c^4/e^2/(e*x+d)^(11 /2)+2/13*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(13/2)/c^4/e^2/(e*x+d)^(13/2)
Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.69 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-48 b^3 e^3 g+8 b^2 c e^2 (13 e f+44 d g+21 e g x)-2 b c^2 e \left (423 d^2 g+7 e^2 x (26 f+27 g x)+d e (390 f+532 g x)\right )+c^3 \left (542 d^3 g+63 e^3 x^2 (13 f+11 g x)+14 d e^2 x (169 f+144 g x)+d^2 e (1963 f+1897 g x)\right )\right )}{9009 c^4 e^2 \sqrt {d+e x}} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/Sqrt[d + e*x],x]
Output:
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^ 3*e^3*g + 8*b^2*c*e^2*(13*e*f + 44*d*g + 21*e*g*x) - 2*b*c^2*e*(423*d^2*g + 7*e^2*x*(26*f + 27*g*x) + d*e*(390*f + 532*g*x)) + c^3*(542*d^3*g + 63*e ^3*x^2*(13*f + 11*g*x) + 14*d*e^2*x*(169*f + 144*g*x) + d^2*e*(1963*f + 18 97*g*x))))/(9009*c^4*e^2*Sqrt[d + e*x])
Time = 0.84 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1221, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-6 b e g-c d g+13 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{\sqrt {d+e x}}dx}{13 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g-c d g+13 c e f) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g-c d g+13 c e f) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right ) (-6 b e g-c d g+13 c e f)}{13 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/Sqrt[d + e*x], x]
Output:
(-2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*c*e^2*Sqrt[d + e*x] ) + ((13*c*e*f - c*d*g - 6*b*e*g)*((-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*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 = 2.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{3} \left (-693 e^{3} g \,x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 d^{2} f \,c^{3} e \right )}{9009 \sqrt {e x +d}\, c^{4} e^{2}}\) | \(229\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-693 e^{3} g \,x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 d^{2} f \,c^{3} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 c^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(235\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-693 e^{3} g \,x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 d^{2} f \,c^{3} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 c^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(235\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x,method= _RETURNVERBOSE)
Output:
-2/9009*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(e*x+d)^(1/2)*(c*e*x+b*e-c*d)^3*( -693*c^3*e^3*g*x^3+378*b*c^2*e^3*g*x^2-2016*c^3*d*e^2*g*x^2-819*c^3*e^3*f* x^2-168*b^2*c*e^3*g*x+1064*b*c^2*d*e^2*g*x+364*b*c^2*e^3*f*x-1897*c^3*d^2* e*g*x-2366*c^3*d*e^2*f*x+48*b^3*e^3*g-352*b^2*c*d*e^2*g-104*b^2*c*e^3*f+84 6*b*c^2*d^2*e*g+780*b*c^2*d*e^2*f-542*c^3*d^3*g-1963*c^3*d^2*e*f)/c^4/e^2
Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (243) = 486\).
Time = 0.11 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.53 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (693 \, c^{6} e^{6} g x^{6} + 63 \, {\left (13 \, c^{6} e^{6} f - {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} g\right )} x^{5} - 7 \, {\left (13 \, {\left (c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} f + {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - {\left (13 \, {\left (206 \, c^{6} d^{2} e^{4} - 192 \, b c^{5} d e^{5} - 113 \, b^{2} c^{4} e^{6}\right )} f - {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} + 3 \, {\left (13 \, {\left (10 \, c^{6} d^{3} e^{3} - 118 \, b c^{5} d^{2} e^{4} + 107 \, b^{2} c^{4} d e^{5} + b^{3} c^{3} e^{6}\right )} f + {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} - 13 \, {\left (151 \, c^{6} d^{5} e - 513 \, b c^{5} d^{4} e^{2} + 641 \, b^{2} c^{4} d^{3} e^{3} - 355 \, b^{3} c^{3} d^{2} e^{4} + 84 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} f - 2 \, {\left (271 \, c^{6} d^{6} - 1236 \, b c^{5} d^{5} e + 2258 \, b^{2} c^{4} d^{4} e^{2} - 2092 \, b^{3} c^{3} d^{3} e^{3} + 1023 \, b^{4} c^{2} d^{2} e^{4} - 248 \, b^{5} c d e^{5} + 24 \, b^{6} e^{6}\right )} g + {\left (13 \, {\left (271 \, c^{6} d^{4} e^{2} - 512 \, b c^{5} d^{3} e^{3} + 207 \, b^{2} c^{4} d^{2} e^{4} + 38 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6}\right )} f - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{9009 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/9009*(693*c^6*e^6*g*x^6 + 63*(13*c^6*e^6*f - (c^6*d*e^5 - 27*b*c^5*e^6)* g)*x^5 - 7*(13*(c^6*d*e^5 - 23*b*c^5*e^6)*f + (296*c^6*d^2*e^4 - 280*b*c^5 *d*e^5 - 159*b^2*c^4*e^6)*g)*x^4 - (13*(206*c^6*d^2*e^4 - 192*b*c^5*d*e^5 - 113*b^2*c^4*e^6)*f - (206*c^6*d^3*e^3 - 3114*b*c^5*d^2*e^4 + 2893*b^2*c^ 4*d*e^5 + 15*b^3*c^3*e^6)*g)*x^3 + 3*(13*(10*c^6*d^3*e^3 - 118*b*c^5*d^2*e ^4 + 107*b^2*c^4*d*e^5 + b^3*c^3*e^6)*f + (683*c^6*d^4*e^2 - 1328*b*c^5*d^ 3*e^3 + 601*b^2*c^4*d^2*e^4 + 50*b^3*c^3*d*e^5 - 6*b^4*c^2*e^6)*g)*x^2 - 1 3*(151*c^6*d^5*e - 513*b*c^5*d^4*e^2 + 641*b^2*c^4*d^3*e^3 - 355*b^3*c^3*d ^2*e^4 + 84*b^4*c^2*d*e^5 - 8*b^5*c*e^6)*f - 2*(271*c^6*d^6 - 1236*b*c^5*d ^5*e + 2258*b^2*c^4*d^4*e^2 - 2092*b^3*c^3*d^3*e^3 + 1023*b^4*c^2*d^2*e^4 - 248*b^5*c*d*e^5 + 24*b^6*e^6)*g + (13*(271*c^6*d^4*e^2 - 512*b*c^5*d^3*e ^3 + 207*b^2*c^4*d^2*e^4 + 38*b^3*c^3*d*e^5 - 4*b^4*c^2*e^6)*f - (271*c^6* d^5*e - 965*b*c^5*d^4*e^2 + 1293*b^2*c^4*d^3*e^3 - 799*b^3*c^3*d^2*e^4 + 2 24*b^4*c^2*d*e^5 - 24*b^5*c*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^4*e^3*x + c^4*d*e^2)
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\sqrt {d + e x}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(1/ 2),x)
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/sqrt(d + e*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (243) = 486\).
Time = 0.07 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.39 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 151 \, c^{5} d^{5} + 513 \, b c^{4} d^{4} e - 641 \, b^{2} c^{3} d^{3} e^{2} + 355 \, b^{3} c^{2} d^{2} e^{3} - 84 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (206 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (10 \, c^{5} d^{3} e^{2} - 118 \, b c^{4} d^{2} e^{3} + 107 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} + {\left (271 \, c^{5} d^{4} e - 512 \, b c^{4} d^{3} e^{2} + 207 \, b^{2} c^{3} d^{2} e^{3} + 38 \, b^{3} c^{2} d e^{4} - 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{693 \, c^{3} e} + \frac {2 \, {\left (693 \, c^{6} e^{6} x^{6} - 542 \, c^{6} d^{6} + 2472 \, b c^{5} d^{5} e - 4516 \, b^{2} c^{4} d^{4} e^{2} + 4184 \, b^{3} c^{3} d^{3} e^{3} - 2046 \, b^{4} c^{2} d^{2} e^{4} + 496 \, b^{5} c d e^{5} - 48 \, b^{6} e^{6} - 63 \, {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} x^{5} - 7 \, {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} x^{4} + {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \, {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} x^{2} - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{9009 \, c^{4} e^{2}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/693*(63*c^5*e^5*x^5 - 151*c^5*d^5 + 513*b*c^4*d^4*e - 641*b^2*c^3*d^3*e^ 2 + 355*b^3*c^2*d^2*e^3 - 84*b^4*c*d*e^4 + 8*b^5*e^5 - 7*(c^5*d*e^4 - 23*b *c^4*e^5)*x^4 - (206*c^5*d^2*e^3 - 192*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^3 + 3*(10*c^5*d^3*e^2 - 118*b*c^4*d^2*e^3 + 107*b^2*c^3*d*e^4 + b^3*c^2*e^5) *x^2 + (271*c^5*d^4*e - 512*b*c^4*d^3*e^2 + 207*b^2*c^3*d^2*e^3 + 38*b^3*c ^2*d*e^4 - 4*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^3*e) + 2/9009*(69 3*c^6*e^6*x^6 - 542*c^6*d^6 + 2472*b*c^5*d^5*e - 4516*b^2*c^4*d^4*e^2 + 41 84*b^3*c^3*d^3*e^3 - 2046*b^4*c^2*d^2*e^4 + 496*b^5*c*d*e^5 - 48*b^6*e^6 - 63*(c^6*d*e^5 - 27*b*c^5*e^6)*x^5 - 7*(296*c^6*d^2*e^4 - 280*b*c^5*d*e^5 - 159*b^2*c^4*e^6)*x^4 + (206*c^6*d^3*e^3 - 3114*b*c^5*d^2*e^4 + 2893*b^2* c^4*d*e^5 + 15*b^3*c^3*e^6)*x^3 + 3*(683*c^6*d^4*e^2 - 1328*b*c^5*d^3*e^3 + 601*b^2*c^4*d^2*e^4 + 50*b^3*c^3*d*e^5 - 6*b^4*c^2*e^6)*x^2 - (271*c^6*d ^5*e - 965*b*c^5*d^4*e^2 + 1293*b^2*c^4*d^3*e^3 - 799*b^3*c^3*d^2*e^4 + 22 4*b^4*c^2*d*e^5 - 24*b^5*c*e^6)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^4*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 5116 vs. \(2 (243) = 486\).
Time = 0.46 (sec) , antiderivative size = 5116, normalized size of antiderivative = 19.16 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")
Output:
-2/45045*(15015*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d^4*e*f - 30030*(-(e* x + d)*c + 2*c*d - b*e)^(3/2)*b*d^3*e^2*f + 15015*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*d^2*e^3*f/c - 6006*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c* d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b* e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b*d^2*e^2*f/c + 6006*(5*(-(e*x + d) *c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3 *((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b^2*d*e^3 *f/c^2 + 3003*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d) *c + 2*c*d - b*e))*d^4*g - 6006*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e) ^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b*d^3*e*g/c + 3003*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e *x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b^2*d^2*e^2*g /c^2 - 858*(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))*d^2 *e*f/c + 858*(35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x...
Time = 11.70 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.84 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^4\,\left (159\,g\,b^2\,e^2+280\,g\,b\,c\,d\,e+299\,f\,b\,c\,e^2-296\,g\,c^2\,d^2-13\,f\,c^2\,d\,e\right )}{1287}+\frac {2\,c^2\,e^4\,g\,x^6}{13}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (-6\,g\,b^3\,e^3+44\,g\,b^2\,c\,d\,e^2+13\,f\,b^2\,c\,e^3+645\,g\,b\,c^2\,d^2\,e+1404\,f\,b\,c^2\,d\,e^2-683\,g\,c^3\,d^3-130\,f\,c^3\,d^2\,e\right )}{3003\,c^2}+\frac {x^3\,\left (30\,g\,b^3\,c^3\,e^6+5786\,g\,b^2\,c^4\,d\,e^5+2938\,f\,b^2\,c^4\,e^6-6228\,g\,b\,c^5\,d^2\,e^4+4992\,f\,b\,c^5\,d\,e^5+412\,g\,c^6\,d^3\,e^3-5356\,f\,c^6\,d^2\,e^4\right )}{9009\,c^4\,e^2}+\frac {2\,c\,e^3\,x^5\,\left (27\,b\,e\,g-c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (-48\,g\,b^3\,e^3+352\,g\,b^2\,c\,d\,e^2+104\,f\,b^2\,c\,e^3-846\,g\,b\,c^2\,d^2\,e-780\,f\,b\,c^2\,d\,e^2+542\,g\,c^3\,d^3+1963\,f\,c^3\,d^2\,e\right )}{9009\,c^4\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (24\,g\,b^3\,e^3-176\,g\,b^2\,c\,d\,e^2-52\,f\,b^2\,c\,e^3+423\,g\,b\,c^2\,d^2\,e+390\,f\,b\,c^2\,d\,e^2-271\,g\,c^3\,d^3+3523\,f\,c^3\,d^2\,e\right )}{9009\,c^3\,e}\right )}{\sqrt {d+e\,x}} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(1/2 ),x)
Output:
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^2*x^4*(159*b^2*e^2*g - 296*c^2*d^2*g + 299*b*c*e^2*f - 13*c^2*d*e*f + 280*b*c*d*e*g))/1287 + (2*c ^2*e^4*g*x^6)/13 + (2*x^2*(b*e - c*d)*(13*b^2*c*e^3*f - 683*c^3*d^3*g - 6* b^3*e^3*g - 130*c^3*d^2*e*f + 1404*b*c^2*d*e^2*f + 645*b*c^2*d^2*e*g + 44* b^2*c*d*e^2*g))/(3003*c^2) + (x^3*(2938*b^2*c^4*e^6*f + 30*b^3*c^3*e^6*g - 5356*c^6*d^2*e^4*f + 412*c^6*d^3*e^3*g + 4992*b*c^5*d*e^5*f - 6228*b*c^5* d^2*e^4*g + 5786*b^2*c^4*d*e^5*g))/(9009*c^4*e^2) + (2*c*e^3*x^5*(27*b*e*g - c*d*g + 13*c*e*f))/143 + (2*(b*e - c*d)^3*(542*c^3*d^3*g - 48*b^3*e^3*g + 104*b^2*c*e^3*f + 1963*c^3*d^2*e*f - 780*b*c^2*d*e^2*f - 846*b*c^2*d^2* e*g + 352*b^2*c*d*e^2*g))/(9009*c^4*e^2) + (2*x*(b*e - c*d)^2*(24*b^3*e^3* g - 271*c^3*d^3*g - 52*b^2*c*e^3*f + 3523*c^3*d^2*e*f + 390*b*c^2*d*e^2*f + 423*b*c^2*d^2*e*g - 176*b^2*c*d*e^2*g))/(9009*c^3*e)))/(d + e*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.64 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (693 c^{6} e^{6} g \,x^{6}+1701 b \,c^{5} e^{6} g \,x^{5}-63 c^{6} d \,e^{5} g \,x^{5}+819 c^{6} e^{6} f \,x^{5}+1113 b^{2} c^{4} e^{6} g \,x^{4}+1960 b \,c^{5} d \,e^{5} g \,x^{4}+2093 b \,c^{5} e^{6} f \,x^{4}-2072 c^{6} d^{2} e^{4} g \,x^{4}-91 c^{6} d \,e^{5} f \,x^{4}+15 b^{3} c^{3} e^{6} g \,x^{3}+2893 b^{2} c^{4} d \,e^{5} g \,x^{3}+1469 b^{2} c^{4} e^{6} f \,x^{3}-3114 b \,c^{5} d^{2} e^{4} g \,x^{3}+2496 b \,c^{5} d \,e^{5} f \,x^{3}+206 c^{6} d^{3} e^{3} g \,x^{3}-2678 c^{6} d^{2} e^{4} f \,x^{3}-18 b^{4} c^{2} e^{6} g \,x^{2}+150 b^{3} c^{3} d \,e^{5} g \,x^{2}+39 b^{3} c^{3} e^{6} f \,x^{2}+1803 b^{2} c^{4} d^{2} e^{4} g \,x^{2}+4173 b^{2} c^{4} d \,e^{5} f \,x^{2}-3984 b \,c^{5} d^{3} e^{3} g \,x^{2}-4602 b \,c^{5} d^{2} e^{4} f \,x^{2}+2049 c^{6} d^{4} e^{2} g \,x^{2}+390 c^{6} d^{3} e^{3} f \,x^{2}+24 b^{5} c \,e^{6} g x -224 b^{4} c^{2} d \,e^{5} g x -52 b^{4} c^{2} e^{6} f x +799 b^{3} c^{3} d^{2} e^{4} g x +494 b^{3} c^{3} d \,e^{5} f x -1293 b^{2} c^{4} d^{3} e^{3} g x +2691 b^{2} c^{4} d^{2} e^{4} f x +965 b \,c^{5} d^{4} e^{2} g x -6656 b \,c^{5} d^{3} e^{3} f x -271 c^{6} d^{5} e g x +3523 c^{6} d^{4} e^{2} f x -48 b^{6} e^{6} g +496 b^{5} c d \,e^{5} g +104 b^{5} c \,e^{6} f -2046 b^{4} c^{2} d^{2} e^{4} g -1092 b^{4} c^{2} d \,e^{5} f +4184 b^{3} c^{3} d^{3} e^{3} g +4615 b^{3} c^{3} d^{2} e^{4} f -4516 b^{2} c^{4} d^{4} e^{2} g -8333 b^{2} c^{4} d^{3} e^{3} f +2472 b \,c^{5} d^{5} e g +6669 b \,c^{5} d^{4} e^{2} f -542 c^{6} d^{6} g -1963 c^{6} d^{5} e f \right )}{9009 c^{4} e^{2}} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(1/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*( - 48*b**6*e**6*g + 496*b**5*c*d*e**5*g + 1 04*b**5*c*e**6*f + 24*b**5*c*e**6*g*x - 2046*b**4*c**2*d**2*e**4*g - 1092* b**4*c**2*d*e**5*f - 224*b**4*c**2*d*e**5*g*x - 52*b**4*c**2*e**6*f*x - 18 *b**4*c**2*e**6*g*x**2 + 4184*b**3*c**3*d**3*e**3*g + 4615*b**3*c**3*d**2* e**4*f + 799*b**3*c**3*d**2*e**4*g*x + 494*b**3*c**3*d*e**5*f*x + 150*b**3 *c**3*d*e**5*g*x**2 + 39*b**3*c**3*e**6*f*x**2 + 15*b**3*c**3*e**6*g*x**3 - 4516*b**2*c**4*d**4*e**2*g - 8333*b**2*c**4*d**3*e**3*f - 1293*b**2*c**4 *d**3*e**3*g*x + 2691*b**2*c**4*d**2*e**4*f*x + 1803*b**2*c**4*d**2*e**4*g *x**2 + 4173*b**2*c**4*d*e**5*f*x**2 + 2893*b**2*c**4*d*e**5*g*x**3 + 1469 *b**2*c**4*e**6*f*x**3 + 1113*b**2*c**4*e**6*g*x**4 + 2472*b*c**5*d**5*e*g + 6669*b*c**5*d**4*e**2*f + 965*b*c**5*d**4*e**2*g*x - 6656*b*c**5*d**3*e **3*f*x - 3984*b*c**5*d**3*e**3*g*x**2 - 4602*b*c**5*d**2*e**4*f*x**2 - 31 14*b*c**5*d**2*e**4*g*x**3 + 2496*b*c**5*d*e**5*f*x**3 + 1960*b*c**5*d*e** 5*g*x**4 + 2093*b*c**5*e**6*f*x**4 + 1701*b*c**5*e**6*g*x**5 - 542*c**6*d* *6*g - 1963*c**6*d**5*e*f - 271*c**6*d**5*e*g*x + 3523*c**6*d**4*e**2*f*x + 2049*c**6*d**4*e**2*g*x**2 + 390*c**6*d**3*e**3*f*x**2 + 206*c**6*d**3*e **3*g*x**3 - 2678*c**6*d**2*e**4*f*x**3 - 2072*c**6*d**2*e**4*g*x**4 - 91* c**6*d*e**5*f*x**4 - 63*c**6*d*e**5*g*x**5 + 819*c**6*e**6*f*x**5 + 693*c* *6*e**6*g*x**6))/(9009*c**4*e**2)