Integrand size = 30, antiderivative size = 418 \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac {\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}+\frac {\left (b^2-4 a c\right )^2 \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \]
1/384*(48*c^3*d*g-9*b^3*f*h-8*c^2*(a*e*h+a*f*g+3*b*d*h+3*b*e*g)+2*b*c*(6*a *f*h+7*b*(e*h+f*g)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/7*f*(h*x+g)^2*(c* x^2+b*x+a)^(5/2)/c/h+1/840*(63*b^2*f*h^2-24*c^2*(5*f*g^2-7*h*(d*h+e*g))-2* c*h*(24*a*f*h+49*b*(e*h+f*g))-10*c*h*(9*b*f*h-14*c*e*h+10*c*f*g)*x)*(c*x^2 +b*x+a)^(5/2)/c^3/h+1/2048*(-4*a*c+b^2)^2*(48*c^3*d*g-9*b^3*f*h-8*c^2*(a*e *h+a*f*g+3*b*d*h+3*b*e*g)+2*b*c*(6*a*f*h+7*b*(e*h+f*g)))*arctanh(1/2*(2*c* x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)-1/1024*(-4*a*c+b^2)*(48*c^3*d*g -9*b^3*f*h-8*c^2*(a*e*h+a*f*g+3*b*d*h+3*b*e*g)+2*b*c*(6*a*f*h+7*b*(e*h+f*g )))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5
Time = 7.71 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.44 \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^6 f h-210 b^5 c (7 f g+7 e h+3 f h x)+28 b^4 c \left (-270 a f h+c \left (90 e g+90 d h+35 f g x+35 e h x+18 f h x^2\right )\right )-16 b^3 c^2 (105 c d (3 g+h x)-7 a (95 f g+95 e h+39 f h x)+c x (7 e (15 g+7 h x)+f x (49 g+27 h x)))+48 b^2 c^2 \left (343 a^2 f h-2 a c (175 d h+7 e (25 g+9 h x)+f x (63 g+31 h x))+2 c^2 x (7 d (5 g+2 h x)+x (7 e (2 g+h x)+f x (7 g+4 h x)))\right )+32 b c^3 \left (-3 a^2 (189 f g+189 e h+73 f h x)+6 a c (7 d (25 g+7 h x)+x (7 e (7 g+3 h x)+f x (21 g+11 h x)))+4 c^2 x^2 (21 d (15 g+11 h x)+x (7 e (33 g+26 h x)+2 f x (91 g+75 h x)))\right )+64 c^3 \left (-96 a^3 f h+3 a^2 c (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))+4 c^3 x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))+2 a c^2 x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))\right )\right )-105 \left (b^2-4 a c\right )^2 \left (-48 c^3 d g+9 b^3 f h+8 c^2 (3 b e g+a f g+3 b d h+a e h)-2 b c (6 a f h+7 b (f g+e h))\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{107520 c^{11/2}} \]
(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^6*f*h - 210*b^5*c*(7*f*g + 7*e*h + 3 *f*h*x) + 28*b^4*c*(-270*a*f*h + c*(90*e*g + 90*d*h + 35*f*g*x + 35*e*h*x + 18*f*h*x^2)) - 16*b^3*c^2*(105*c*d*(3*g + h*x) - 7*a*(95*f*g + 95*e*h + 39*f*h*x) + c*x*(7*e*(15*g + 7*h*x) + f*x*(49*g + 27*h*x))) + 48*b^2*c^2*( 343*a^2*f*h - 2*a*c*(175*d*h + 7*e*(25*g + 9*h*x) + f*x*(63*g + 31*h*x)) + 2*c^2*x*(7*d*(5*g + 2*h*x) + x*(7*e*(2*g + h*x) + f*x*(7*g + 4*h*x)))) + 32*b*c^3*(-3*a^2*(189*f*g + 189*e*h + 73*f*h*x) + 6*a*c*(7*d*(25*g + 7*h*x ) + x*(7*e*(7*g + 3*h*x) + f*x*(21*g + 11*h*x))) + 4*c^2*x^2*(21*d*(15*g + 11*h*x) + x*(7*e*(33*g + 26*h*x) + 2*f*x*(91*g + 75*h*x)))) + 64*c^3*(-96 *a^3*f*h + 3*a^2*c*(112*d*h + 7*e*(16*g + 5*h*x) + f*x*(35*g + 16*h*x)) + 4*c^3*x^3*(21*d*(5*g + 4*h*x) + 2*x*(7*e*(6*g + 5*h*x) + 5*f*x*(7*g + 6*h* x))) + 2*a*c^2*x*(21*d*(25*g + 16*h*x) + x*(7*e*(48*g + 35*h*x) + f*x*(245 *g + 192*h*x))))) - 105*(b^2 - 4*a*c)^2*(-48*c^3*d*g + 9*b^3*f*h + 8*c^2*( 3*b*e*g + a*f*g + 3*b*d*h + a*e*h) - 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*Ar cTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(107520*c^(11/2))
Time = 0.58 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.77, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2184, 27, 1225, 1087, 1087, 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.
| \(\displaystyle \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\int -\frac {1}{2} h (g+h x) (5 b f g-14 c d h+4 a f h+(10 c f g-14 c e h+9 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{7 c h^2}+\frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {\int (g+h x) (5 b f g-14 c d h+4 a f h+(10 c f g-14 c e h+9 b f h) x) \left (c x^2+b x+a\right )^{3/2}dx}{14 c h}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {-\frac {7 h \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{60 c^2}}{14 c h}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {-\frac {7 h \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{60 c^2}}{14 c h}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {-\frac {7 h \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{60 c^2}}{14 c h}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {-\frac {7 h \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{60 c^2}}{14 c h}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}-\frac {-\frac {7 h \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right ) \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{24 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{60 c^2}}{14 c h}\) |
(f*(g + h*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c*h) - (-1/60*((63*b^2*f*h^2 - 24*c^2*(5*f*g^2 - 7*h*(e*g + d*h)) - 2*c*h*(24*a*f*h + 49*b*(f*g + e*h)) - 10*c*h*(10*c*f*g - 14*c*e*h + 9*b*f*h)*x)*(a + b*x + c*x^2)^(5/2))/c^2 - (7*h*(48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2)) /(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b ^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^( 3/2))))/(16*c)))/(24*c^2))/(14*c*h)
3.2.98.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[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. \(935\) vs. \(2(392)=784\).
Time = 0.77 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.24
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(936\) |
| risch | \(-\frac {\left (-15360 c^{6} f h \,x^{6}-19200 b \,c^{5} f h \,x^{5}-17920 c^{6} e h \,x^{5}-17920 c^{6} f g \,x^{5}-24576 a \,c^{5} f h \,x^{4}-384 b^{2} c^{4} f h \,x^{4}-23296 b \,c^{5} e h \,x^{4}-23296 b \,c^{5} f g \,x^{4}-21504 c^{6} d h \,x^{4}-21504 c^{6} e g \,x^{4}-2112 a b \,c^{4} f h \,x^{3}-31360 a \,c^{5} e h \,x^{3}-31360 a \,c^{5} f g \,x^{3}+432 b^{3} c^{3} f h \,x^{3}-672 b^{2} c^{4} e h \,x^{3}-672 b^{2} c^{4} f g \,x^{3}-29568 b \,c^{5} d h \,x^{3}-29568 b \,c^{5} e g \,x^{3}-26880 c^{6} d g \,x^{3}-3072 a^{2} c^{4} f h \,x^{2}+2976 a \,b^{2} c^{3} f h \,x^{2}-4032 a b \,c^{4} e h \,x^{2}-4032 a b \,c^{4} f g \,x^{2}-43008 a \,c^{5} d h \,x^{2}-43008 a \,c^{5} e g \,x^{2}-504 b^{4} c^{2} f h \,x^{2}+784 b^{3} c^{3} e h \,x^{2}+784 b^{3} c^{3} f g \,x^{2}-1344 b^{2} c^{4} d h \,x^{2}-1344 b^{2} c^{4} e g \,x^{2}-40320 b \,c^{5} d g \,x^{2}+7008 a^{2} b \,c^{3} f h x -6720 a^{2} c^{4} e h x -6720 a^{2} c^{4} f g x -4368 a \,b^{3} c^{2} f h x +6048 a \,b^{2} c^{3} e h x +6048 a \,b^{2} c^{3} f g x -9408 a b \,c^{4} d h x -9408 a b \,c^{4} e g x -67200 a \,c^{5} d g x +630 b^{5} c f h x -980 b^{4} c^{2} e h x -980 b^{4} c^{2} f g x +1680 b^{3} c^{3} d h x +1680 b^{3} c^{3} e g x -3360 b^{2} c^{4} d g x +6144 a^{3} c^{3} f h -16464 a^{2} b^{2} c^{2} f h +18144 a^{2} b \,c^{3} e h +18144 a^{2} b \,c^{3} f g -21504 a^{2} c^{4} d h -21504 a^{2} c^{4} e g +7560 a \,b^{4} c f h -10640 a \,b^{3} c^{2} e h -10640 a \,b^{3} c^{2} f g +16800 a \,b^{2} c^{3} d h +16800 a \,b^{2} c^{3} e g -33600 a b \,c^{4} d g -945 b^{6} f h +1470 b^{5} c e h +1470 b^{5} c f g -2520 b^{4} c^{2} d h -2520 b^{4} c^{2} e g +5040 b^{3} c^{3} d g \right ) \sqrt {c \,x^{2}+b x +a}}{107520 c^{5}}+\frac {\left (192 a^{3} b \,c^{3} f h -128 a^{3} c^{4} e h -128 a^{3} c^{4} f g -240 a^{2} b^{3} c^{2} f h +288 a^{2} b^{2} c^{3} e h +288 a^{2} b^{2} c^{3} f g -384 a^{2} b \,c^{4} d h -384 a^{2} b \,c^{4} e g +768 a^{2} c^{5} d g +84 a \,b^{5} c f h -120 a \,b^{4} c^{2} e h -120 a \,b^{4} c^{2} f g +192 a \,b^{3} c^{3} d h +192 a \,b^{3} c^{3} e g -384 a \,b^{2} c^{4} d g -9 b^{7} f h +14 b^{6} c e h +14 b^{6} c f g -24 b^{5} c^{2} d h -24 b^{5} c^{2} e g +48 b^{4} c^{3} d g \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {11}{2}}}\) | \(974\) |
d*g*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b) /c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x ^2+b*x+a)^(1/2))))+f*h*(1/7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x ^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+ b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a) ^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)) )))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*( 2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1 /2)+(c*x^2+b*x+a)^(1/2)))))-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/ 8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x ^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+ a)^(1/2))))))+(e*h+f*g)*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+ b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^ 2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2* b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x +a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a *c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+(d*h+e*g)*( 1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/1 6*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/ 2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (392) = 784\).
Time = 0.65 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.39 \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
[1/430080*(105*(2*(24*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 12*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e + (7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c ^3 - 64*a^3*c^4)*f)*g - (24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - 2*( 7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*e + 3*(3*b^7 - 28*a *b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*f)*h)*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*(153 60*c^7*f*h*x^6 + 1280*(14*c^7*f*g + (14*c^7*e + 15*b*c^6*f)*h)*x^5 + 128*( 14*(12*c^7*e + 13*b*c^6*f)*g + (168*c^7*d + 182*b*c^6*e + 3*(b^2*c^5 + 64* a*c^6)*f)*h)*x^4 + 16*(14*(120*c^7*d + 132*b*c^6*e + (3*b^2*c^5 + 140*a*c^ 6)*f)*g + (1848*b*c^6*d + 14*(3*b^2*c^5 + 140*a*c^6)*e - 3*(9*b^3*c^4 - 44 *a*b*c^5)*f)*h)*x^3 + 8*(14*(360*b*c^6*d + 12*(b^2*c^5 + 32*a*c^6)*e - (7* b^3*c^4 - 36*a*b*c^5)*f)*g + (168*(b^2*c^5 + 32*a*c^6)*d - 14*(7*b^3*c^4 - 36*a*b*c^5)*e + 3*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*f)*h)*x^2 - 14*(120*(3*b^3*c^4 - 20*a*b*c^5)*d - 12*(15*b^4*c^3 - 100*a*b^2*c^4 + 128* a^2*c^5)*e + (105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*f)*g + (168*(1 5*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d - 14*(105*b^5*c^2 - 760*a*b^3*c ^3 + 1296*a^2*b*c^4)*e + 3*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*f)*h + 2*(14*(120*(b^2*c^5 + 20*a*c^6)*d - 12*(5*b^3*c^4 - 28*a*b*c^5)*e + (35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*f)*g - (168*(5 *b^3*c^4 - 28*a*b*c^5)*d - 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5...
Leaf count of result is larger than twice the leaf count of optimal. 3990 vs. \(2 (435) = 870\).
Time = 1.13 (sec) , antiderivative size = 3990, normalized size of antiderivative = 9.55 \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
Piecewise((sqrt(a + b*x + c*x**2)*(c*f*h*x**6/7 + x**5*(15*b*c*f*h/14 + c* *2*e*h + c**2*f*g)/(6*c) + x**4*(8*a*c*f*h/7 + b**2*f*h + 2*b*c*e*h + 2*b* c*f*g - 11*b*(15*b*c*f*h/14 + c**2*e*h + c**2*f*g)/(12*c) + c**2*d*h + c** 2*e*g)/(5*c) + x**3*(2*a*b*f*h + 2*a*c*e*h + 2*a*c*f*g - 5*a*(15*b*c*f*h/1 4 + c**2*e*h + c**2*f*g)/(6*c) + b**2*e*h + b**2*f*g + 2*b*c*d*h + 2*b*c*e *g - 9*b*(8*a*c*f*h/7 + b**2*f*h + 2*b*c*e*h + 2*b*c*f*g - 11*b*(15*b*c*f* h/14 + c**2*e*h + c**2*f*g)/(12*c) + c**2*d*h + c**2*e*g)/(10*c) + c**2*d* g)/(4*c) + x**2*(a**2*f*h + 2*a*b*e*h + 2*a*b*f*g + 2*a*c*d*h + 2*a*c*e*g - 4*a*(8*a*c*f*h/7 + b**2*f*h + 2*b*c*e*h + 2*b*c*f*g - 11*b*(15*b*c*f*h/1 4 + c**2*e*h + c**2*f*g)/(12*c) + c**2*d*h + c**2*e*g)/(5*c) + b**2*d*h + b**2*e*g + 2*b*c*d*g - 7*b*(2*a*b*f*h + 2*a*c*e*h + 2*a*c*f*g - 5*a*(15*b* c*f*h/14 + c**2*e*h + c**2*f*g)/(6*c) + b**2*e*h + b**2*f*g + 2*b*c*d*h + 2*b*c*e*g - 9*b*(8*a*c*f*h/7 + b**2*f*h + 2*b*c*e*h + 2*b*c*f*g - 11*b*(15 *b*c*f*h/14 + c**2*e*h + c**2*f*g)/(12*c) + c**2*d*h + c**2*e*g)/(10*c) + c**2*d*g)/(8*c))/(3*c) + x*(a**2*e*h + a**2*f*g + 2*a*b*d*h + 2*a*b*e*g + 2*a*c*d*g - 3*a*(2*a*b*f*h + 2*a*c*e*h + 2*a*c*f*g - 5*a*(15*b*c*f*h/14 + c**2*e*h + c**2*f*g)/(6*c) + b**2*e*h + b**2*f*g + 2*b*c*d*h + 2*b*c*e*g - 9*b*(8*a*c*f*h/7 + b**2*f*h + 2*b*c*e*h + 2*b*c*f*g - 11*b*(15*b*c*f*h/14 + c**2*e*h + c**2*f*g)/(12*c) + c**2*d*h + c**2*e*g)/(10*c) + c**2*d*g)/( 4*c) + b**2*d*g - 5*b*(a**2*f*h + 2*a*b*e*h + 2*a*b*f*g + 2*a*c*d*h + 2...
Exception generated. \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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. 925 vs. \(2 (392) = 784\).
Time = 0.31 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.21 \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, c f h x + \frac {14 \, c^{7} f g + 14 \, c^{7} e h + 15 \, b c^{6} f h}{c^{6}}\right )} x + \frac {168 \, c^{7} e g + 182 \, b c^{6} f g + 168 \, c^{7} d h + 182 \, b c^{6} e h + 3 \, b^{2} c^{5} f h + 192 \, a c^{6} f h}{c^{6}}\right )} x + \frac {1680 \, c^{7} d g + 1848 \, b c^{6} e g + 42 \, b^{2} c^{5} f g + 1960 \, a c^{6} f g + 1848 \, b c^{6} d h + 42 \, b^{2} c^{5} e h + 1960 \, a c^{6} e h - 27 \, b^{3} c^{4} f h + 132 \, a b c^{5} f h}{c^{6}}\right )} x + \frac {5040 \, b c^{6} d g + 168 \, b^{2} c^{5} e g + 5376 \, a c^{6} e g - 98 \, b^{3} c^{4} f g + 504 \, a b c^{5} f g + 168 \, b^{2} c^{5} d h + 5376 \, a c^{6} d h - 98 \, b^{3} c^{4} e h + 504 \, a b c^{5} e h + 63 \, b^{4} c^{3} f h - 372 \, a b^{2} c^{4} f h + 384 \, a^{2} c^{5} f h}{c^{6}}\right )} x + \frac {1680 \, b^{2} c^{5} d g + 33600 \, a c^{6} d g - 840 \, b^{3} c^{4} e g + 4704 \, a b c^{5} e g + 490 \, b^{4} c^{3} f g - 3024 \, a b^{2} c^{4} f g + 3360 \, a^{2} c^{5} f g - 840 \, b^{3} c^{4} d h + 4704 \, a b c^{5} d h + 490 \, b^{4} c^{3} e h - 3024 \, a b^{2} c^{4} e h + 3360 \, a^{2} c^{5} e h - 315 \, b^{5} c^{2} f h + 2184 \, a b^{3} c^{3} f h - 3504 \, a^{2} b c^{4} f h}{c^{6}}\right )} x - \frac {5040 \, b^{3} c^{4} d g - 33600 \, a b c^{5} d g - 2520 \, b^{4} c^{3} e g + 16800 \, a b^{2} c^{4} e g - 21504 \, a^{2} c^{5} e g + 1470 \, b^{5} c^{2} f g - 10640 \, a b^{3} c^{3} f g + 18144 \, a^{2} b c^{4} f g - 2520 \, b^{4} c^{3} d h + 16800 \, a b^{2} c^{4} d h - 21504 \, a^{2} c^{5} d h + 1470 \, b^{5} c^{2} e h - 10640 \, a b^{3} c^{3} e h + 18144 \, a^{2} b c^{4} e h - 945 \, b^{6} c f h + 7560 \, a b^{4} c^{2} f h - 16464 \, a^{2} b^{2} c^{3} f h + 6144 \, a^{3} c^{4} f h}{c^{6}}\right )} - \frac {{\left (48 \, b^{4} c^{3} d g - 384 \, a b^{2} c^{4} d g + 768 \, a^{2} c^{5} d g - 24 \, b^{5} c^{2} e g + 192 \, a b^{3} c^{3} e g - 384 \, a^{2} b c^{4} e g + 14 \, b^{6} c f g - 120 \, a b^{4} c^{2} f g + 288 \, a^{2} b^{2} c^{3} f g - 128 \, a^{3} c^{4} f g - 24 \, b^{5} c^{2} d h + 192 \, a b^{3} c^{3} d h - 384 \, a^{2} b c^{4} d h + 14 \, b^{6} c e h - 120 \, a b^{4} c^{2} e h + 288 \, a^{2} b^{2} c^{3} e h - 128 \, a^{3} c^{4} e h - 9 \, b^{7} f h + 84 \, a b^{5} c f h - 240 \, a^{2} b^{3} c^{2} f h + 192 \, a^{3} b c^{3} f h\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \]
1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*c*f*h*x + (14*c^7*f*g + 14*c^7*e*h + 15*b*c^6*f*h)/c^6)*x + (168*c^7*e*g + 182*b*c^6*f*g + 168*c^ 7*d*h + 182*b*c^6*e*h + 3*b^2*c^5*f*h + 192*a*c^6*f*h)/c^6)*x + (1680*c^7* d*g + 1848*b*c^6*e*g + 42*b^2*c^5*f*g + 1960*a*c^6*f*g + 1848*b*c^6*d*h + 42*b^2*c^5*e*h + 1960*a*c^6*e*h - 27*b^3*c^4*f*h + 132*a*b*c^5*f*h)/c^6)*x + (5040*b*c^6*d*g + 168*b^2*c^5*e*g + 5376*a*c^6*e*g - 98*b^3*c^4*f*g + 5 04*a*b*c^5*f*g + 168*b^2*c^5*d*h + 5376*a*c^6*d*h - 98*b^3*c^4*e*h + 504*a *b*c^5*e*h + 63*b^4*c^3*f*h - 372*a*b^2*c^4*f*h + 384*a^2*c^5*f*h)/c^6)*x + (1680*b^2*c^5*d*g + 33600*a*c^6*d*g - 840*b^3*c^4*e*g + 4704*a*b*c^5*e*g + 490*b^4*c^3*f*g - 3024*a*b^2*c^4*f*g + 3360*a^2*c^5*f*g - 840*b^3*c^4*d *h + 4704*a*b*c^5*d*h + 490*b^4*c^3*e*h - 3024*a*b^2*c^4*e*h + 3360*a^2*c^ 5*e*h - 315*b^5*c^2*f*h + 2184*a*b^3*c^3*f*h - 3504*a^2*b*c^4*f*h)/c^6)*x - (5040*b^3*c^4*d*g - 33600*a*b*c^5*d*g - 2520*b^4*c^3*e*g + 16800*a*b^2*c ^4*e*g - 21504*a^2*c^5*e*g + 1470*b^5*c^2*f*g - 10640*a*b^3*c^3*f*g + 1814 4*a^2*b*c^4*f*g - 2520*b^4*c^3*d*h + 16800*a*b^2*c^4*d*h - 21504*a^2*c^5*d *h + 1470*b^5*c^2*e*h - 10640*a*b^3*c^3*e*h + 18144*a^2*b*c^4*e*h - 945*b^ 6*c*f*h + 7560*a*b^4*c^2*f*h - 16464*a^2*b^2*c^3*f*h + 6144*a^3*c^4*f*h)/c ^6) - 1/2048*(48*b^4*c^3*d*g - 384*a*b^2*c^4*d*g + 768*a^2*c^5*d*g - 24*b^ 5*c^2*e*g + 192*a*b^3*c^3*e*g - 384*a^2*b*c^4*e*g + 14*b^6*c*f*g - 120*a*b ^4*c^2*f*g + 288*a^2*b^2*c^3*f*g - 128*a^3*c^4*f*g - 24*b^5*c^2*d*h + 1...
Timed out. \[ \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\int \left (g+h\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \]