Integrand size = 32, antiderivative size = 504 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (12 c^2 d-6 b c e+7 b^2 f-16 a c f\right ) h (g+h x)^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {h \left (192 c^4 d g^2+105 b^4 f h^2-10 b^2 c h (46 a f h+9 b (3 f g+e h))-16 c^3 \left (3 b g (2 e g+3 d h)+4 a \left (7 f g^2+9 e g h+3 d h^2\right )\right )+8 c^2 \left (32 a^2 f h^2+39 a b h (3 f g+e h)+b^2 \left (20 f g^2+9 h (3 e g+d h)\right )\right )+2 c h \left (48 c^3 d g-35 b^3 f h-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)+2 b c (17 b f g+15 b e h+58 a f h)\right ) x\right ) \sqrt {a+b x+c x^2}}{24 c^4 \left (b^2-4 a c\right )}-\frac {\left (35 b^3 f h^3-30 b c h^2 (3 b f g+b e h+2 a f h)-16 c^3 g \left (f g^2+3 h (e g+d h)\right )+24 c^2 h \left (a h (3 f g+e h)+b \left (3 f g^2+3 e g h+d h^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{9/2}} \]
-1/16*(35*b^3*f*h^3-30*b*c*h^2*(2*a*f*h+b*e*h+3*b*f*g)-16*c^3*g*(f*g^2+3*h *(d*h+e*g))+24*c^2*h*(a*h*(e*h+3*f*g)+b*(d*h^2+3*e*g*h+3*f*g^2)))*arctanh( 1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)+2*(c*(2*a*e-b*(d+a*f/c) )-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x)*(h*x+g)^3/c/(-4*a*c+b^2)/(c*x^2+b*x+a) ^(1/2)+1/3*(-16*a*c*f+7*b^2*f-6*b*c*e+12*c^2*d)*h*(h*x+g)^2*(c*x^2+b*x+a)^ (1/2)/c^2/(-4*a*c+b^2)+1/24*h*(192*c^4*d*g^2+105*b^4*f*h^2-10*b^2*c*h*(46* a*f*h+9*b*(e*h+3*f*g))-16*c^3*(3*b*g*(3*d*h+2*e*g)+4*a*(3*d*h^2+9*e*g*h+7* f*g^2))+8*c^2*(32*a^2*f*h^2+39*a*b*h*(e*h+3*f*g)+b^2*(20*f*g^2+9*h*(d*h+3* e*g)))+2*c*h*(48*c^3*d*g-35*b^3*f*h-8*c^2*(9*a*e*h+11*a*f*g+3*b*d*h+3*b*e* g)+2*b*c*(58*a*f*h+15*b*e*h+17*b*f*g))*x)*(c*x^2+b*x+a)^(1/2)/c^4/(-4*a*c+ b^2)
Time = 4.72 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.38 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {-105 b^5 f h^3 x-5 b^4 h^2 (21 a f h+c x (-54 f g-18 e h+7 f h x))+2 b^3 c h \left (5 a h (27 f g+9 e h+53 f h x)+c x \left (3 h (-36 e g-12 d h+5 e h x)+f \left (-108 g^2+45 g h x+7 h^2 x^2\right )\right )\right )+16 c^2 \left (-16 a^3 f h^3+6 c^3 d g^3 x+a c^2 \left (6 d h \left (-3 g^2-3 g h x+h^2 x^2\right )-3 e \left (2 g^3+6 g^2 h x-6 g h^2 x^2-h^3 x^3\right )+f x \left (-6 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )\right )+a^2 c h \left (f \left (36 g^2+27 g h x-8 h^2 x^2\right )+3 h (4 d h+3 e (4 g+h x))\right )\right )+8 b c^2 \left (-6 c^2 g^2 (-d g+e g x+3 d h x)-a^2 h^2 (117 f g+39 e h+61 f h x)+a c \left (f \left (6 g^3+90 g^2 h x-45 g h^2 x^2-7 h^3 x^3\right )+3 h \left (2 d h (3 g+5 h x)+e \left (6 g^2+30 g h x-5 h^2 x^2\right )\right )\right )\right )+4 b^2 c \left (115 a^2 f h^3-a c h \left (3 h (18 e g+6 d h+31 e h x)+f \left (54 g^2+279 g h x-43 h^2 x^2\right )\right )-c^2 x \left (f \left (-12 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )+3 h \left (2 d h (-6 g+h x)+e \left (-12 g^2+6 g h x+h^2 x^2\right )\right )\right )\right )}{24 c^4 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {\left (-35 b^3 f h^3+30 b c h^2 (3 b f g+b e h+2 a f h)+16 c^3 \left (f g^3+3 g h (e g+d h)\right )-24 c^2 h \left (3 b f g^2+b h (3 e g+d h)+a h (3 f g+e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{8 c^{9/2}} \]
-1/24*(-105*b^5*f*h^3*x - 5*b^4*h^2*(21*a*f*h + c*x*(-54*f*g - 18*e*h + 7* f*h*x)) + 2*b^3*c*h*(5*a*h*(27*f*g + 9*e*h + 53*f*h*x) + c*x*(3*h*(-36*e*g - 12*d*h + 5*e*h*x) + f*(-108*g^2 + 45*g*h*x + 7*h^2*x^2))) + 16*c^2*(-16 *a^3*f*h^3 + 6*c^3*d*g^3*x + a*c^2*(6*d*h*(-3*g^2 - 3*g*h*x + h^2*x^2) - 3 *e*(2*g^3 + 6*g^2*h*x - 6*g*h^2*x^2 - h^3*x^3) + f*x*(-6*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3)) + a^2*c*h*(f*(36*g^2 + 27*g*h*x - 8*h^2*x^2) + 3*h*(4*d*h + 3*e*(4*g + h*x)))) + 8*b*c^2*(-6*c^2*g^2*(-(d*g) + e*g*x + 3 *d*h*x) - a^2*h^2*(117*f*g + 39*e*h + 61*f*h*x) + a*c*(f*(6*g^3 + 90*g^2*h *x - 45*g*h^2*x^2 - 7*h^3*x^3) + 3*h*(2*d*h*(3*g + 5*h*x) + e*(6*g^2 + 30* g*h*x - 5*h^2*x^2)))) + 4*b^2*c*(115*a^2*f*h^3 - a*c*h*(3*h*(18*e*g + 6*d* h + 31*e*h*x) + f*(54*g^2 + 279*g*h*x - 43*h^2*x^2)) - c^2*x*(f*(-12*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3) + 3*h*(2*d*h*(-6*g + h*x) + e*(-12* g^2 + 6*g*h*x + h^2*x^2)))))/(c^4*(b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) + ( (-35*b^3*f*h^3 + 30*b*c*h^2*(3*b*f*g + b*e*h + 2*a*f*h) + 16*c^3*(f*g^3 + 3*g*h*(e*g + d*h)) - 24*c^2*h*(3*b*f*g^2 + b*h*(3*e*g + d*h) + a*h*(3*f*g + e*h)))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(8*c^(9/ 2))
Time = 1.15 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2175, 27, 1236, 27, 1225, 1092, 219}
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.
| \(\displaystyle \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {(g+h x)^2 \left (f g b^2+6 (c d+a f) h b-4 a c (f g+3 e h)+c \left (\frac {7 f b^2}{c}-6 e b+12 c d-16 a f\right ) h x\right )}{2 c \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(g+h x)^2 \left (f g b^2+6 (c d+a f) h b-4 a c (f g+3 e h)+\left (7 f b^2-6 c e b+12 c^2 d-16 a c f\right ) h x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\frac {\int -\frac {(g+h x) \left (7 f g h b^3+\left (28 a f h^2-6 c g (f g+e h)\right ) b^2-4 c h (6 c d g+13 a f g+6 a e h) b-8 a c \left (8 a f h^2-3 c \left (f g^2+3 e h g+2 d h^2\right )\right )-h \left (-35 f h b^3+2 c (17 b f g+15 b e h+58 a f h) b+48 c^3 d g-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {h (g+h x)^2 \sqrt {a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {h (g+h x)^2 \sqrt {a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c}-\frac {\int \frac {(g+h x) \left (7 f g h b^3+\left (28 a f h^2-6 c g (f g+e h)\right ) b^2-4 c h (6 c d g+13 a f g+6 a e h) b-8 a c \left (8 a f h^2-3 c \left (f g^2+3 e h g+2 d h^2\right )\right )-h \left (-35 f h b^3+2 c (17 b f g+15 b e h+58 a f h) b+48 c^3 d g-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {h (g+h x)^2 \sqrt {a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 \left (3 g h (d h+e g)+f g^3\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h \left (32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left (9 h (d h+3 e g)+20 f g^2\right )\right )+2 c h^2 x \left (-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right )-10 b^2 c h^2 (46 a f h+9 b (e h+3 f g))-16 c^3 h \left (4 a \left (3 d h^2+9 e g h+7 f g^2\right )+3 b g (3 d h+2 e g)\right )+105 b^4 f h^3+192 c^4 d g^2 h\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {h (g+h x)^2 \sqrt {a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 \left (3 g h (d h+e g)+f g^3\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h \left (32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left (9 h (d h+3 e g)+20 f g^2\right )\right )+2 c h^2 x \left (-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right )-10 b^2 c h^2 (46 a f h+9 b (e h+3 f g))-16 c^3 h \left (4 a \left (3 d h^2+9 e g h+7 f g^2\right )+3 b g (3 d h+2 e g)\right )+105 b^4 f h^3+192 c^4 d g^2 h\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {h (g+h x)^2 \sqrt {a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 \left (3 g h (d h+e g)+f g^3\right )\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (8 c^2 h \left (32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left (9 h (d h+3 e g)+20 f g^2\right )\right )+2 c h^2 x \left (-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right )-10 b^2 c h^2 (46 a f h+9 b (e h+3 f g))-16 c^3 h \left (4 a \left (3 d h^2+9 e g h+7 f g^2\right )+3 b g (3 d h+2 e g)\right )+105 b^4 f h^3+192 c^4 d g^2 h\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}+\frac {2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x)*( g + h*x)^3)/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (((12*c^2*d - 6*b*c* e + 7*b^2*f - 16*a*c*f)*h*(g + h*x)^2*Sqrt[a + b*x + c*x^2])/(3*c) - (-1/4 *((192*c^4*d*g^2*h + 105*b^4*f*h^3 - 10*b^2*c*h^2*(46*a*f*h + 9*b*(3*f*g + e*h)) - 16*c^3*h*(3*b*g*(2*e*g + 3*d*h) + 4*a*(7*f*g^2 + 9*e*g*h + 3*d*h^ 2)) + 8*c^2*h*(32*a^2*f*h^2 + 39*a*b*h*(3*f*g + e*h) + b^2*(20*f*g^2 + 9*h *(3*e*g + d*h))) + 2*c*h^2*(48*c^3*d*g - 35*b^3*f*h - 8*c^2*(3*b*e*g + 11* a*f*g + 3*b*d*h + 9*a*e*h) + 2*b*c*(17*b*f*g + 15*b*e*h + 58*a*f*h))*x)*Sq rt[a + b*x + c*x^2])/c^2 + (3*(b^2 - 4*a*c)*(35*b^3*f*h^3 - 30*b*c*h^2*(3* b*f*g + b*e*h + 2*a*f*h) - 16*c^3*(f*g^3 + 3*g*h*(e*g + d*h)) + 24*c^2*h*( 3*b*f*g^2 + b*h*(3*e*g + d*h) + a*h*(3*f*g + e*h)))*ArcTanh[(b + 2*c*x)/(2 *Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))/(c*(b^2 - 4*a*c))
3.3.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
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[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[Polyno mialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((R*b - 2*a*S + (2*c*R - b*S)*x)/((p + 1)*(b^2 - 4*a*c))), x ] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2 )^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Qx + S*(2*a*e*m + b*d *(2*p + 3)) - R*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*R - b*S)*(m + 2*p + 3)*x , x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a *c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (Inte gerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(996\) vs. \(2(484)=968\).
Time = 1.28 (sec) , antiderivative size = 997, normalized size of antiderivative = 1.98
| method | result | size |
| risch | \(-\frac {h \left (-8 f \,h^{2} c^{2} x^{2}+22 b c f \,h^{2} x -12 c^{2} e \,h^{2} x -36 c^{2} f g h x +40 a c f \,h^{2}-57 b^{2} f \,h^{2}+42 b c e \,h^{2}+126 b c f g h -24 c^{2} d \,h^{2}-72 c^{2} e g h -72 c^{2} f \,g^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{4}}+\frac {\frac {32 c^{4} d \,g^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {16 a^{2} c^{2} e \,h^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {38 a \,b^{3} f \,h^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {16 a b \,c^{2} d \,h^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {56 a^{2} b c f \,h^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {48 a^{2} c^{2} f g \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {28 a \,b^{2} c e \,h^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {84 a \,b^{2} c f g \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {48 a b \,c^{2} e g \,h^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {48 a b \,c^{2} f \,g^{2} h \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (60 a b \,c^{2} f \,h^{3}-24 a \,c^{3} e \,h^{3}-72 a \,c^{3} f g \,h^{2}-35 b^{3} c f \,h^{3}+30 b^{2} c^{2} e \,h^{3}+90 b^{2} c^{2} f g \,h^{2}-24 b \,c^{3} d \,h^{3}-72 b \,c^{3} e g \,h^{2}-72 b \,c^{3} f \,g^{2} h +48 c^{4} d g \,h^{2}+48 c^{4} e \,g^{2} h +16 c^{4} f \,g^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (16 a^{2} c^{2} f \,h^{3}+12 a \,b^{2} c f \,h^{3}+8 a b \,c^{2} e \,h^{3}+24 a b \,c^{2} f g \,h^{2}-16 a \,c^{3} d \,h^{3}-48 a \,c^{3} e g \,h^{2}-48 a \,c^{3} f \,g^{2} h -19 b^{4} f \,h^{3}+14 b^{3} c e \,h^{3}+42 b^{3} c f g \,h^{2}-8 b^{2} c^{2} d \,h^{3}-24 b^{2} c^{2} e g \,h^{2}-24 b^{2} c^{2} f \,g^{2} h +48 c^{4} d \,g^{2} h +16 c^{4} e \,g^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{16 c^{4}}\) | \(997\) |
| default | \(\text {Expression too large to display}\) | \(1366\) |
-1/24*h*(-8*c^2*f*h^2*x^2+22*b*c*f*h^2*x-12*c^2*e*h^2*x-36*c^2*f*g*h*x+40* a*c*f*h^2-57*b^2*f*h^2+42*b*c*e*h^2+126*b*c*f*g*h-24*c^2*d*h^2-72*c^2*e*g* h-72*c^2*f*g^2)*(c*x^2+b*x+a)^(1/2)/c^4+1/16/c^4*(32*c^4*d*g^3*(2*c*x+b)/( 4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-16*a^2*c^2*e*h^3*(2*c*x+b)/(4*a*c-b^2)/(c*x ^2+b*x+a)^(1/2)-38*a*b^3*f*h^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1 6*a*b*c^2*d*h^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+56*a^2*b*c*f*h^3 *(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-48*a^2*c^2*f*g*h^2*(2*c*x+b)/(4 *a*c-b^2)/(c*x^2+b*x+a)^(1/2)+28*a*b^2*c*e*h^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^ 2+b*x+a)^(1/2)+84*a*b^2*c*f*g*h^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2 )-48*a*b*c^2*e*g*h^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-48*a*b*c^2* f*g^2*h*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(60*a*b*c^2*f*h^3-24*a*c ^3*e*h^3-72*a*c^3*f*g*h^2-35*b^3*c*f*h^3+30*b^2*c^2*e*h^3+90*b^2*c^2*f*g*h ^2-24*b*c^3*d*h^3-72*b*c^3*e*g*h^2-72*b*c^3*f*g^2*h+48*c^4*d*g*h^2+48*c^4* e*g^2*h+16*c^4*f*g^3)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a )^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b +c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(16*a^2*c^2*f*h^3+12*a*b^2*c*f*h^3+8*a *b*c^2*e*h^3+24*a*b*c^2*f*g*h^2-16*a*c^3*d*h^3-48*a*c^3*e*g*h^2-48*a*c^3*f *g^2*h-19*b^4*f*h^3+14*b^3*c*e*h^3+42*b^3*c*f*g*h^2-8*b^2*c^2*d*h^3-24*b^2 *c^2*e*g*h^2-24*b^2*c^2*f*g^2*h+48*c^4*d*g^2*h+16*c^4*e*g^3)*(-1/c/(c*x^2+ b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1467 vs. \(2 (482) = 964\).
Time = 7.72 (sec) , antiderivative size = 2937, normalized size of antiderivative = 5.83 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/96*(3*(16*(a*b^2*c^3 - 4*a^2*c^4)*f*g^3 + 24*(2*(a*b^2*c^3 - 4*a^2*c^4) *e - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*f)*g^2*h + 6*(8*(a*b^2*c^3 - 4*a^2*c^4)*d - 12*(a*b^3*c^2 - 4*a^2*b*c^3)*e + 3*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16*a^3 *c^3)*f)*g*h^2 - (24*(a*b^3*c^2 - 4*a^2*b*c^3)*d - 6*(5*a*b^4*c - 24*a^2*b ^2*c^2 + 16*a^3*c^3)*e + 5*(7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*f)*h^3 + (16*(b^2*c^4 - 4*a*c^5)*f*g^3 + 24*(2*(b^2*c^4 - 4*a*c^5)*e - 3*(b^3*c^3 - 4*a*b*c^4)*f)*g^2*h + 6*(8*(b^2*c^4 - 4*a*c^5)*d - 12*(b^3*c^3 - 4*a*b* c^4)*e + 3*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*f)*g*h^2 - (24*(b^3*c^3 - 4*a*b*c^4)*d - 6*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + 5*(7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*f)*h^3)*x^2 + (16*(b^3*c^3 - 4*a*b*c^4)*f* g^3 + 24*(2*(b^3*c^3 - 4*a*b*c^4)*e - 3*(b^4*c^2 - 4*a*b^2*c^3)*f)*g^2*h + 6*(8*(b^3*c^3 - 4*a*b*c^4)*d - 12*(b^4*c^2 - 4*a*b^2*c^3)*e + 3*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*f)*g*h^2 - (24*(b^4*c^2 - 4*a*b^2*c^3)*d - 6*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*e + 5*(7*b^6 - 40*a*b^4*c + 48*a ^2*b^2*c^2)*f)*h^3)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x ^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(8*(b^2*c^4 - 4*a*c^5)*f*h^ 3*x^4 - 48*(b*c^5*d - 2*a*c^5*e + a*b*c^4*f)*g^3 + 72*(4*a*c^5*d - 2*a*b*c ^4*e + (3*a*b^2*c^3 - 8*a^2*c^4)*f)*g^2*h - 18*(8*a*b*c^4*d - 4*(3*a*b^2*c ^3 - 8*a^2*c^4)*e + (15*a*b^3*c^2 - 52*a^2*b*c^3)*f)*g*h^2 + (24*(3*a*b^2* c^3 - 8*a^2*c^4)*d - 6*(15*a*b^3*c^2 - 52*a^2*b*c^3)*e + (105*a*b^4*c -...
\[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right )^{3} \left (d + e x + f x^{2}\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (482) = 964\).
Time = 0.30 (sec) , antiderivative size = 1028, normalized size of antiderivative = 2.04 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {4 \, {\left (b^{2} c^{3} f h^{3} - 4 \, a c^{4} f h^{3}\right )} x}{b^{2} c^{4} - 4 \, a c^{5}} + \frac {18 \, b^{2} c^{3} f g h^{2} - 72 \, a c^{4} f g h^{2} + 6 \, b^{2} c^{3} e h^{3} - 24 \, a c^{4} e h^{3} - 7 \, b^{3} c^{2} f h^{3} + 28 \, a b c^{3} f h^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {72 \, b^{2} c^{3} f g^{2} h - 288 \, a c^{4} f g^{2} h + 72 \, b^{2} c^{3} e g h^{2} - 288 \, a c^{4} e g h^{2} - 90 \, b^{3} c^{2} f g h^{2} + 360 \, a b c^{3} f g h^{2} + 24 \, b^{2} c^{3} d h^{3} - 96 \, a c^{4} d h^{3} - 30 \, b^{3} c^{2} e h^{3} + 120 \, a b c^{3} e h^{3} + 35 \, b^{4} c f h^{3} - 172 \, a b^{2} c^{2} f h^{3} + 128 \, a^{2} c^{3} f h^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x - \frac {96 \, c^{5} d g^{3} - 48 \, b c^{4} e g^{3} + 48 \, b^{2} c^{3} f g^{3} - 96 \, a c^{4} f g^{3} - 144 \, b c^{4} d g^{2} h + 144 \, b^{2} c^{3} e g^{2} h - 288 \, a c^{4} e g^{2} h - 216 \, b^{3} c^{2} f g^{2} h + 720 \, a b c^{3} f g^{2} h + 144 \, b^{2} c^{3} d g h^{2} - 288 \, a c^{4} d g h^{2} - 216 \, b^{3} c^{2} e g h^{2} + 720 \, a b c^{3} e g h^{2} + 270 \, b^{4} c f g h^{2} - 1116 \, a b^{2} c^{2} f g h^{2} + 432 \, a^{2} c^{3} f g h^{2} - 72 \, b^{3} c^{2} d h^{3} + 240 \, a b c^{3} d h^{3} + 90 \, b^{4} c e h^{3} - 372 \, a b^{2} c^{2} e h^{3} + 144 \, a^{2} c^{3} e h^{3} - 105 \, b^{5} f h^{3} + 530 \, a b^{3} c f h^{3} - 488 \, a^{2} b c^{2} f h^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x - \frac {48 \, b c^{4} d g^{3} - 96 \, a c^{4} e g^{3} + 48 \, a b c^{3} f g^{3} - 288 \, a c^{4} d g^{2} h + 144 \, a b c^{3} e g^{2} h - 216 \, a b^{2} c^{2} f g^{2} h + 576 \, a^{2} c^{3} f g^{2} h + 144 \, a b c^{3} d g h^{2} - 216 \, a b^{2} c^{2} e g h^{2} + 576 \, a^{2} c^{3} e g h^{2} + 270 \, a b^{3} c f g h^{2} - 936 \, a^{2} b c^{2} f g h^{2} - 72 \, a b^{2} c^{2} d h^{3} + 192 \, a^{2} c^{3} d h^{3} + 90 \, a b^{3} c e h^{3} - 312 \, a^{2} b c^{2} e h^{3} - 105 \, a b^{4} f h^{3} + 460 \, a^{2} b^{2} c f h^{3} - 256 \, a^{3} c^{2} f h^{3}}{b^{2} c^{4} - 4 \, a c^{5}}}{24 \, \sqrt {c x^{2} + b x + a}} - \frac {{\left (16 \, c^{3} f g^{3} + 48 \, c^{3} e g^{2} h - 72 \, b c^{2} f g^{2} h + 48 \, c^{3} d g h^{2} - 72 \, b c^{2} e g h^{2} + 90 \, b^{2} c f g h^{2} - 72 \, a c^{2} f g h^{2} - 24 \, b c^{2} d h^{3} + 30 \, b^{2} c e h^{3} - 24 \, a c^{2} e h^{3} - 35 \, b^{3} f h^{3} + 60 \, a b c f h^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {9}{2}}} \]
1/24*(((2*(4*(b^2*c^3*f*h^3 - 4*a*c^4*f*h^3)*x/(b^2*c^4 - 4*a*c^5) + (18*b ^2*c^3*f*g*h^2 - 72*a*c^4*f*g*h^2 + 6*b^2*c^3*e*h^3 - 24*a*c^4*e*h^3 - 7*b ^3*c^2*f*h^3 + 28*a*b*c^3*f*h^3)/(b^2*c^4 - 4*a*c^5))*x + (72*b^2*c^3*f*g^ 2*h - 288*a*c^4*f*g^2*h + 72*b^2*c^3*e*g*h^2 - 288*a*c^4*e*g*h^2 - 90*b^3* c^2*f*g*h^2 + 360*a*b*c^3*f*g*h^2 + 24*b^2*c^3*d*h^3 - 96*a*c^4*d*h^3 - 30 *b^3*c^2*e*h^3 + 120*a*b*c^3*e*h^3 + 35*b^4*c*f*h^3 - 172*a*b^2*c^2*f*h^3 + 128*a^2*c^3*f*h^3)/(b^2*c^4 - 4*a*c^5))*x - (96*c^5*d*g^3 - 48*b*c^4*e*g ^3 + 48*b^2*c^3*f*g^3 - 96*a*c^4*f*g^3 - 144*b*c^4*d*g^2*h + 144*b^2*c^3*e *g^2*h - 288*a*c^4*e*g^2*h - 216*b^3*c^2*f*g^2*h + 720*a*b*c^3*f*g^2*h + 1 44*b^2*c^3*d*g*h^2 - 288*a*c^4*d*g*h^2 - 216*b^3*c^2*e*g*h^2 + 720*a*b*c^3 *e*g*h^2 + 270*b^4*c*f*g*h^2 - 1116*a*b^2*c^2*f*g*h^2 + 432*a^2*c^3*f*g*h^ 2 - 72*b^3*c^2*d*h^3 + 240*a*b*c^3*d*h^3 + 90*b^4*c*e*h^3 - 372*a*b^2*c^2* e*h^3 + 144*a^2*c^3*e*h^3 - 105*b^5*f*h^3 + 530*a*b^3*c*f*h^3 - 488*a^2*b* c^2*f*h^3)/(b^2*c^4 - 4*a*c^5))*x - (48*b*c^4*d*g^3 - 96*a*c^4*e*g^3 + 48* a*b*c^3*f*g^3 - 288*a*c^4*d*g^2*h + 144*a*b*c^3*e*g^2*h - 216*a*b^2*c^2*f* g^2*h + 576*a^2*c^3*f*g^2*h + 144*a*b*c^3*d*g*h^2 - 216*a*b^2*c^2*e*g*h^2 + 576*a^2*c^3*e*g*h^2 + 270*a*b^3*c*f*g*h^2 - 936*a^2*b*c^2*f*g*h^2 - 72*a *b^2*c^2*d*h^3 + 192*a^2*c^3*d*h^3 + 90*a*b^3*c*e*h^3 - 312*a^2*b*c^2*e*h^ 3 - 105*a*b^4*f*h^3 + 460*a^2*b^2*c*f*h^3 - 256*a^3*c^2*f*h^3)/(b^2*c^4 - 4*a*c^5))/sqrt(c*x^2 + b*x + a) - 1/16*(16*c^3*f*g^3 + 48*c^3*e*g^2*h -...
Timed out. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (g+h\,x\right )}^3\,\left (f\,x^2+e\,x+d\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]